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b/09E5T4oBgHgl3EQfOw41/content/tmp_files/2301.05499v1.pdf.txt @@ -0,0 +1,1208 @@ +CLIP the Gap: A Single Domain Generalization Approach for Object Detection +Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2 +CVLab, EPFL1, ClearSpace SA2 +firstname.lastname@epfl.ch +Abstract +Single Domain Generalization (SDG) tackles the prob- +lem of training a model on a single source domain so that +it generalizes to any unseen target domain. While this has +been well studied for image classification, the literature on +SDG object detection remains almost non-existent. To ad- +dress the challenges of simultaneously learning robust ob- +ject localization and representation, we propose to leverage +a pre-trained vision-language model to introduce semantic +domain concepts via textual prompts. We achieve this via +a semantic augmentation strategy acting on the features ex- +tracted by the detector backbone, as well as a text-based +classification loss. Our experiments evidence the benefits of +our approach, outperforming by 10% the only existing SDG +object detection method, Single-DGOD [49], on their own +diverse weather-driving benchmark. +1. Introduction +As for most machine learning models, the performance +of object detectors degrades when the test data distribu- +tion deviates from the training data one. +Domain adap- +tation techniques [3, 5, 8, 30, 41, 43] try to alleviate this +problem by learning domain invariant features between a +source and a known target domain. In practice, however, +it is not always possible to obtain target data, even un- +labeled, precluding the use of such techniques. +Domain +generalization tackles this by seeking to learn representa- +tions that generalize to any target domain. +While early +approaches [1, 10, 25, 26, 28, 47, 57] focused on the sce- +nario where multiple source domains are available during +training, many recent methods tackle the more challenging, +yet more realistic, case of Single Domain Generalization +(SDG), aiming to learn to generalize from a single source +dataset. While this has been well studied for image clas- +sification [13, 35, 45, 48, 56], it remains a nascent topic in +object detection. To the best of our knowledge, a single ex- +isting approach, Single-DGOD [49], uses disentanglement +and self-distillation [22] to learn domain-invariant features. +In this paper, we introduce a fundamentally different ap- +Figure 1. Semantic Augmentation: We compare the PCA pro- +jections of CLIP [36] image embeddings obtained in two different +manners: (Top) The embeddings were directly obtained from the +real images from 5 domains corresponding to different weather +conditions. (Bottom) The embeddings were obtained from the day +images only and modified with our semantic augmentation strat- +egy based on text prompts to reflect the other 4 domains. Note that +the relative positions of the clusters in the bottom plot resembles +that of the top one, showing that our augmentations let us gener- +alize to different target domains. The principal components used +are the same for both the figures. +proach to SDG for object detection. To this end, we build +on two observations: (i) Unsupervised/self-supervised pre- +training facilitates the transfer of a model to new tasks [2, +1 +arXiv:2301.05499v1 [cs.CV] 13 Jan 2023 + +Imageday +Imagenight +Imagefoggy +Image rainyday +Image rainy night +Imageday +Semanticaugmentationnight +Semantic augmentation foggy +Semanticaugmentationrainy day +Semantic augmentation rainy night4, 18]; (ii) Exploiting language supervision to train vision +models allows them to generalize more easily to new cat- +egories and concepts [9, 36]. Inspired by this, we there- +fore propose to leverage a self-supervised vision-language +model, CLIP [36], to guide the training of an object detec- +tor so that it generalizes to unseen target domains. Since the +visual CLIP representation has been jointly learned with the +textual one, we transfer text-based domain variations to the +image representation during training, thus increasing the di- +versity of the source data. +Specifically, we define textual prompts describing po- +tential target domain concepts, such as weather and day- +time variations for road scene understanding, and use these +prompts to perform semantic augmentations of the images. +These augmentations, however, are done in feature space, +not in image space, which is facilitated by the joint image- +text CLIP latent space. This is illustrated in Fig. 1, which +shows that, even though we did not use any target data +for semantic augmentation, the resulting augmented embed- +dings reflect the distributions of the true image embeddings +from different target domains. +We show the effectiveness of our method on the SDG +driving dataset of [49], which reflects a practical scenario +where the training (source) images were captured on a +clear day whereas the test (target) ones were acquired in +rainy, foggy, night, and dusk conditions. Our experiments +demonstrate the benefits of our approach over the Single- +DGOD [49] one. +To summarize our contributions, we employ a vision- +language model to improve the generalizability of an object +detector; during training, we introduce domain concepts via +text-prompts to augment the diversity of the learned image +features and make them more robust to an unseen target do- +main. This enables us to achieve state-of-the-art results on +the diverse weather SDG driving benchmark of [49]. +2. Related Work +Domain Adaptation for Object Detection. +Domain +adaptation methods seek to align the source domain distri- +bution to a particular target domain. To bridge the global +and instance-level domain gaps, [3, 5, 41, 43] learn feature +alignment via [15] adversarial training; [58] and [46] utilize +category-level centroids and attention maps, respectively, to +better align instances in the two domains; [8, 30] generate +pseudo-labels in the target domain and use them for target- +aware training. Domain adaptation, however, assumes that +images from the target domain are available during training. +In contrast, domain generalization aims to learn models that +generalize to domains that were not seen at all during train- +ing. Below, we focus on the domain generalization methods +that, as us, use a single source domain to do so. +Single Domain Generalization (SDG). +Several image +classification works [13,35,45,48,56] have proposed strate- +gies to improve the performance on unseen domains while +training on a single source domain. In particular, [35,45,48] +introduce data augmentation strategies where diverse input +images are generated via adversarial training; [13, 56] pro- +pose normalization techniques to adapt the feature distri- +bution to unseen domains. While SDG has been reason- +ably well studied for image classification, the case of ob- +ject detection remains largely unexplored, and poses addi- +tional challenges related to the need to further localize the +objects of interest. This was recently tackled by Single- +DGOD [49] with an approach relying on learning domain- +specific and domain-invariant features. +Specifically, this +was achieved by exploiting contrastive learning to disentan- +gle the features and self-distillation [22] to further improve +the network’s generalizability. Here, we introduce a fun- +damentally different approach that leverages the CLIP [36] +pre-trained model and semantically augments the data us- +ing textual prompts. As will be shown by our results, our +method outperforms the state-of-the-art Single-DGOD [49]. +Vision-Language Models. +Jointly learning a representa- +tion of images and text has been studied in many works [9, +11,12,14,24,27,36,55]. They use image-text pairs to train +visual-semantic embeddings which can be used not only +for image classification, captioning or retrieval but also for +zero-shot prediction on unseen labels. VirTex [9] relies on +image-caption-based pre-training to learn a rich visual em- +bedding from a small amount of data. CLIP [36] proposes a +scalable contrastive pre-training method for joint text and +image feature learning. CLIP leverages a corpus of 400 +million image-text pairs and a large language model [37] to +learn a joint embedding space, which was shown to have su- +perior zero-shot learning ability on classification tasks. The +image-text-based training is also useful for Open Vocabu- +lary Detection (OVD) [53], where the objects are detected +using arbitrary textual descriptions. To address this task, +[53] train their own visual-semantic representation, whereas +[16, 39] employ CLIP embeddings. Recently, [29, 54] in- +troduced a phrase-grounding-based pre-training for better +OVD and zero-shot object detection. In contrast to these +works, whose objective is to generalize to novel categories +or objects, we seek to generalize to new domains depicting +the same object categories as the source one. +3. Method +Let us now introduce our approach to exploiting a vision- +language model for single-domain generalization in object +detection. Below, we first present our semantic augmenta- +tion strategy aiming to facilitate generalization to new do- +mains. We then describe the architecture and training strat- +egy for our object detector. +2 + +Figure 2. Our Approach: (Left) We first estimate a set of semantic augmentations A using a set of textual domain prompts {Pt, ps} +and source domain images. The goal of these semantic augmentations is to translate source domain image embeddings to the domain +specified by the prompts. We can do this because of the CLIP’s joint embedding space and its ability to encode semantic relationships via +algebraic operations. Lopt is minimized w.r.t A over random image crops of the same size as CLIP [36]. (Right) The optimized semantic +augmentations are used to train our modified detector which minimizes a text-based classification loss Lclip�t. Here, we train with the full +image and add a randomly sampled Aj after average pooling. This pooling operation allows us to use A on extracted feature maps of the +arbitrary-sized image. We initialize the detector with the pre-trained CLIP [36] V and T encoders to leverage their general representations. +3.1. Semantic Augmentation +In SDG, we have access to images from only a single +domain. To enable generalization, we seek to learn object +representations that are robust to domain shifts. Here, we +do so by introducing such shifts while training the model +on the source data. Specifically, we exploit CLIP’s joint +representation to estimate shifts in the visual domain using +textual prompts, as illustrated in Fig. 1. This corresponds to +the optimization step shown in the left portion of Fig. 2. +Formally, let T denote CLIP’s text encoder and V its im- +age one. For reasons that will become clear later, we further +split V into a feature extractor Va and a projector to the em- +bedding space Vb. The CLIP [36] model is trained to bring +image features closer to their textual captions. In essence, +this means that, for an image I and a corresponding prompt +p, it seeks to minimize the distance between Vb(Va(I)) and +T (p). +A useful property of the text embedding space is that +algebraic operations can be used to estimate semantically +related concepts. Word2Vec [31] had demonstrated such a +learned relationship (e.g. king-man+woman approaches the +word representation of queen). Such a relationship exists +with CLIP embeddings as well [38]. +To exploit this for SDG, we define a generic textual +prompt ps related to the source domain, such as An image +taken during the day, and a set of prompts Pt = +{pt +j}M +1 +encompassing variations that can be expected to +occur in different target domains, e.g, describing different +weather conditions or times of the day. Our objective then +is to define augmentations {Aj} of the features extracted +from a source image such that the shift incurred by Aj cor- +responds to the semantic difference between ps and pt +j. +To achieve this, we first compute the embeddings qs = +T (ps) and qt +j = T (pt +j) of the textual prompt. We then take +multiple random crops from a source image. For each such +crop Icrop, we create a target image embedding +z∗ +j = z + +qt +j − qs +∥qt +j − qs∥2 +, +(1) +where z = V(Icrop). We then search for an augmentation +Aj ∈ RH×W ×C such that +¯zj = Vb(Va(Icrop) + Aj) +(2) +is as similar as possible to z∗ +j , which we measure with the +cosine similarity. Ultimately, we estimate the augmenta- +tions {Aj}M +1 +through an optimization process using only +source domain images. Specifically, we minimize the loss +function +Lopt = +� +Icrop +� +j +D(z∗ +j , ¯zj) + ∥¯zj − z∥1 , +(3) +where +D(a, b) = 1 − +a − b +∥a − b∥2 +(4) +is the cosine distance. The loss also includes an l1 regu- +larizer that prevents the embeddings from deviating too far +from their initial values, so as to preserve the image content. +As the objective is to estimate the meaningful fea- +ture augmentation while preserving the original CLIP pre- +training, we keep the image crop size the same as the orig- +inal CLIP training. Note that the optimization of the aug- +mentations is done once in an offline stage, and we then use +the resulting augmentations to train our detector. +3 + +Semantic Augmentations +A = [A1,A2, .., AM] +RPN +va ++) +ROI +ROI ++ +vb +Align +head +Avgpool +2 +zj +Random Crops +A; = Sample(A) +Lclip-t +q +qf +K classes +CLIP Init. +Car + Source Domain prompt +Bus +CLIP Frozen +a photo of +Person +Domain prompts +pt +pt =- (pi, p2,...PM] +Random Init. +Truck + CLIP Frozen +Optimization Step +Training StepFigure 3. Diverse Weather Dataset [49]: Day-Clear acts as our source domain while the other weather condition are our target domains. +In these domains, the objects’ appearance drastically changes from the Day-Clear scenario. As we do not utilize any target domain images, +learning generalizable features on source images is crucial for the SDG task. +3.2. Architecture +Let us now describe our detector architecture. As shown +in the right portion of Fig. 2, it follows a standard Faster- +RCNN [40] structure but departs from it in two ways. First, +to exploit the augmentations optimized as discussed in the +previous section, we initialize the blocks before and af- +ter the ROI align one with the corresponding Va and Vb +modules of the ResNet-based trained CLIP model. Second, +to further leverage the vision-language model, we incorpo- +rate a text-based classifier in our model’s head. Note that, +in contrast to OVD [16, 39] where a text-based classifier +is used to handle novel categories, we employ it to keep +the image features close to the pre-trained joint embedding +space. +Specifically, we define textual prompts that represent the +individual categories we seek to detect, and extract corre- +sponding embeddings Q ∈ R(K+1)×Dclip, for K categories +and the background class, using the text encoder T . For +a candidate image region r proposed by the Region Pro- +posal Network(RPN) [40], we then compute the cosine sim- +ilarities between the text embeddings Q and the features +Fr ∈ RDclip obtained by projection to the embedding space +using Vb after ROI-Align [19] and the text embeddings Q. +These cosine similarities, sim(Fr, Q) ∈ RK+1, act as log- +its to the softmax based cross-entropy loss +Lclip�t = +� +r +LCE +� +esim(Fr,Qk) +�K +k=0 esim(Fr,Qk) +� +. +(5) +Similarly to [36], we formulate prompts of the form a +photo of a {category name} to obtain our text +embeddings. +3.3. Training with Augmentation +Following the standard detector training [40], we use the +full image as our input. This subsequently increases the +output feature map size of Va, hence we use average pool- +ing operation and obtain channel-wise augmentations which +can work for arbitrary-sized feature maps. The training of +our modified object detector with the semantic augmenta- +tions is as follows, first, we randomly sample an augmenta- +tion Aj from the full set and collapse its spatial dimension +using average pooling. We then add the resulting vector to +every element in the feature map extracted by Va. In prac- +tice, we apply augmentations to a batch with a probability +θ. +The detector is then trained with the loss +Ldet = Lrpn + Lreg + Lclip�t , +(6) +which combines the Lclip�t loss of Eq. (5) with the standard +RPN and regression losses [40]. During inference, we use +the detector without any augmentation of the feature maps. +4. Experiments +4.1. Experimental setup +Datasets. +To evaluate our model, we use the same +datasets as [49]. They include five sets, each containing +images with different weather conditions: daytime sunny, +night clear, dusk rainy, night rainy, and daytime foggy. +The images have been selected from three primary datasets, +Berkeley Deep Drive 100K (BBD-100K) [52], Cityscapes +[7] and Adverse-Weather [17]. Additionally, rainy images +are rendered by [50], and some of the foggy images are syn- +thetically generated from [42]. Our model is trained on the +daytime sunny scenes, consisting of 19,395 training images, +the remaining 8,313 daytime sunny images are used for val- +idation and model selection. The four other weather condi- +tions are only used during testing. They consist of 26,158 +images of clear night scenes, 3501 images of rainy scenes +at dusk, 2494 images of rainy scenes at night, and 3775 im- +ages of foggy scenes during daytime. All the datasets con- +tain bounding box annotations for the objects bus, bike, car, +motorbike, person, rider and truck. Fig. 3 shows examples +from this dataset. +Metric. +In all our experiments, we use the Mean Average +Precision (mAP) as our metric. Specifically, following [49], +we report the mAP@0.5, which considers a prediction as a +true positive if it matches the ground-truth label and has an +intersection over union (IOU) score of more than 0.5 with +the ground-truth bounding box. +4 + +Day - Clear +Day - Foggy +Dusk-Rainy +Night - Clear +Night - RainyFigure 4. Qualitative Results. We visualize the predictions of the detectors trained only with day-clear images. (Top) FasterRCNN [40] +predictions. (Bottom) The predictions with our approach. Night-Clear and Night-Rainy contain scenes that are taken under low light +conditions. Due to this, the appearance of the object is obscure and deviates from the daytime case. FasterRCNN fails to detect most of +the objects. As shown in the Night-Clear, it misclassifies a car to bus. By contrast, we can still detect car under such a big shift. For +Dusk-Rainy scenes, the rain pattern on the windscreen and the wet ground causes an appearance shift. As shown FasterRCNN fails to +detect several cars and misclassifies person on the bottom-left. +Figure 5. Qualitative Results. In the foggy scenes, the objects +further away w.r.t the camera are more obscure than the near ones. +Due to this FasterRCNN (Top) struggles to detect them. car and +person missed by FasterRCNN are successfully recovered by our +approach (Bottom). +4.2. Implementation Details +We use the Detectron2 [51] implementation of Faster- +RCNN with a ResNet101 [20] backbone. We initialize the +detector with CLIP [36] pre-trained weights, where ResNet +convolution blocks 1-3 act as Va, and block-4 along with +the CLIP attention pooling act as Vb. This follows from the +standard FasterRCNN implementation with ResNet back- +bone. +Optimization Step. +As the benchmark dataset evalu- +ates the method on different weather conditions, we cu- +rated a list of domain prompts Pt matching the concept +weather. +To this end, we take all the hyponyms of the +term weather from WordNet [44] and generate their text +embeddings using the CLIP text encoder T . +We prune +away the words whose cosine similarity with the term +weather is lower than 0.5. Additionally, we filter out the +words that are not in the top 10k frequent words in GloVe +wordlist [34]. After combining the synonyms, we get to +a list of six words: snow, fog, cloudy, rain, stormy, sun- +shine. We remove sunshine as it corresponds to our source +domain concept. +Furthermore, we consider three times +of the day: day, night, evening. +This lets us generate +M = 15 prompts using the template an image taken +on a {weather} {time of the day}. We use an +image taken during the day as the source do- +main prompt ps. We provide more details in our supple- +mentary material. +To optimize the augmentations with these prompts, we +generated random crops from the source images and re- +sized them to 224 × 224 pixels. The resulting output fea- +ture map of Va and Aj are in R14×14×1024. We initial- +ize Aj ∀ 1 ≥ j ≥ M with zeros and train it using the +Adam [23] optimizer while keeping the CLIP encoder, V +and T , frozen. Optimization was done for 1000 iterations +with a learning rate of 0.01. +Detector Training with Augmentation. +When training +the detector, the input image is resized to 600 × 1067 and V +and T are initialized with CLIP pre-trained weights. While +T is kept frozen during the training, the ResNet blocks 3- +4 and attention pooling of V, along with the other Faster- +RCNN learnable blocks, are trained with Stochastic Gra- +dient Descent (SGD) for 100k iterations. We train with a +learning rate of 1e−3, scaled down by a factor of 0.1 after +40k iterations. We use a batch size of 4 and apply Aj to +the features with probability θ = 0.5. We also use random +5 + +Night-Clear +Dusk-RainyDay-Foggy +Day-FoggymAP +Method +Day +Clear +Night +Clear +Dusk +Rainy +Night +Rainy +Day +Foggy +FR [40] +48.1 +34.4 +26.0 +12.4 +32.0 +SW [33] +50.6 +33.4 +26.3 +13.7 +30.8 +IBN-Net [32] +49.7 +32.1 +26.1 +14.3 +29.6 +IterNorm [21] +43.9 +29.6 +22.8 +12.6 +28.4 +ISW [6] +51.3 +33.2 +25.9 +14.1 +31.8 +S-DGOD [49] +56.1 +36.6 +28.2 +16.6 +33.5 +Ours +51.3 +36.9 +32.3 +18.7 +38.5 +Table 1. Single domain generalization results. We show consis- +tent improvements across all the target domains. S-DGOD boosts +the source domain results, but at the cost of reduced generalization +ability. By contrast, our approach is robust to domain changes. +The numbers for S-DGOD, SW, IBN-Net, IterNorm, ISW are +taken from [49]. +horizontal flipping augmentation as in Single-DGOD [49]. +Dclip is set to 512 as in [36] and background class is initial- +ized by zeros in Q. All of our training was done on a single +NVIDIA A100 GPU. Our code will be made public upon +acceptance. +4.3. Comparison with the State of the Art +We compare our method trained with semantic augmen- +tations against the state-of-the-art Single-DGOD [49]. Sim- +ilar to them, we also show comparisons with feature nor- +malization methods, SW [33], IBN-Net [32], IterNorm [21], +and ISW [6]. These methods improve network generaliza- +tion by using better feature normalization. We addition- +ally report the performance of FasterRCNN (FR) initialized +with ImageNet pre-trained weights. For the SDG task, we +evaluate the generalization performance on unseen target +domains, hence we compare the mAP scores on the out- +of-domain datasets: day-foggy, night-rainy, dusk-rainy, and +night-clear. +Our approach of combining CLIP pre-training and se- +mantic augmentation outperforms the baselines on all of the +target domains. Tab. 1 shows a consistent improvement in +all domains with close to 15% improvement on day-foggy +and dusk-rainy compared to Single-DGOD. In the challeng- +ing scenario with Night conditions, we improve by 12.6% +on night-rainy while being comparable with Single-DGOD +on night-clear. On the source domain, both our method and +Single-DGOD are better than the FR baseline. However, +while Single-DGOD gains improvement at the cost of los- +AP +mAP +Method +Bus Bike Car Motor Person Rider Truck +All +FR [40] 28.1 29.7 49.7 +26.3 +33.2 +35.5 +21.5 +32.0 +S-DGOD [49] 32.9 28.0 48.8 +29.8 +32.5 +38.2 +24.1 +33.5 +Ours 36.1 34.3 58.0 +33.1 +39.0 +43.9 +25.1 +38.5 +Table 2. Per-class results on Daytime Clear to Day Foggy. Our +method consistently performs better on all categories for the dif- +ficult foggy domain. This shows that CLIP initialization and our +semantic augmentations improve the detector’s generalizability. +AP +mAP +Method +Bus Bike Car Motor Person Rider Truck +All +FR [40] 28.5 20.3 58.2 +6.5 +23.4 +11.3 +33.9 +26.0 +S-DGOD [49] 37.1 19.6 50.9 +13.4 +19.7 +16.3 +40.7 +28.2 +Ours 37.8 22.8 60.7 +16.8 +26.8 +18.7 +42.4 +32.3 +Table 3. Per-class results on Daytime Clear to Dusk Rainy. +Our approach generalizes to rainy road conditions along with the +low light conditions of the dusk hours. The car category sees the +biggest improvement, but we nonetheless also boost the perfor- +mance of all the other classes. +ing out for domain generalization, we improve on both the +source and target domains. The failure of feature normal- +ization baselines suggests a large domain gap between the +source and target domains. Fig. 4 and Fig. 5 provide a qual- +itative results on different weather-datasets. +In the remainder of this section, we discuss the per-class +results on the individual target domains. +Daytime Clear to Day Foggy. +The object appearance +drastically changes in the foggy images compared to the +day-clear scenario. As shown in Tab. 2, our method brings +in a large improvement for the car, person, and bike cat- +egories, while still being consistently better than Single- +DGOD and FR on the others. +Daytime Clear to Dusk Rainy. +Dusk Rainy scenes re- +flect a low light condition and along with the rainy pat- +tern. +The image distribution is thus further away from +the daytime clear images. +As shown in Tab. 3, our +method improves the AP of each class, with the biggest +improvement in the car and person categories. Since we +leverage CLIP pre-training and bring in concepts such as +rain/cloudy/stormy and evening/night hours through our se- +mantic augmentation, the learnt detector generalizes better. +6 + +AP +mAP +Method +Bus Bike Car Motor Person Rider Truck +All +FR [40] 34.7 32.0 56.6 +13.6 +37.4 +27.6 +38.6 +34.4 +S-DGOD [49] 40.6 35.1 50.7 +19.7 +34.7 +32.1 +43.4 +36.6 +Ours 37.7 34.3 58.0 +19.2 +37.6 +28.5 +42.9 +36.9 +Table 4. Per-class results on Daytime Clear to Night Clear. +While being comparable to S-DGOD on most of the categories, +we improve on car and person. +AP +mAP +Method +Bus Bike Car Motor Person Rider Truck +All +FR [40] 16.8 +6.9 +26.3 +0.6 +11.6 +9.4 +15.4 +12.4 +S-DGOD [49] 24.4 11.6 29.5 +9.8 +10.5 +11.4 +19.2 +16.6 +Ours 28.6 12.1 36.1 +9.2 +12.3 +9.6 +22.9 +18.7 +Table 5. Per-class results on Daytime Clear to Night Rainy. +This dataset presents the most challenging scenario, where the low +light and rainy conditions obscure the objects. We still perform +better than the baseline on most of the categories. +Daytime Clear to Night Clear. +The Night Clear dataset +shows a challenging night driving scene under severe low- +light conditions. In Tab. 4, we show that while being com- +parable to Single-DGOD, we bring in a larger improvement +in the car and person categories. Night scenes are partic- +ularly challenging as the low light condition leads to more +confusion among visually closer categories such as bus and +truck. +Daytime Clear to Night Rainy. +This is the most chal- +lenging scenario where dark night conditions are exacer- +bated by patterns occurring due to rain. Tab. 5 shows consis- +tent improvement by our approach for most of the classes. +The car class sees the biggest improvement with an increase +in AP of more than 22% compared to Single-DGOD. The +lower performance of the class rider can be attributed to an +increase in the confusion between the visually similar per- +son and rider classes under adverse conditions. +4.4. Ablation Study +To understand how each element of the proposed method +contributes to the overall performance, we conduct an ab- +lation study. +We test five individual components of our +model. Specifically, we remove semantic augmentation, re- +place CLIP attention pooling in Vb with average pooling, +replace Lclip�t with the FasterRCNN classification loss, and +change the weight initialization from the CLIP model to +an ImageNet classification model. +Removing those five +components turns our model back into the standard Faster- +RCNN. The ablation study results are provided in Tab. 6 +and discussed below. +CLIP initialization. +When the FasterRCNN backbone +V is initialized with CLIP pre-trained weights, the model +performance consistently increases both in the in-domain +and out-of-domain scenarios, as shown in the second row +of Tab. 6. This setting itself already outperforms Single- +DGOD (penultimate row of Tab. 1). This goes to show that, +for the generalization task, model weight initialization plays +a crucial role. We further improve this performance with se- +mantic augmentations. +Attention pooling and Lclip�t. +Next we test the impact +of the text-embedding-based loss Lclip�t for classification. +As visible in the third row of Tab. 6, when combined with +CLIP initialization, it improves the generalization perfor- +mance for the rainy scenarios, but degrades it for the other +ones. Replacing average pooling in Vb with CLIP attention +pooling helps to mitigate the detrimental effect of Lclip�t +and exhibits consistent improvement on all datasets. +Semantic augmentation. +Finally, adding semantic aug- +mentation gives us the best results, as shown in the last row +of Tab. 6. Exposing the visual encoder V to targeted seman- +tic augmentations helps the overall model to better gener- +alize when exposed to new domains sharing similarity with +the augmentations. +4.5. Additional Analyses +Study of semantic augmentation. +Our proposed method +involves translating feature maps by semantic augmenta- +tions learned using plausible domain prompts. To further +study the utility of our approach, we replace the augmen- +tation strategy in our training pipeline with (a) no-aug: no +augmentation; (b) random: A is initialized with a normal +distribution; (c) clip-random: we define Pt with concepts +that are not specific to weather. We generate prompts with +a template an image of {word}, where the words are +desert, ocean, forest, and mountain. Tab. 7 illustrates the +importance of the semantics in our augmentation strategy. +The random augmentation performs worse than the no-aug +strategy. clip-random is comparable to no-aug and doesn’t +show any consistent trend but is mostly better than random. +Our semantic augmentation strategy provides a consistent +improvement over no-aug because the translations are per- +formed with prompts from the relevant weather concept. +7 + +Model Component +mAP +Source +Target +CLIP init +Lclip�t +Attn. Pool +Sem. Aug +Day +Clear +Night +Clear +Dusk +Rainy +Night +Rainy +Day +Foggy +48.1 +34.4 +26.0 +12.4 +32.0 +✓ +51.2 +37.0 +31.0 +15.7 +37.5 +✓ +✓ +50.7 +36.0 +31.3 +16.3 +36.9 +✓ +✓ +✓ +51.0 +35.9 +31.3 +16.7 +37.7 +✓ +✓ +✓ +✓ +51.3 +36.9 +32.3 +18.7 +38.5 +Table 6. Ablation study. We study the influence of five different components of our approach: the backbone weight initialization strategy, +the classification loss, the attention pooling, and the semantic augmentation. When those five components are removed (first row of the +table) the model is equivalent to the standard FasterRCNN. Initializing the detector with CLIP weights (second row) largely improves the +generalization performance; on its own it already outperforms Single-DGOD (penultimate row of Tab. 1) on most of the datasets, hence +suggesting that CLIP has better generalizability than ImageNet pre-trained weights. Combining this with the text embedding-based loss +Lclip�t (third row) improves the results on the challenging scenarios of dusk rainy and night rainy, but has a detrimental effect for the other +weather conditions. Adding attention pooling to the architecture (fourth row) helps to mitigate these detrimental effects as it brings the +visual features closer to the joint embedding space. Finally, the best results are obtained when the semantic augmentation is added (last +row), greatly helping with adverse weather, rainy and foggy, scenarios. +mAP +Aug. Type +Day +Clear +Night +Clear +Dusk +Rainy +Night +Rainy +Day +Foggy +no-aug. +51.0 +35.9 +31.3 +16.7 +37.7 +random +51.2 +36.0 +30.4 +15.3 +37.3 +clip-random +51.5 +36.4 +30.2 +15.9 +37.9 +Ours w/ seg.aug +51.3 +36.9 +32.3 +18.7 +38.5 +Table 7. Semantic Augmentation. Our semantic augmentation +consistently outperforms other augmentation strategies. +While +random augmentations are worse than no-aug., clip-random is +comparable to no-aug.. Only when we give relevant prompts, there +is a consistent improvement across datasets. +5. Limitations +Our method augments visual features using textual +prompts. To generate these prompts, it is assumed that some +information about the domain gap is known. In our experi- +ments, we assumed that the domain gap was due to changes +in weather and daytime conditions. In practice, we only +used the word weather and time of the day to derive all the +prompts used in our augmentation; nonetheless, some extra +information was used. In most applications, however, the +domain gap can be known in advance, and providing a few +keywords characterizing it shouldn’t be an issue. In the rare +cases where no information can be known, our approach +still has the potential to be used by using multiple broad +concept keywords such as weather, ambiance, or location. +6. Conclusion +We have proposed an approach to improving the gener- +alization of object detectors on unseen target domains. Our +approach fundamentally departs from existing method by +leveraging a pre-trained vision-language model, CLIP, to +help the detector to generalize. Specifically, we have ex- +ploited textual prompts to develop a semantic augmentation +strategy that alters image embeddings so that they reflect +potential target domains, and to design a text-based image +classifier. 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In Proceedings of the IEEE/CVF Con- +ference on Computer Vision and Pattern Recognition, pages +687–696, 2019. 2 +11 + diff --git a/09E5T4oBgHgl3EQfOw41/content/tmp_files/load_file.txt b/09E5T4oBgHgl3EQfOw41/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..17026da1f834c8d75c59dec24afa71bd5108e420 --- /dev/null +++ b/09E5T4oBgHgl3EQfOw41/content/tmp_files/load_file.txt @@ -0,0 +1,714 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf,len=713 +page_content='CLIP the Gap: A Single Domain Generalization Approach for Object Detection Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2 CVLab, EPFL1, ClearSpace SA2 firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='lastname@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='ch Abstract Single Domain Generalization (SDG) tackles the prob- lem of training a model on a single source domain so that it generalizes to any unseen target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While this has been well studied for image classification, the literature on SDG object detection remains almost non-existent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To ad- dress the challenges of simultaneously learning robust ob- ject localization and representation, we propose to leverage a pre-trained vision-language model to introduce semantic domain concepts via textual prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We achieve this via a semantic augmentation strategy acting on the features ex- tracted by the detector backbone, as well as a text-based classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our experiments evidence the benefits of our approach, outperforming by 10% the only existing SDG object detection method, Single-DGOD [49], on their own diverse weather-driving benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Introduction As for most machine learning models, the performance of object detectors degrades when the test data distribu- tion deviates from the training data one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Domain adap- tation techniques [3, 5, 8, 30, 41, 43] try to alleviate this problem by learning domain invariant features between a source and a known target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In practice, however, it is not always possible to obtain target data, even un- labeled, precluding the use of such techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Domain generalization tackles this by seeking to learn representa- tions that generalize to any target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While early approaches [1, 10, 25, 26, 28, 47, 57] focused on the sce- nario where multiple source domains are available during training, many recent methods tackle the more challenging, yet more realistic, case of Single Domain Generalization (SDG), aiming to learn to generalize from a single source dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While this has been well studied for image clas- sification [13, 35, 45, 48, 56], it remains a nascent topic in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To the best of our knowledge, a single ex- isting approach, Single-DGOD [49], uses disentanglement and self-distillation [22] to learn domain-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In this paper, we introduce a fundamentally different ap- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Semantic Augmentation: We compare the PCA pro- jections of CLIP [36] image embeddings obtained in two different manners: (Top) The embeddings were directly obtained from the real images from 5 domains corresponding to different weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (Bottom) The embeddings were obtained from the day images only and modified with our semantic augmentation strat- egy based on text prompts to reflect the other 4 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Note that the relative positions of the clusters in the bottom plot resembles that of the top one, showing that our augmentations let us gener- alize to different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The principal components used are the same for both the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' proach to SDG for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To this end, we build on two observations: (i) Unsupervised/self-supervised pre- training facilitates the transfer of a model to new tasks [2, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='05499v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='CV] 13 Jan 2023 Imageday Imagenight Imagefoggy Image rainyday Image rainy night Imageday Semanticaugmentationnight Semantic augmentation foggy Semanticaugmentationrainy day Semantic augmentation rainy night4, 18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (ii) Exploiting language supervision to train vision models allows them to generalize more easily to new cat- egories and concepts [9, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Inspired by this, we there- fore propose to leverage a self-supervised vision-language model, CLIP [36], to guide the training of an object detec- tor so that it generalizes to unseen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Since the visual CLIP representation has been jointly learned with the textual one, we transfer text-based domain variations to the image representation during training, thus increasing the di- versity of the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we define textual prompts describing po- tential target domain concepts, such as weather and day- time variations for road scene understanding, and use these prompts to perform semantic augmentations of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' These augmentations, however, are done in feature space, not in image space, which is facilitated by the joint image- text CLIP latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1, which shows that, even though we did not use any target data for semantic augmentation, the resulting augmented embed- dings reflect the distributions of the true image embeddings from different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We show the effectiveness of our method on the SDG driving dataset of [49], which reflects a practical scenario where the training (source) images were captured on a clear day whereas the test (target) ones were acquired in rainy, foggy, night, and dusk conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our experiments demonstrate the benefits of our approach over the Single- DGOD [49] one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To summarize our contributions, we employ a vision- language model to improve the generalizability of an object detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' during training, we introduce domain concepts via text-prompts to augment the diversity of the learned image features and make them more robust to an unseen target do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This enables us to achieve state-of-the-art results on the diverse weather SDG driving benchmark of [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Related Work Domain Adaptation for Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Domain adaptation methods seek to align the source domain distri- bution to a particular target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To bridge the global and instance-level domain gaps, [3, 5, 41, 43] learn feature alignment via [15] adversarial training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' [58] and [46] utilize category-level centroids and attention maps, respectively, to better align instances in the two domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' [8, 30] generate pseudo-labels in the target domain and use them for target- aware training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Domain adaptation, however, assumes that images from the target domain are available during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In contrast, domain generalization aims to learn models that generalize to domains that were not seen at all during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Below, we focus on the domain generalization methods that, as us, use a single source domain to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Single Domain Generalization (SDG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Several image classification works [13,35,45,48,56] have proposed strate- gies to improve the performance on unseen domains while training on a single source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In particular, [35,45,48] introduce data augmentation strategies where diverse input images are generated via adversarial training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' [13, 56] pro- pose normalization techniques to adapt the feature distri- bution to unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While SDG has been reason- ably well studied for image classification, the case of ob- ject detection remains largely unexplored, and poses addi- tional challenges related to the need to further localize the objects of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This was recently tackled by Single- DGOD [49] with an approach relying on learning domain- specific and domain-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, this was achieved by exploiting contrastive learning to disentan- gle the features and self-distillation [22] to further improve the network’s generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Here, we introduce a fun- damentally different approach that leverages the CLIP [36] pre-trained model and semantically augments the data us- ing textual prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As will be shown by our results, our method outperforms the state-of-the-art Single-DGOD [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Vision-Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Jointly learning a representa- tion of images and text has been studied in many works [9, 11,12,14,24,27,36,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' They use image-text pairs to train visual-semantic embeddings which can be used not only for image classification, captioning or retrieval but also for zero-shot prediction on unseen labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' VirTex [9] relies on image-caption-based pre-training to learn a rich visual em- bedding from a small amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' CLIP [36] proposes a scalable contrastive pre-training method for joint text and image feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' CLIP leverages a corpus of 400 million image-text pairs and a large language model [37] to learn a joint embedding space, which was shown to have su- perior zero-shot learning ability on classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The image-text-based training is also useful for Open Vocabu- lary Detection (OVD) [53], where the objects are detected using arbitrary textual descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To address this task, [53] train their own visual-semantic representation, whereas [16, 39] employ CLIP embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Recently, [29, 54] in- troduced a phrase-grounding-based pre-training for better OVD and zero-shot object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In contrast to these works, whose objective is to generalize to novel categories or objects, we seek to generalize to new domains depicting the same object categories as the source one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Method Let us now introduce our approach to exploiting a vision- language model for single-domain generalization in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Below, we first present our semantic augmenta- tion strategy aiming to facilitate generalization to new do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We then describe the architecture and training strat- egy for our object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our Approach: (Left) We first estimate a set of semantic augmentations A using a set of textual domain prompts {Pt, ps} and source domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The goal of these semantic augmentations is to translate source domain image embeddings to the domain specified by the prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We can do this because of the CLIP’s joint embedding space and its ability to encode semantic relationships via algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Lopt is minimized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='t A over random image crops of the same size as CLIP [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (Right) The optimized semantic augmentations are used to train our modified detector which minimizes a text-based classification loss Lclip�t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Here, we train with the full image and add a randomly sampled Aj after average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This pooling operation allows us to use A on extracted feature maps of the arbitrary-sized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We initialize the detector with the pre-trained CLIP [36] V and T encoders to leverage their general representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Semantic Augmentation In SDG, we have access to images from only a single domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To enable generalization, we seek to learn object representations that are robust to domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Here, we do so by introducing such shifts while training the model on the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we exploit CLIP’s joint representation to estimate shifts in the visual domain using textual prompts, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This corresponds to the optimization step shown in the left portion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Formally, let T denote CLIP’s text encoder and V its im- age one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' For reasons that will become clear later, we further split V into a feature extractor Va and a projector to the em- bedding space Vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The CLIP [36] model is trained to bring image features closer to their textual captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In essence, this means that, for an image I and a corresponding prompt p, it seeks to minimize the distance between Vb(Va(I)) and T (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' A useful property of the text embedding space is that algebraic operations can be used to estimate semantically related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Word2Vec [31] had demonstrated such a learned relationship (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' king-man+woman approaches the word representation of queen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Such a relationship exists with CLIP embeddings as well [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To exploit this for SDG, we define a generic textual prompt ps related to the source domain, such as An image taken during the day, and a set of prompts Pt = {pt j}M 1 encompassing variations that can be expected to occur in different target domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='g, describing different weather conditions or times of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our objective then is to define augmentations {Aj} of the features extracted from a source image such that the shift incurred by Aj cor- responds to the semantic difference between ps and pt j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To achieve this, we first compute the embeddings qs = T (ps) and qt j = T (pt j) of the textual prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We then take multiple random crops from a source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' For each such crop Icrop, we create a target image embedding z∗ j = z + qt j − qs ∥qt j − qs∥2 , (1) where z = V(Icrop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We then search for an augmentation Aj ∈ RH×W ×C such that ¯zj = Vb(Va(Icrop) + Aj) (2) is as similar as possible to z∗ j , which we measure with the cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Ultimately, we estimate the augmenta- tions {Aj}M 1 through an optimization process using only source domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we minimize the loss function Lopt = � Icrop � j D(z∗ j , ¯zj) + ∥¯zj − z∥1 , (3) where D(a, b) = 1 − a − b ∥a − b∥2 (4) is the cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The loss also includes an l1 regu- larizer that prevents the embeddings from deviating too far from their initial values, so as to preserve the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As the objective is to estimate the meaningful fea- ture augmentation while preserving the original CLIP pre- training, we keep the image crop size the same as the orig- inal CLIP training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Note that the optimization of the aug- mentations is done once in an offline stage, and we then use the resulting augmentations to train our detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3 Semantic Augmentations A = [A1,A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='., AM] RPN va +) ROI ROI + vb Align head Avgpool 2 zj Random Crops A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' = Sample(A) Lclip-t q qf K classes CLIP Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Car Source Domain prompt Bus CLIP Frozen a photo of Person Domain prompts pt pt =- (pi, p2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='PM] Random Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Truck CLIP Frozen Optimization Step Training StepFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Diverse Weather Dataset [49]: Day-Clear acts as our source domain while the other weather condition are our target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In these domains, the objects’ appearance drastically changes from the Day-Clear scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As we do not utilize any target domain images, learning generalizable features on source images is crucial for the SDG task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Architecture Let us now describe our detector architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As shown in the right portion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 2, it follows a standard Faster- RCNN [40] structure but departs from it in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' First, to exploit the augmentations optimized as discussed in the previous section, we initialize the blocks before and af- ter the ROI align one with the corresponding Va and Vb modules of the ResNet-based trained CLIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Second, to further leverage the vision-language model, we incorpo- rate a text-based classifier in our model’s head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Note that, in contrast to OVD [16, 39] where a text-based classifier is used to handle novel categories, we employ it to keep the image features close to the pre-trained joint embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we define textual prompts that represent the individual categories we seek to detect, and extract corre- sponding embeddings Q ∈ R(K+1)×Dclip, for K categories and the background class, using the text encoder T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' For a candidate image region r proposed by the Region Pro- posal Network(RPN) [40], we then compute the cosine sim- ilarities between the text embeddings Q and the features Fr ∈ RDclip obtained by projection to the embedding space using Vb after ROI-Align [19] and the text embeddings Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' These cosine similarities, sim(Fr, Q) ∈ RK+1, act as log- its to the softmax based cross-entropy loss Lclip�t = � r LCE � esim(Fr,Qk) �K k=0 esim(Fr,Qk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (5) Similarly to [36], we formulate prompts of the form a photo of a {category name} to obtain our text embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Training with Augmentation Following the standard detector training [40], we use the full image as our input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This subsequently increases the output feature map size of Va, hence we use average pool- ing operation and obtain channel-wise augmentations which can work for arbitrary-sized feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The training of our modified object detector with the semantic augmenta- tions is as follows, first, we randomly sample an augmenta- tion Aj from the full set and collapse its spatial dimension using average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We then add the resulting vector to every element in the feature map extracted by Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In prac- tice, we apply augmentations to a batch with a probability θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The detector is then trained with the loss Ldet = Lrpn + Lreg + Lclip�t , (6) which combines the Lclip�t loss of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (5) with the standard RPN and regression losses [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' During inference, we use the detector without any augmentation of the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Experimental setup Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To evaluate our model, we use the same datasets as [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' They include five sets, each containing images with different weather conditions: daytime sunny, night clear, dusk rainy, night rainy, and daytime foggy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The images have been selected from three primary datasets, Berkeley Deep Drive 100K (BBD-100K) [52], Cityscapes [7] and Adverse-Weather [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Additionally, rainy images are rendered by [50], and some of the foggy images are syn- thetically generated from [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our model is trained on the daytime sunny scenes, consisting of 19,395 training images, the remaining 8,313 daytime sunny images are used for val- idation and model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The four other weather condi- tions are only used during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' They consist of 26,158 images of clear night scenes, 3501 images of rainy scenes at dusk, 2494 images of rainy scenes at night, and 3775 im- ages of foggy scenes during daytime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' All the datasets con- tain bounding box annotations for the objects bus, bike, car, motorbike, person, rider and truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3 shows examples from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In all our experiments, we use the Mean Average Precision (mAP) as our metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, following [49], we report the mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5, which considers a prediction as a true positive if it matches the ground-truth label and has an intersection over union (IOU) score of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 with the ground-truth bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4 Day - Clear Day - Foggy Dusk-Rainy Night - Clear Night - RainyFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We visualize the predictions of the detectors trained only with day-clear images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (Top) FasterRCNN [40] predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (Bottom) The predictions with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Night-Clear and Night-Rainy contain scenes that are taken under low light conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Due to this, the appearance of the object is obscure and deviates from the daytime case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' FasterRCNN fails to detect most of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As shown in the Night-Clear, it misclassifies a car to bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' By contrast, we can still detect car under such a big shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' For Dusk-Rainy scenes, the rain pattern on the windscreen and the wet ground causes an appearance shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As shown FasterRCNN fails to detect several cars and misclassifies person on the bottom-left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In the foggy scenes, the objects further away w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='t the camera are more obscure than the near ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Due to this FasterRCNN (Top) struggles to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' car and person missed by FasterRCNN are successfully recovered by our approach (Bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Implementation Details We use the Detectron2 [51] implementation of Faster- RCNN with a ResNet101 [20] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We initialize the detector with CLIP [36] pre-trained weights, where ResNet convolution blocks 1-3 act as Va, and block-4 along with the CLIP attention pooling act as Vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This follows from the standard FasterRCNN implementation with ResNet back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Optimization Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As the benchmark dataset evalu- ates the method on different weather conditions, we cu- rated a list of domain prompts Pt matching the concept weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To this end, we take all the hyponyms of the term weather from WordNet [44] and generate their text embeddings using the CLIP text encoder T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We prune away the words whose cosine similarity with the term weather is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Additionally, we filter out the words that are not in the top 10k frequent words in GloVe wordlist [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' After combining the synonyms, we get to a list of six words: snow, fog, cloudy, rain, stormy, sun- shine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We remove sunshine as it corresponds to our source domain concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Furthermore, we consider three times of the day: day, night, evening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This lets us generate M = 15 prompts using the template an image taken on a {weather} {time of the day}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We use an image taken during the day as the source do- main prompt ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We provide more details in our supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To optimize the augmentations with these prompts, we generated random crops from the source images and re- sized them to 224 × 224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The resulting output fea- ture map of Va and Aj are in R14×14×1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We initial- ize Aj ∀ 1 ≥ j ≥ M with zeros and train it using the Adam [23] optimizer while keeping the CLIP encoder, V and T , frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Optimization was done for 1000 iterations with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Detector Training with Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' When training the detector, the input image is resized to 600 × 1067 and V and T are initialized with CLIP pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While T is kept frozen during the training, the ResNet blocks 3- 4 and attention pooling of V, along with the other Faster- RCNN learnable blocks, are trained with Stochastic Gra- dient Descent (SGD) for 100k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We train with a learning rate of 1e−3, scaled down by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 after 40k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We use a batch size of 4 and apply Aj to the features with probability θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We also use random 5 Night-Clear Dusk-RainyDay-Foggy Day-FoggymAP Method Day Clear Night Clear Dusk Rainy Night Rainy Day Foggy FR [40] 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 SW [33] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 IBN-Net [32] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 IterNorm [21] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 ISW [6] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 S-DGOD [49] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Ours 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Single domain generalization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We show consis- tent improvements across all the target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' S-DGOD boosts the source domain results, but at the cost of reduced generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' By contrast, our approach is robust to domain changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The numbers for S-DGOD, SW, IBN-Net, IterNorm, ISW are taken from [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' horizontal flipping augmentation as in Single-DGOD [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Dclip is set to 512 as in [36] and background class is initial- ized by zeros in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' All of our training was done on a single NVIDIA A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our code will be made public upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Comparison with the State of the Art We compare our method trained with semantic augmen- tations against the state-of-the-art Single-DGOD [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Sim- ilar to them, we also show comparisons with feature nor- malization methods, SW [33], IBN-Net [32], IterNorm [21], and ISW [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' These methods improve network generaliza- tion by using better feature normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We addition- ally report the performance of FasterRCNN (FR) initialized with ImageNet pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' For the SDG task, we evaluate the generalization performance on unseen target domains, hence we compare the mAP scores on the out- of-domain datasets: day-foggy, night-rainy, dusk-rainy, and night-clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our approach of combining CLIP pre-training and se- mantic augmentation outperforms the baselines on all of the target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1 shows a consistent improvement in all domains with close to 15% improvement on day-foggy and dusk-rainy compared to Single-DGOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In the challeng- ing scenario with Night conditions, we improve by 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6% on night-rainy while being comparable with Single-DGOD on night-clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' On the source domain, both our method and Single-DGOD are better than the FR baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' However, while Single-DGOD gains improvement at the cost of los- AP mAP Method Bus Bike Car Motor Person Rider Truck All FR [40] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 S-DGOD [49] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Ours 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Per-class results on Daytime Clear to Day Foggy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our method consistently performs better on all categories for the dif- ficult foggy domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This shows that CLIP initialization and our semantic augmentations improve the detector’s generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' AP mAP Method Bus Bike Car Motor Person Rider Truck All FR [40] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 S-DGOD [49] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 Ours 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Per-class results on Daytime Clear to Dusk Rainy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our approach generalizes to rainy road conditions along with the low light conditions of the dusk hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The car category sees the biggest improvement, but we nonetheless also boost the perfor- mance of all the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' ing out for domain generalization, we improve on both the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The failure of feature normal- ization baselines suggests a large domain gap between the source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 5 provide a qual- itative results on different weather-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In the remainder of this section, we discuss the per-class results on the individual target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Daytime Clear to Day Foggy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The object appearance drastically changes in the foggy images compared to the day-clear scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 2, our method brings in a large improvement for the car, person, and bike cat- egories, while still being consistently better than Single- DGOD and FR on the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Daytime Clear to Dusk Rainy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Dusk Rainy scenes re- flect a low light condition and along with the rainy pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The image distribution is thus further away from the daytime clear images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 3, our method improves the AP of each class, with the biggest improvement in the car and person categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Since we leverage CLIP pre-training and bring in concepts such as rain/cloudy/stormy and evening/night hours through our se- mantic augmentation, the learnt detector generalizes better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6 AP mAP Method Bus Bike Car Motor Person Rider Truck All FR [40] 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 S-DGOD [49] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 Ours 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Per-class results on Daytime Clear to Night Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While being comparable to S-DGOD on most of the categories, we improve on car and person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' AP mAP Method Bus Bike Car Motor Person Rider Truck All FR [40] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 S-DGOD [49] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 Ours 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Per-class results on Daytime Clear to Night Rainy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This dataset presents the most challenging scenario, where the low light and rainy conditions obscure the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We still perform better than the baseline on most of the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Daytime Clear to Night Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The Night Clear dataset shows a challenging night driving scene under severe low- light conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4, we show that while being com- parable to Single-DGOD, we bring in a larger improvement in the car and person categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Night scenes are partic- ularly challenging as the low light condition leads to more confusion among visually closer categories such as bus and truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Daytime Clear to Night Rainy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This is the most chal- lenging scenario where dark night conditions are exacer- bated by patterns occurring due to rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 5 shows consis- tent improvement by our approach for most of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The car class sees the biggest improvement with an increase in AP of more than 22% compared to Single-DGOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The lower performance of the class rider can be attributed to an increase in the confusion between the visually similar per- son and rider classes under adverse conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Ablation Study To understand how each element of the proposed method contributes to the overall performance, we conduct an ab- lation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We test five individual components of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we remove semantic augmentation, re- place CLIP attention pooling in Vb with average pooling, replace Lclip�t with the FasterRCNN classification loss, and change the weight initialization from the CLIP model to an ImageNet classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Removing those five components turns our model back into the standard Faster- RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The ablation study results are provided in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6 and discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' CLIP initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' When the FasterRCNN backbone V is initialized with CLIP pre-trained weights, the model performance consistently increases both in the in-domain and out-of-domain scenarios, as shown in the second row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This setting itself already outperforms Single- DGOD (penultimate row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' This goes to show that, for the generalization task, model weight initialization plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We further improve this performance with se- mantic augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Attention pooling and Lclip�t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Next we test the impact of the text-embedding-based loss Lclip�t for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' As visible in the third row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6, when combined with CLIP initialization, it improves the generalization perfor- mance for the rainy scenarios, but degrades it for the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Replacing average pooling in Vb with CLIP attention pooling helps to mitigate the detrimental effect of Lclip�t and exhibits consistent improvement on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Semantic augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Finally, adding semantic aug- mentation gives us the best results, as shown in the last row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Exposing the visual encoder V to targeted seman- tic augmentations helps the overall model to better gener- alize when exposed to new domains sharing similarity with the augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Additional Analyses Study of semantic augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our proposed method involves translating feature maps by semantic augmenta- tions learned using plausible domain prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To further study the utility of our approach, we replace the augmen- tation strategy in our training pipeline with (a) no-aug: no augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (b) random: A is initialized with a normal distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' (c) clip-random: we define Pt with concepts that are not specific to weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We generate prompts with a template an image of {word}, where the words are desert, ocean, forest, and mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 7 illustrates the importance of the semantics in our augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' The random augmentation performs worse than the no-aug strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' clip-random is comparable to no-aug and doesn’t show any consistent trend but is mostly better than random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our semantic augmentation strategy provides a consistent improvement over no-aug because the translations are per- formed with prompts from the relevant weather concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 7 Model Component mAP Source Target CLIP init Lclip�t Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Pool Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Aug Day Clear Night Clear Dusk Rainy Night Rainy Day Foggy 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='1 34.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 ✓ ✓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 ✓ ✓ ✓ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 ✓ ✓ ✓ ✓ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We study the influence of five different components of our approach: the backbone weight initialization strategy, the classification loss, the attention pooling, and the semantic augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' When those five components are removed (first row of the table) the model is equivalent to the standard FasterRCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Initializing the detector with CLIP weights (second row) largely improves the generalization performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' on its own it already outperforms Single-DGOD (penultimate row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 1) on most of the datasets, hence suggesting that CLIP has better generalizability than ImageNet pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Combining this with the text embedding-based loss Lclip�t (third row) improves the results on the challenging scenarios of dusk rainy and night rainy, but has a detrimental effect for the other weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Adding attention pooling to the architecture (fourth row) helps to mitigate these detrimental effects as it brings the visual features closer to the joint embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Finally, the best results are obtained when the semantic augmentation is added (last row), greatly helping with adverse weather, rainy and foggy, scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' mAP Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Type Day Clear Night Clear Dusk Rainy Night Rainy Day Foggy no-aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 random 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 clip-random 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 Ours w/ seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='aug 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='5 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Semantic Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our semantic augmentation consistently outperforms other augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' While random augmentations are worse than no-aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=', clip-random is comparable to no-aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content='. Only when we give relevant prompts, there is a consistent improvement across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Limitations Our method augments visual features using textual prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' To generate these prompts, it is assumed that some information about the domain gap is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In our experi- ments, we assumed that the domain gap was due to changes in weather and daytime conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In practice, we only used the word weather and time of the day to derive all the prompts used in our augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' nonetheless, some extra information was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In most applications, however, the domain gap can be known in advance, and providing a few keywords characterizing it shouldn’t be an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In the rare cases where no information can be known, our approach still has the potential to be used by using multiple broad concept keywords such as weather, ambiance, or location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Conclusion We have proposed an approach to improving the gener- alization of object detectors on unseen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Our approach fundamentally departs from existing method by leveraging a pre-trained vision-language model, CLIP, to help the detector to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Specifically, we have ex- ploited textual prompts to develop a semantic augmentation strategy that alters image embeddings so that they reflect potential target domains, and to design a text-based image classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' We have shown that our approach outperforms the state of the art on four adverse-weather target datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' In future work, we plan to extend our approach to learning the prompts to further improve generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' 8 References [1] Yogesh Balaji, Swami Sankaranarayanan, and Rama Chel- lappa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E5T4oBgHgl3EQfOw41/content/2301.05499v1.pdf'} +page_content=' Metareg: Towards domain generalization using meta- regularization.' metadata={'source': 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+{kavinkang, yurong}@gdut.edu.cn, yuanwu@um.edu.mo, mpan2@uh.edu +Abstract—In this work, we investigate the challenging prob- +lem of on-demand federated learning (FL) over heterogeneous +edge devices with diverse resource constraints. We propose a +cost-adjustable FL framework, named AnycostFL, that enables +diverse edge devices to efficiently perform local updates under +a wide range of efficiency constraints. To this end, we design +the model shrinking to support local model training with elastic +computation cost, and the gradient compression to allow param- +eter transmission with dynamic communication overhead. An +enhanced parameter aggregation is conducted in an element-wise +manner to improve the model performance. Focusing on Any- +costFL, we further propose an optimization design to minimize +the global training loss with personalized latency and energy +constraints. By revealing the theoretical insights of the conver- +gence analysis, personalized training strategies are deduced for +different devices to match their locally available resources. Ex- +periment results indicate that, when compared to the state-of-the- +art efficient FL algorithms, our learning framework can reduce +up to 1.9 times of the training latency and energy consumption +for realizing a reasonable global testing accuracy. Moreover, +the results also demonstrate that, our approach significantly +improves the converged global accuracy. +Index Terms—Federated learning, edge intelligence, mobile +computing, resource management. +I. INTRODUCTION +Federated learning (FL) is an emerging distributed learning +paradigm that enables multiple edge devices to train a common +global model without sharing individual data [1]. This privacy- +friendly data analytics technique over massive devices is envi- +sioned as a promising solution to realize pervasive intelligence +[2]. However, in many real-world application areas, mobile de- +vices are often equipped with different local resources, which +raises the emerging challenges for locally on-demand training +[3]. Given different local resources status (e.g., computing +capability and communication channel state) and personalized +efficiency constraints (e.g., latency and energy), it is crucial to +customize training strategies for heterogeneous edge devices. +We perform an in-depth analysis on the time delay and the +energy consumption for performing the local model updates at +edge devices. Specifically, we evaluate and record the cost of +local training on three different NVIDIA Jetson family plat- +forms (i.e., Nano, NX AGX, and Xavier AGX) under different +channel states (i.e., good, medium, and poor). On the one hand, +we observe that the learning efficiency differs significantly +Xavier-Good +NX-Medium +Nano-Poor +0 +1 +2 +3 +4 +5 +6 +7 +8 +time delay (in second) of single-round local update +local model training +parameter transmission +4.0 times +0.7 times +Xavier-Good +NX-Medium +Nano-Poor +0 +2 +4 +6 +8 +10 +energy consumption (in joule) of single-round local update +local model training +parameter transmission +Fig. 1. +The time delay (top) and energy consumption (bottom) of single- +round local update on different hardware platforms with varying communica- +tion conditions. +with diverse learning scenarios. As shown in Fig. 1, the single- +epoch training on Nano with poor communication condition +consumes about 4.0 times training latency than that of Xavier +AGX with good communication condition, while its energy +consumption is about 0.7 times less than the latter one’s. On +the other hand, we observe that the bottlenecks of latency and +energy are induced by parameter transmission and local model +training, respectively. +The above observations provide insights for a proper design +of the on-demand FL system. To handle the resource hetero- +geneity, it is suggested to alleviate the energy and the latency +cost of the local device. More importantly, the computation +and communication costs should be jointly reduced to achieve +efficient local training. In the literature, most existing studies +either employ resource allocation and device scheduling to +mitigate the system cost [4]–[10], or design gradient com- +pression to accelerate the parameter transmission procedure +[11]–[17]. The former method inherits the ideas of traditional +design for mobile edge systems and takes no account of the +optimization for neural networks, while the latter overlooks +the computation cost of local model training. +In this paper, we propose “anycost” FL, named AnycostFL, +to break the latency and energy bottlenecks for on-demand +distributed training over heterogeneous edge devices. Our goal +is to develop a cost-adjustable FL framework that enables +edge devices to perform local updates under diverse learning +scenarios. To this end, we first design the model shrinking and + +gradient compression to enable adaptive local updates with +different computation and communication costs. Meanwhile, +an enhanced parameter aggregation scheme is proposed to +fuse the knowledge of the local updates. Following that, +we investigate the on-demand learning of AnycostFL by +regulating the local model structure, gradient compression +policy and computing frequency under personalized latency +and energy constraints. However, customizing training strategy +for different learning scenarios is a non-trivial task, since how +the global accuracy is affected by the local model structure and +compression rate is still unknown. To address this issue, we +theoretically reveal the convergence insights of our framework, +which are further leveraged to guide optimization analysis. +Finally, the optimal training strategy is derived for each device +according to its locally available resource. +Our main contributions are summarized as follows. +• We propose a novel FL framework, named AnycostFL, +that enables the local updates with elastic computation +cost and communication overhead. +• We theoretically present the optimal aggregation scheme +and convergence analysis for AnycostFL. +• We investigate the on-demand training problem of Any- +costFL, and the optimal training strategy is devised to +adapt the locally available resource. +• Extensive experiments indicate that the proposed Any- +costFL outperforms the state-of-the-art efficient FL meth- +ods in terms of resource utilization and learning accuracy. +The remainder of this paper is organized as follows. Section +II describes related studies. In Section III, we detail the main +operations of AnycostFL to fulfill the single-round training. +The problem formulation, theoretical analysis and the corre- +sponding solution are provided in Section IV. The experiment +evaluations are presented in Section V, and we finally conclude +the paper in Section VI and discuss the future directions. +II. RELATED WORK +Resource Management Methods. Resource management +methods aim to reduce the FL system cost by arranging +the local and system resources. Resource allocation methods +employ frequency scheduling [18], transmission power control +[19], and bandwidth allocation [20] to balance the cost of local +training. Recent device selection methods directly exclude +those weak devices with poor computation or communication +capabilities to accelerate the convergence time [21]–[23]. +Besides, topology-aware management is another very effective +method to mitigate the network throughput [18], [24], [25]. +However, these methods inherit the ideas of the efficient design +for traditional mobile systems and overlook the optimization +of neural networks. +Neuron-aware Techniques. Neuron-aware techniques focus +on revealing the black box of neural networks to improve the +training efficiency of the FL system. Early gradient compres- +sion utilizes sparsification [11], [26], and quantization [14], +[27], [28] to reduce the transmission cost of FL system. In +addition, feature maps fusion and knowledge distillation can be +carried out to improve the information aggregation [29], [30]. +Besides, FedMask proposes to train a personalized mask for +each device to improve the test accuracy on the local dataset +[31]. Recently, model structure pruning enables multiple de- +vices with different model architectures to train a shared global +model [32], [33]. Such methods can reduce the cost of local +training, but how to customize optimal training strategies (e.g., +gradient compression and model pruning policy) for different +learning scenarios is still unknown. +III. TRAINING WITH ANYCOSTFL +In this section, we first outline the overall design of Any- +costFL. Next, we detail the key techniques of our framework, +including elastic model shrinking (EMS), flexible gradient +compression (FGC), and all-in-one aggregation (AIO). +A. Outline of AnycostFL +We consider a generic application scenario of FL with a set +of I edge devices I = {1, 2, · · · , I}. We use Di to denote the +local training data of the device i, and D = ∪I +i=1Di indicates +the global data. Let Fi(w) = ℓ(w, Di) represent the local +training loss of device i with respect to model weight w, where +ℓ(·, ·) is the predetermined loss function. The objective of the +FL system is to minimize the following global loss function +F(w) +∆= +I +� +i=1 +|Di| +|D| Fi(w), +(1) +where |Di| is the size of Di. Given the specified learning task, +the original training workload of single sample W and the data +size of uncompressed gradient S can be empirically measured. +As shown in Fig. 2(a), to reduce the computational com- +plexity of the local model training and the communication cost +of gradient update transmission, we propose AnycostFL with +two device-side techniques, i.e., model shrinking and gradient +compression. At the t-th global iteration of AnycostFL, the +device i is enabled to adjust its training workload and gradient +size as Wt,i = αt,iW and St,i = βt,iS, respectively. Here, +αt,i ∈ (0, 1] and βt,i ∈ (0, 1] are defined as the model shrink- +ing factor and the gradient compression rate, respectively. The +training procedure of AnycostFL is summarized as follows. +1) Elastic local training: At the t-th global round, the +device i downloads the latest global model wt from the pa- +rameter server. With the pre-calculated model shrinking factor +αt,i, the specialized sub-model wα +t,i = shrink(wt, αt,i) can +be efficiently derived, where function shrink(·, ·) indicates +the operations for model shrinking. Then, the local training +is conducted with sub-model wα +t,i and local data Di, and the +updated local sub-model wα +t+1,i is obtained. Furthermore, the +local gradient update can be acquired as ut,i = wα +t,i −wα +t+1,i. +2) Flexible gradient upload: To further reduce the uplink +traffic, the local device i is motivated to compress the gradient +update ut,i before the parameter transmission. With the given +compression rate βt,i, the compressed gradient update ˜ut,i = +cmprs(ut,i, βt,i) is uploaded to the server, where cmprs(·, ·) +is the function for gradient compression. + +computation capacity +communication capacity +device C +device A +device B +local data +compressed update +comp. capacity +comm. capacity +the neural structure becomes larger +the data size of local update becomes larger +C +A +B +param. server +global model +aggregation +model distribution +param. upload +power +the size of each hidden layer is reduced by half, and +the training complexity is reduced by ¼ approximately. +. . . +1 0 0 0 0 0 1 0 +0 0 0 1 0 1 0 0 +0 0 1 1 0 0 0 0 +… … … … … … … … +. . . +sparsification +binary mask +quantization +entropy +encoding +Golomb +encoding +compressed +update +an example of gradient compression (single layer) +(b) optimization for the training strategy +(c) model shrinking & gradient compression +(a) outline of AnycostFL +… +… +global model +sub-model +16 +32 +64 +8 +16 +32 +size: 16x8x3x3 +encoding +an example of model shrinking +Fig. 2. left: AnycostFL over heterogeneous edge devices. middle: the neural structure and gradient compression strategies are customized for diverse devices +according to their locally available resources; the darker color indicates the higher computing complexity for training and the larger marker size denotes the +larger data size of the local update. right: illustrations of the model shrinking for the local model and the gradient compression for the local update. +3) Parameter aggregation: The server collects the com- +pressed local updates {˜ut,i}∀i with different shrinking factors +{αt,i}∀i and compression rates {βt,i}∀i. After that, the global +update is calculated by ˜ut += aioagg({˜ut,i}∀i), where +aioagg(·) is the server-side all-in-one aggregation. Then, the +updated global model is computed as wt+1 = wt − ˜ut. +After the T -round training of the above three-step iterations, +the final global model wT is obtained. Before introducing how +to customize the values of {αt,i}∀i and {βt,i}∀i in Section +IV, we illustrate the details of model shrinking, gradient +compression and update aggregation in the rest of this section. +B. Elastic Model Shrinking +We aim to derive the sub-model wα +t,i with training complex- +ity of αt,iW from global model wt by reducing the width of +the global model. The shrinking operations work as follows. +1) Server-side channel sorting: To avoid incurring extra +memory cost for the edge devices, the server first sorts +the channels of the latest global model before the model +distribution. Given one layer of the weight of the global +model, the server sorts the output channels in the current +layer in descending order according to their values of L2 +norm, and meanwhile, the input channels of the next layer +should be sorted accordingly in the same order to maintain +the permutation invariance of the whole model [34]. +2) Layer-wise uniform shrinking: Next, the server broad- +casts the weight of each layer of the global model in a channel- +by-channel manner. Instead of downloading the full global +model, each device only receives those important parameters +from the global model to assemble the local sub-model. Here, +we utilize the fixed shrinking ratio for each layer in the same +sub-model. Empirically, given model shrinking factor αt,i, we +can reduce the size of the hidden layer by √αt,i to acquire +the sub-model. For example, as shown in Fig. 2(c), when +shrinking a global model with hidden sizes of {16, 32, 64} +under αt,i = +1 +4, we approximately reduce the size of each +hidden layer by half as {8, 16, 32} to form the sub-model. +At the beginning of the t-th global round, all device ini- +tialize their local sub-models {wα +t,i}∀i by choosing the most +important channels from the global model wt. In this way, the +training complexity is significantly reduced while maintaining +the performance of local sub-models. After that, the local +training of device k is conducted with sub-model wα +t,i, which +produces the local gradient ut,i with data size of αt,iS. +C. Flexible Gradient Compression +Given the local update ut,i with the desired compression +rate βt,i, we aim to obtain the compressed update ˜ut,i with +data size of αt,iβt,iS. Let ρt,i and Lt,i denote the sparsity +rate and the number of quantization levels, respectively. The +gradient compression scheme works as follows. +1) Kernel-wise sparsification: Without loss of generality, +we take the convolution neural network (CNN) as an example +to illustrate the sparsification procedure. We aim to acquire the +sparse update ˆut,i from ut,i. Let ut,i[k] denote the k-th kernel +of ut,i, and ut,i = {ut,i[k]}∀k. We measure the importance +of each kernel and obtain N = {∥ut,i[k]∥2}∀k, where ∥ · ∥2 +denotes the L2 norm operation. Next, by selecting the ⌈ρt,iK⌉- +th largest value in N as the threshold Π, the kernel-wise +sparsification is expressed as +ˆut,i[k] = +� +0 +if ∥ut,i[k]∥2 < Π, +ut,i[k] +otherwise. +(2) +Meanwhile, the binary mask of ˆut,i is denoted as mt,i. +2) Probabilistic quantization: Motivated by the studies +in [35], [36], we aim to obtain the quantized update ˜ut,i +with the given sparse ˆut,i and the quantization level Lt,i. +Let u ∈ ˆut,i be a scalar value. To begin with, we first +calculate the magnitude range of the non-zero elements of +ˆut,i, denoted as [umin, umax], where umin = min{|u|}∀u̸=0, +and umax = max{|u|}∀u̸=0. Next, let Q = {Ql}Lt,i +l=1 denote +the set of quantization points, where Ql is computed by +Ql = l (umax − umin) +Lt,i ++ umin. +(3) + +1 +1 +1 +2 +2 +3 +2 1 +1 +1 +3 +1 +2 +3 +1 +1 +2 +1 +1 +1 +2 +2 +3 +1 +2 +1 +2 +model structure +update of each layer +2 2 +2 +2 +2 2 +2 +2 +2 +2 +2 2 2 +2 2 +3 3 +3 +3 3 +3 +3 3 +3 3 +1 1 1 +1 1 +1 1 +1 1 +1 +1 +1 +1 +1 +1 1 +1 1 +1 +1 +local compressed updates +global update +legend +normal update +zero update +1 +2 +3 +1 +2 +1 +3 +2 +3 +not existing +zero update +update by device 1 +update by all devices +update by devices 1&2 +update by device 3 +update by devices 1&3 +update by devices 2&3 +update by device 2 +�ut,1 +�ut,2 +�ut,3 +�ut +Fig. 3. An illustration of the all-in-one aggregation. +For any u ∈ ˆut,i and u ̸= 0, we can always find a quantization +interval [Ql, Ql+1] such that Ql ≤ |u| ≤ Ql+1, and its +corresponding quantized value ˜u is further computed by +˜u = +� +sgn(u) · Ql +with probability Ql+1−|u| +Ql+1−Ql , +sgn(u) · Ql+1 +otherwise, +(4) +where sgn(·) calculates the sign of the given scalar. Fur- +thermore, the set of the quantization indices of all ˜u ∈ ˜ut,i +is denoted as Lt,i = {l, Ql = ˜u}∀˜u̸=0. Now, ˜ut,i can be +represented by a tuple of {umin, umax, Lt,i, mt,i, Lt,i}. +3) Lossless encoding: Due to the distribution characteristics +of Lt,i that smaller indices may occur more frequently, we +apply entropy coding to reduce the data size [14], [37]. +Besides, the sparse binary matrix mt,i can be compressed by +Golomb encoding [11], [38]. +After determining the compression scheme, we can vary +the combinations of {ρt,i, Lt,i} and record the corresponding +compression rates. Based on the results, we can build a +piecewise linear function to predict the compression strategy +{ρt,i, Lt,i} with the given βt,i. Notably, this function can be +efficiently fitted by the server with a rather small amount of +public training data (e.g., 16 samples) in an offline manner. +D. All-in-One Aggregation +After all the devices upload their encoded updates, the +server receives, decodes and then reconstructs the compressed +local updates {˜ut,i}∀i. Our goal is to obtain the global +update ˜ut by aggregating {˜ut,i}∀i. However, the aggregation +of local updates in our framework cannot be supported by +conventional FedAvg [1], since the local updates are produced +by different model structures with different levels of precision +(i.e., different quantization levels and sparsity). +To tackle the above challenge, we propose an all-in-one +aggregation scheme that fuses the local updates in an element- +wise manner. Let the set {1, 2, · · · , J} index elements of the +global update ˜ut, and ˜u[j] +t +denote the j-th element of ˜ut. To +accomplish the aggregation for ˜u[j] +t , we first determine the +subset of devices Ij ⊆ I whose local model structure also +contains the j-th element. Then, we have +˜u[j] +t += + + + + + + + +0 +if � +i∈Ij +m[j] +t,i = 0, +1 +� +i∈Ij +pt,im[j] +t,i +� +i∈Ij +pt,im[j] +t,iu[j] +t,i +otherwise, +(5) +where pt,i is the aggregation coefficient for the j-th device at +the t-th global round. The optimal values of {pt,i}∀i will be +further analyzed in Section IV. Fig. 3 gives an example to illus- +trate the aggregation details. Specifically, different elements in +the global update are updated by different subsets of devices, +and more important elements will “absorb” knowledge from +more devices. When the j-th element is zeroed out by all the +devices in Ij, we have ˜u[j] +t += 0. +IV. THEORETICAL ANALYSIS AND OPTIMIZATION +In this section, we focus on the optimization of our frame- +work by customizing the training strategies for diverse devices. +We first formulate the on-demand training problem of Any- +costFL. Then, we derive the upper bound of the convergence +rate and reveal the key insights to improve the performance of +AnycostFL. Based on the analysis, the optimization problem is +transformed into a tractable form, and the closed-form solution +is derived. +A. AnycostFL over Wireless Networks +In this subsection, we formulate the computation and com- +munication models for our framework. After that, we build +up an on-demand learning problem that minimizes the global +training loss with given delay and energy constraints. +1) Computation model: For the device i at the t-th global +round, given the model shrinking factor αt,i and computing +frequency ft,i, the time consumption of local model training +can be measured by +T cmp +t,i += τ|Di|αt,iW +ft,i +, +(6) +where τ denotes the number of local epochs. Meanwhile, the +corresponding energy consumption can be given by +Ecmp +t,i = ǫif 2 +t,iτ|Di|αt,iW, +(7) +where ǫi is the hardware energy coefficient of the device i. +2) Communication model: We consider the frequency divi- +sion multiple access (FDMA) scheme for the transmission of +the local gradient update. For the device i at the t-th global +round, the achievable transmitting rate can be estimated by +rt,i = bilog2 +� +1 + |ht,i|P com +t,i +N0bi +� +, +(8) +where P com +t,i +is the transmitting power; bi is the achievable +bandwidth; |ht,i| denotes the path loss of wireless channel; N0 +is the power spectral density of the additive white Gaussian +noise. For the device i at t-th global round, given the update +˜ut,i generated by the local model with a shrinking factor +of αt,i and compression rate of βt,i, the required time T com +t,i +and energy consumption Ecom +t,i +of uplink transmission can be +respectively measured by +T com +t,i += αt,iβt,iS +rt,i +, and Ecom +t,i += T com +t,i P com +t,i . +(9) +With the above computation and communication models, +we next focus on the optimization problem of AnycostFL. + +3) Problem formulation: To optimize AnycostFL, we study +an on-demand training problem. Specifically, the shared max- +imal latency for each round T max is determined by the server. +The local energy consumption budget for each round Emax +t,i +is +customized by the device itself. Given multiple devices with +diverse local resources (e.g., computation, communication and +data), our goal is to customize the training strategy for each +device to minimize the global training loss with personalized +constraints (e.g., latency and energy). To sum up, at the t-th +global round, we aim to optimize the following problem. +(P1) +min F +� +wt; {αt,i}∀i, {βt,i}∀i +� +(10) +subject to: +T cmp +t,i + T com +t,i +≤ T max, ∀i, +(10a) +Ecmp +t,i + Ecom +t,i ≤ Emax +t,i , ∀i, +(10b) +αmin ≤ αt,i ≤ 1, ∀i, +(10c) +0 ≤ βt,i ≤ βmax, ∀i, +(10d) +f min +i +≤ ft,i ≤ f max +i +, ∀i, +(10e) +variables: +{αt,i, βt,i, ft,i}∀i, +where F +� +wt; {αt,i}∀i, {βt,i}∀i +� +denotes the global loss of the +t-th round with given the global model weight wt under the +training strategies of {αt,i}∀i and {βt,i}∀i. In the rest of this +section, we analyze the relationship between training loss and +training strategies. After that, Problem (P1) is further solved +based on the theoretical insights. +B. Assumptions and Key Lemmas +Being in line with the studies in [5], [39], we make the +following assumptions for the local loss function Fi, ∀i. +Assumption 1. Fi is λ-Lipschitz: ∥Fi(w) − Fi(w′)∥ +≤ +λ ∥w − w′∥, where λ > 0. +Assumption 2. Fi is ν-strongly convex: Fi(w) ≥ Fi(w′) + +(w − w′)⊤∇Fi(w′) + ν +2 ∥w − w′∥2. +Assumption 3. Fi is twice-continuously differentiable. Based +on Assumptions 1 and 2, we have νI ⪯ ∇2Fi(w) ⪯ λI. +Assumption 4. The ratios between the norms of ∇Fi(w) and +∇F(w) are bounded: ∥∇Fi(w)∥2 ≤ ε ∥∇F(w)∥2, where ε ≥ +0 is a positive constant. +Assumption 5. For the moderate shrinking factor α ≥ αmin, +the first-shrinking-then-training can be approximated as first- +training-then-shrinking: ∇Fi(wα) = [∇Fi(w)]α. Here, we +use [∇Fi(w)]α to denote the shrinking operation for ∇Fi(w). +Next, we give the following two definitions. +Definition 1 (Local gradient divergence). The local gradient +divergence δt,i is defined as the difference between ut,i and +˜ut,i, which is given by δt,i = ∥ut,i − ˜ut,i∥. +Definition 2 (Global gradient divergence). The global gra- +dient divergence ∆t is defined as the difference between +ut and ˜ut, which is measured by ∆t = ∥ut − ˜ut∥ = +��� +I� +i=1 +pt,iut,i − +I� +i=1 +pt,i˜ut,i +���. +Notably, in Definition 1, ut,i and ˜ut,i may have different +dimensions. We pad the missing elements in ˜ut,i with zeros +before the arithmetic operation. Next, we are interested in how +the training strategies {αt,i, βt,i}∀i affect {δt,i}∀i and ∆t. We +derive the following two lemmas. +Lemma 1. For the local training with the model shrinking +factor αt,i and compression rate βt,i. The square of the local +gradient divergence is bounded by +E∥δt,i∥2 ≤ +� +1 − αt,i(2 − αt,i) +� +βt,i +�2E∥ut,i∥2. +(11) +Proof. See Appendix A. +Lemma 2. For the local update {˜ut,i, ∀i} with the corre- +sponding training strategies {αt,i, βt,i}∀i and aggregation co- +efficients {pt,i}∀i, the square of the global gradient divergence +is bounded by +E∥∆t∥2 ≤ Iεη2 +I +� +i=1 +p2 +t,i +� +1 − αt,i(2 − αt,i) +� +βt,i +�2E∥∇F(wt)∥2. +(12) +Proof. See Appendix B. +C. Optimal Aggregation Scheme and Convergence Analysis +Intuitively, the local update ut,i generated with larger +{αt,i, βt,i} may carry more accurate information, and thus a +larger pt,i should be assigned during the aggregation. Based +on Lemma 2, we deduce the following theorem. +Theorem 1 (Optimal aggregation scheme). Given the lo- +cal updates {˜ut,i}∀i with corresponding training strategies +{αt,i, βt,i}∀i, the optimal aggregation coefficients are +p∗ +t,i = +1 +� +1−αt,i(2−αt,i)√ +βt,i +�2 +� +i +1 +� +1−αt,i(2−αt,i)√ +βt,i +�2 +, ∀i. +(13) +Proof. Based on Lemma 2, we study the following optimiza- +tion problem to minimize the global gradient divergence. +(P2) +min +{pt,i}∀i +I +� +i=1 +p2 +t,i +� +1 − αt,i(2 − αt,i) +� +βt,i +�2 +(14) +subject to: +pt,i ≥0, ∀i, +(14a) +I +� +i=1 +pt,i = 1. +(14b) +It can be verified that Problem (P2) is a convex op- +timization problem. We further solve the problem by the +Karush–Kuhn–Tucker (KKT) conditions. Let {̟}∀i and θ +be the Lagrange multipliers for Constraints (14a) and (14b), +respectively. Then, we obtain +̟i ≥ 0, ̟ipt,i = 0, pt,i ≥ 0, +I +� +i=1 +pt,i = 1, +2pt,i +� +1 − αt,i(2 − αt,i) +� +βt,i +�2 − ̟i + θ = 0, ∀i. +(15) +Being in line with the study in [40], we can obtain +pt,i = − +θ +2 +� +1 − αt,i(2 − αt,i) +� +βt,i +�2 . +(16) +By putting Eqn. (16) into Eqn. (14b), we obtain +θ = − +2 +� +k +1 +� +1−αt,i(2−αt,i)√ +βt,i +�2 +. +(17) +Putting Eqn. (17) into Eqn. (16) completes the proof. + +With the optimal aggregation scheme, we investigate the +upper bound of the convergence rate of AnycostFL. +Definition 3 (Local and global learning gains). The local +and global learning gains are defined as gt,i = α4 +t,iβt,i and +gt = � +i gt,i/I, respectively. Specifically, the local and global +learning gains (i.e., gt,i ∈ [0, 1] and gt ∈ [0, 1]) measure the +amount of effective information carried in the local and global +updates, respectively. +Theorem 2 (Convergence rate of AnycostFL). Let gmin = +min{gt}∀t be the minimal global learning gain over the T - +round training. The upper bound of the convergence rate of +AnycostFL satisfies +E +� +F(wT ) − F(w∗) +� +≤ ZT −1E +� +F(w0) − F(w∗) +� +, +(18) +where Z = 1 − ν +λ +� +1 − ε(1 − gmin) +� +. Recall that parameters +ν, λ and ǫ are defined in Assumptions 1 to 4 before. +Proof. See Appendix C. +Based on Definition 3 and Theorem 2, we derive the +following proposition. +Proposition 1. The key to minimizing the training loss of +AnycostFL is to maximize the learning gain gt for each global +round. If gt = 1 ∀t, AnycostFL degrades to conventional FL +without model shrinking and gradient compression. +D. Solution for Problem (P1) +Based on Theorem 2 and Proposition 1, Problem (P1) can +be transformed into the following problem. +(P3) +max 1 +I +I +� +t=1 +α4 +t,iβt,i +(19) +subject to: +Constrains (10a) to (10e), +variables: +{αt,i, βt,i, ft,i}∀i. +Based on Constraints (10a) and (10b) for the training latency +and energy, we obtain the following lemma. +Lemma 3. The equality will always hold for Constraints +(10a) and (10b) when confirming the optimal training strategy +{α∗ +t,i, β∗ +t,i, f ∗ +t,i}∀i, and thus T ∗ +t,i = T max and Et,i = E∗ +t,i ∀i. +Proof. The lemma can be proved by showing the contradic- +tion. Suppose that there exists i0 such that T ∗ +t,i0 < T max. +We can find a new solution {α′ +t,i0, β∗ +t,i0, f ′ +t,i0} for device i0 +and α′ +t,i0 > α∗ +t,i0, f ′ +t,i0 < f ∗ +t,i0, such that T ′ +t,i0 = T max +and E′ +t,t0 = Emax +t,i . Since the global learning gain increases +with the increase of αt,i0, we have g′ +t > g∗ +t . Likewise, the +contradiction also appears when E∗ +t,i0 < Emax +t,i0 , and thus we +complete the proof. +Based on Lemma 3, we employ two intermediate variables +(i.e., φt,i and ϕt,i) for each device to reparameterize Problem +(P3). Specifically, φt,i ∈ [0, 1] and ϕt,i ∈ [0, 1] are the splitting +factors for latency and energy, respectively, such that +T cmp +t,i = φt,iT max, T com +t,i = (1 − φt,i)T max, +Ecmp +t,i = ϕt,iEmax +t,i , Ecom +t,i = (1 − ϕt,i)Emax +t,i , ∀i. +(20) +By combining Eqns (6) and (20), the local learning gain of +the device i at the t-th round can be rewritten as +gt,i(φt,i) = κt,i +� +Emax +t,i +− (1 − φt,i)T maxP com +t,i +� +(φ2 +t,i − φ3 +t,i), (21) +where κt,i = rt,i +Sǫi +� T max +τ|Di|W +�3. +Note that Problem (P3) can be transformed into I sub- +problems because the decision-making procedure of each +device is independent. Based on Eqn. (21), the i-th sub- +problem can be expressed as a single-variable optimization +problem with respect to φt,i as follows. +(P4) +max +φt,i +gt,i +� +φt,i +� +(22) +subject to: +φmin +t,i +≤ φt,i ≤φmax +t,i , +where the lower and upper limits of φt,i can be acquired by +φmin +t,i += max +�αminτ |Di| W +f max +i +T max +, 1 − βmaxS +rt,iT max +� +, +φmax +t,i += min +� τ |Di| W +f min +i +T max , 1 − αminβminS +rt,iT max +� +. +(23) +Based on the first-order optimality condition ∂gt,i/φt,i = 0, +we obtain the stationary points as +φs1 +t,i = +� +ψt,i − 3Emax +t,i +8P com +t,i T max ++ 3 +4, φs2 +t,i = − +� +ψt,i + 3Emax +t,i +8P com +t,i T max +− 3 +4, (24) +where ψt,i = 4(P com +t,i T max)2 − 4Emax +t,i P com +t,i T max + 9(Emax +t,i )2. +Let St,i = {φmin +t,i , φmax +t,i , φs1 +t,i, φs2 +t,i} denote the union of the +stationary points and the boundary points for Problem (P4). +Then, S′ +t,i = {φt,i|φt,i ∈ [φmin +t,i , φmax +t,i ], φt,i ∈ St,i} is the set +of the feasible solutions of St,i. The optimal solution for +Problem (P4) can be acquired by +φ∗ +t,i = arg max +φt,i∈S′ +t,i +gt,i(φt,i). +(25) +Furthermore, we obtain the optimal solution for device i at the +t-th global round by putting φ∗ +t,i into the following equations. +ϕ∗ +t,i = 1 − (1 − φ∗ +t,i)T maxP com +t,i +Emax +t,i +, α∗ +t,i = +3 +� +(φ∗ +t,iT max)2ϕ∗ +t,iEmax +t,i +ǫi(τ |Di| W )3 +, +β∗ +t,i = rt,i(1 − φ∗ +t,i)T max +α∗ +t,iS +, f ∗ +t,i = α∗ +t,iτ |Di| W +φ∗ +t,iT max +. +(26) +Notably, the decision-making process of each device does +not involve the auxiliary information of the resource status +from other devices. At the beginning of each global round, +each device can determine its training strategy locally. +V. EXPERIMENT EVALUATIONS +A. Experiment Settings +1) Setup for FL training: We consider the FL application +with image classification on Fashion-MNIST and CIFAR- +10 datasets [41], [42]. For Fashion-MNIST, we use a small +convolutional neural network (CNN) with data size of model +update as 53.22Mb [1]. For the CIFAR-10 dataset, we employ +VGG-9 with data size of model update as 111.7Mb [43]. +For IID and non-IID data settings, we follow the dataset +partition strategy in [34]. For the learning hyper-parameters, +the learning rate, batch size and local epoch are set as {0.01, +32, 1} for Fashion-MNIST and {0.08, 64, 1} for CIFAR-10 +dataset. The maximal latency is set as T max = 10 seconds + +0 +5 +10 +15 +20 +25 +30 +80 +82 +84 +86 +88 +90 +92 +0 +5 +10 +15 +20 +25 +30 +80 +82 +84 +86 +88 +90 +0 +10 +20 +30 +40 +50 +60 +30 +40 +50 +60 +70 +80 +90 +0 +10 +20 +30 +40 +50 +60 +30 +40 +50 +60 +70 +80 +14 +16 +89 +90 +16 +18 +20 +88 +89 +Test accuracy (%) +Time consumption (min) +Time consumption (min) +STC +QSGD +UVeQFed +HeteroFL +FedHQ +AnycostFL +(a) FMNIST IID +(b) FMNIST non-IID +(c) CIFAR-10 IID +(d) CIFAR-10 non-IID +Energy consumption (KJ) +Energy consumption (KJ) +Fig. 4. Performance on various network architectures and datasets. ((a-b): global accuracy vs. time consumption with Fashion MNIST on 2-layer CNN; (c-d): +global accuracy vs. energy consumption with CIFAR-10 on VGG-9.) +TABLE I +PERFORMANCE COMPARISON BETWEEN ANYCOSTFL AND OTHER METHODS ON FASHION-MNIST AND CIFAR-10 DATASETS. +IID +non-IID +Dataset +Method +#Round +Energy +(KJ) +Latency +(min) +Comp. +(TFLOPs) +Comm. +(GB) +Best Acc. +(%) +#Round +Energy +(KJ) +Latency +(min) +Comp. +(TFLOPs) +Comm. +(GB) +Best Acc. +(%) +FMNIST +{90%, 89%}∗ +STC +305 (1.7×) 10.94 (1.4×) 25.42 (1.7×) +152.71 +0.71 +90.28±0.18 283 (1.3×) 10.17 (1.1×) 23.56 (1.3×) +141.53 +0.66 +89.47±0.16 +QSGD +283 (1.6×) 11.40 (1.4×) 23.56 (1.6×) +141.53 +0.80 +90.39±0.04 279 (1.3×) 11.27 (1.2×) 23.28 (1.3×) +139.86 +0.79 +89.49±0.07 +UVeQFed +247 (1.4×) 11.36 (1.4×) 20.58 (1.4×) +123.67 +0.72 +90.44±0.10 266 (1.2×) 12.21 (1.3×) 22.14 (1.2×) +133.01 +0.77 +89.64±0.16 +HeteroFL +233 (1.3×) 12.03 (1.5×) 21.78 (1.5×) +92.21 +0.57 +90.43±0.13 242 (1.1×) 12.51 (1.3×) 22.62 (1.3×) +95.77 +0.59 +89.42±0.10 +FedHQ +288 (1.6×) 13.89 (1.7×) 24.03 (1.6×) +144.36 +0.86 +90.21±0.07 313 (1.5×) 14.96 (1.6×) 26.06 (1.5×) +156.55 +0.93 +89.27±0.19 +AnycostFL 179 (1.0×) 8.07 (1.0×) 14.94 (1.0×) +67.49 +0.35 +91.20±0.09 214 (1.0×) 9.63 (1.0×) 17.83 (1.0×) +80.51 +0.42 +90.32±0.14 +CIFAR-10 +{82%, 80%}∗ +STC +341 (1.2×) 35.39 (1.3×) 56.83 (1.2×) +4160.56 +1.78 +85.38±0.29 412 (1.1×) 42.39 (1.3×) 68.67 (1.1×) +5026.84 +2.15 +83.09±0.53 +QSGD +337 (1.2×) 39.82 (1.5×) 56.17 (1.1×) +4111.76 +2.14 +84.83±0.54 430 (1.2×) 50.29 (1.5×) 71.61 (1.2×) +5242.39 +2.73 +81.94±0.13 +UVeQFed +296 (1.0×) 40.77 (1.5×) 49.28 (1.0×) +3607.45 +2.12 +85.09±0.16 377 (1.0×) 51.59 (1.5×) 62.89 (1.0×) +4603.87 +2.71 +82.30±0.28 +HeteroFL +332 (1.1×) 50.07 (1.9×) 69.14 (1.4×) +3222.26 +1.65 +83.75±0.55 413 (1.1×) 62.88 (1.9×) 85.78 (1.4×) +3990.49 +2.05 +80.68±0.45 +FedHQ +340 (1.2×) 48.95 (1.9×) 56.67 (1.2×) +4148.36 +2.32 +84.02±0.22 435 (1.2×) 61.99 (1.9×) 72.44 (1.2×) +5303.40 +2.96 +81.00±0.41 +AnycostFL 294 (1.0×) 26.43 (1.0×) 48.94 (1.0×) +2459.92 +1.56 +87.72±0.23 372 (1.0×) 33.51 (1.0×) 62.06 (1.0×) +3118.60 +1.98 +84.91±0.51 +*{x, y}: x and y denote the target global model accuracy under IID and non-IID data settings, respectively. +and the energy budget is set as Emax +t,i +∼ U[3, 9] joules for the +CIFAR-10 dataset, and the corresponding hyper-parameters for +the FMNIST dataset are halved by default. Additionally, we +set αmin = 1/4 and βmax = 1/15. +2) Setup for mobile system: We investigate a mobile system +with I = 60 devices located within a circle cell with a +radius of 550 meters, and a base station is situated at the +center. To simulate the mobility, the position of each device +is refreshed randomly at the beginning of each round [44]. +For the computation, the energy coefficient is set as ǫi ∼ +U[5 × 10−27, 1 × 1−26]. For communication, the bandwidth +is set as 1MHz equally for each device, and the path loss +exponent is 3.76. The transmission power is set as 0.1W, and +N0 is set as −114dBm/MHz. +B. Performance Comparisons +We compare the proposed AnycostFL with the following +efficient FL algorithms with three different random seeds. +• STC. The sparse ternary compression (STC) is adapted +to reduce the cost of uplink parameter transmission [11]. +• QSGD. The TopK sparsification and probabilistic quanti- +zation are combined to compress the local gradient [36]. +• UVeQFed. The TopK sparsification and universal vector +quantization are used to compress the local gradient [14]. +• HeteroFL. Each device trains the local sub-model in +different widths to match its computation capacity [32]. +• FedHQ. Each device uses different quantization levels to +compress the gradient according to its channel state [40]. +Fig. 4 shows the performance of the global model over +time consumption and energy consumption under the IID and +the non-IID data setting. With the same training efficiency +(i.e., time and energy consumption), the proposed AnycostFL +consistently outperforms the baseline schemes to improve the +test accuracy of the global model. Meanwhile, Table I provides +the best accuracy and required system cost for achieving +the specified test accuracy. Particularly, when compared with +HeterFL and FedHQ, AnycostFL can reduce up to 1.9 times +the energy consumption to reach the test accuracy of 82% +on CIFAR-10 dataset under the IID setting. When compared +with STC, AnycostFL can reduce up to 1.7 times the time +consumption to reach the test accuracy of 90% on FMNIST +dataset under the IID setting. Moreover, our framework can +significantly improve the best accuracy of the global model +by 2.33% and 1.82% on CIFAR-10 dataset under the IID and +the non-IID settings, respectively. +C. Impact of Key Mechanisms and Hyper-parameters +Fig. 5(a) verifies the advantages of the main techniques of +AnycostFL. We gradually remove the elastic model shrinking +(w/o EMS), the flexible gradient compression (w/o FGC) and +the all-in-one aggregation (w/o AIO), and record the required +system cost to achieve 80% test accuracy with CIFAR-10 +dataset under the IID setting. We observe that the proposed +EMS and FGC can significantly save the energy consumption +and training time, respectively. Besides, AIO contributes to +saving both energy and time. + +AnycostFL w/o EMS w/o FGC +w/o AIO +20 +30 +40 +50 +60 +Energy consumption +Required time +Energy consumption (KJ) +2.4x +1.3x +1.5x +1.3x +35 +40 +45 +50 +55 +60 +65 +70 +75 +Required time (min) +0 +2 +4 +6 +8 +10 +45 +60 +75 +90 +105 +120 +6 +7 +8 +9 +70 +80 +Average time consumption (min) +Level of communication heterogeneity +STC +QSGD +UVeQFed +HeteroFL +FedHQ +AnycostFL +0 +2 +4 +6 +8 +10 +12 +24 +32 +40 +48 +56 +Average energy consumption (KJ) +Level of computation heterogeneity +STC +QSGD +UVeQFed +HeteroFL +FedHQ +AnycostFL +0.06 +0.09 +0.12 +78 +81 +84 +87 +90 +STC +HeteroFL +AnycostFL +Global test accuracy (%) +Computational complexity (GFLOPs) +(a) Impact of key mechanisms +(b) Impact of comm. heterogeneity +(c) Impact of comp. heterogeneity +(d) Performance of sub-models +Fig. 5. The main advantages of AnycostFL. ((a): the impact of key mechanisms; (b-c): the impact of system heterogeneity; (d): the performance of sub-models.) +We next evaluate the impact of resource heterogeneity on +the training efficiency in Fig. 5(b-c). We set the average energy +coefficient ǫi as 7.5×10−27 and the average distance between +the base station and edge devices as 400 meters, and then +change their variances to simulate the computation and com- +munication heterogeneity, respectively. The larger variance +indicates a higher level of system heterogeneity. As we expect, +the proposed AnycostFL shows more resilience than other +baselines to tackle the high level of system heterogeneity. +We also evaluate the performance of sub-models in different +widths in Fig. 5(d). Specifically, We compare AnycostFL with +HeteroFL (i.e., local training with different widths) and STC +(i.e., the best-performing compression-only method). The sub- +models are derived from the well-trained global model without +further re-training. Surprisingly, the sub-models of the global +model trained by AnycostFL can still maintain satisfactory test +accuracy, which provides dynamic inference for diverse edge +devices after the training time. +VI. CONCLUSION +In this paper, we proposed AnycostFL, a joint computation +and communication efficient framework for FL, that enables +edge devices with diverse resources to train a shared global +model. We aimed to minimize the global training loss under +given personalized latency and energy constraints. By leverag- +ing the theoretical insight of AnycostFL, we decomposed the +optimization problem into multiple sub-problems. Following +that, the optimal training strategy is derived for each de- +vice according to its locally available resource. Experiments +demonstrate the advantage of our framework in improving the +system efficiency and model performance compared to the +state-of-the-art methods. +ACKNOWLEDGMENT +Rong Yu and Yuan Wu are the corresponding authors. This +work was supported in part by National Key R&D Program +of China under Grant 2020YFB1807802, in part by National +Natural Science Foundation of China under Grants 61971148, +62102099, U22A2054 and 62001125, in part by Science and +Technology Development Fund of Macau SAR under Grant +0162/2019/A3, in part by FDCT-MOST Joint Project under +Grant 0066/2019/AMJ, in part by the Guangdong Basic and +Applied Basic Research Foundation (2022A1515011287), and +in part by US National Science Foundation under grant CNS- +2107057. +APPENDIX A +PROOF OF LEMMA 1 +Proof. For the given local gradient ˜ut,i with shrinking factor +αt,i and gradient compression rate βt,i, we aim to capture the +divergence between ˜ut,i and ut,i. Suppose that the absolute +value of the element in ut,i follows uniform distribution |u| ∼ +U(0, umax), and umax = max{|u|}∀u∈ut,i. +For clear notation, we sort the element-wise absolute +value of ut,i in ascending order. Then, we obtain ut,i = +[u[1] +t,i, . . . , u[j] +t,i, . . . , u[J] +t,i ]⊤ and |u[j] +t,i| ≤ |u[j+1] +t,i +|. Thus, we have +E∥ut,i∥2 = E +J +� +j=1 +|u[j] +t,i|2 = JE|u[j] +t,i|2 = Ju2 +max +3 +. +(27) +Based on Assumption 5, the update generated from lo- +cal training with wα +t,i is equal to shrink(ut,i, αt,i). The +operation of model shrinking on ut,i with αt,i can be +viewed as removing (1 − αt,i)J elements with the least +value from ut,i. Then, we obtain shrink(ut,i, αt,i) += +[0, . . . , 0, u[(1−αt,i)J+1] +t,i +, . . . , u[J] +t,i ]⊤. Thus, we have +E∥ut,i − shrink(ut,i, αt,i)∥2 = E +(1−αt,i)J +� +j=1 +|u[j] +t,i|2 += J(1 − αt,i)3u2 +max/3 = (1 − αt,i)3E∥ut,i∥2. +(28) +We next focus on the gradient compression. The operation +of gradient sparsification on ut,i with sparsity of ρt,i can +be viewed as removing ρt,iJ elements with the least value +from ut,i. Then, the quantization is conducted on the non- +zero elements of ˆut,i, and we obtain cmprs(ut,i, βt,i) = +[0, . . . , 0, ˜u[ρt,iJ+1] +t,i +, . . . , ˜u[J] +t,i ]⊤. Furthermore, we have +E∥ut,i − cmprs(ut,i, βt,i)∥2 += E +ρt,iJ +� +j=1 +|u[j] +t,i|2 +� +�� +� +(A) ++ E +J +� +j=ρt,iJ+1 +|u[j] +t,i − ˜u[j] +t,i|2 +� +�� +� +(B) +. +(29) +Likewise to Eqn. (28), we have (A) = ρ3 +t,iE∥ut,i∥2. Based on +Eqn. (4) and the statistical feature of ut,i, we obtain (B) = +(1 − ρt,i)3E∥ut,i∥2/(2L2 +t,i). +Given plain update ut,i in 32-bit floating point and the +desired compression rate βt,i, we can set ρt,i = 1− +� +βt,i and +Lt,i = 232√ +βt,i for the analysis. In this way, the operations + +of sparsification and quantization contribute equally to the +gradient compression. Furthermore, we have +E∥ut,i − cmprs(ut,i, βt,i)∥2 ≤ (1 − βt,i)2E∥ut,i∥2. +(30) +Next, we focus on the local divergence δt,i with respect to +αt,i and βt,i. According to the Definition 1, we have +E∥δt,i∥2 = E∥ut,i − cmprs([ut,i]α, βt,i)∥2 += E∥ut,i − [ut,i]α∥2 + E∥[ut,i]α − cmprs([ut,i]α, βt,i)∥2 ++ 2 < ut,i − [ut,i]α, [ut,i]α − cmprs([ut,i]α, βt,i) > +� +�� +� +(C) +. (31) +It can be verified that the two vectors in term (C) are +orthogonal, and we obtain (C) = 0. According to Eqns (28) +and (30), we further obtain +E∥δt,i∥2 ≤ (1 − αt,i)3E∥ut,i∥2 + (1 − +� +βt,i)2E∥[ut,i]α∥2 +(a) +≤ (1 − αt,i)3E∥ut,i∥2 ++ (1 − +� +βt,i)2αt,i(α2 +t,i − 3αt,i + 3)E∥ut,i∥2 +(b) +≤ +� +1 − αt,i(2 − αt,i) +� +βt,i +�2E∥ut,i∥2. +(32) +Likewise to Eqn. (28), inequality (a) stems from the fact that +E∥[ut,i]α∥2 = αt,i(α2 +t,i−3αt,i+3)E∥ut,i∥2. Besides, inequal- +ity (b) holds for all αt,i ∈ [αmin, 1] and βt,i ∈ [0, βmax]. Thus, +we complete the proof. +APPENDIX B +PROOF OF LEMMA 2 +Proof. Based on Definition 2 and Lemma 1, we have +E∥∆t∥2 = E +��� +I +� +i=1 +pt,iut,i − +I +� +i=1 +pt,i˜ut,i +��� +2 +≤ E +� I +� +i=1 +pt,i +� +1 − αt,i(2 − αt,i) +� +βt,i +� +∥ut,i∥ +�2 +. +(33) +We use η to denote the learning rate, and ut,i = η∇Fi(wt). +Based on Assumption 4, we obtain +E∥∆t∥2 ≤ εη2� +I +� +i=1 +pt,i +� +1 − αt,i(2 − αt,i) +� +βt,i +��2 +E∥∇F(wt)∥2. +(34) +According to Cauchy–Schwarz inequality, we obtain +E∥∆t∥2 ≤ Iεη2 +I +� +i=1 +p2 +t,i +� +1 − αt,i(2 − αt,i) +� +βt,i +�2E∥∇F(wt)∥2. +(35) +Thus, we complete the proof. +APPENDIX C +ON THE CONVERGENCE OF ANYCOSTFL +Proof. Inspired by the studies in [5], [39], we deduce the +convergence analysis of AnycostFL. According to Taylor +expansion and Assumption 3, we have +F(wt+1) ≤ F(wt) + (wt+1 − wt)⊤∇F(wt) + λ +2 ∥wt+1 − wt∥2 += F(wt) − ˜u⊤ +t ∇F(wt) + λ +2 +��˜ut +��2. +(36) +By using learning rate η = 1 +λ, we obtain +E +� +F(wt+1) +� +≤ E +� +F(wt) − λ (ut − ∆t)⊤ut + λ +2 ∥ut − ∆t∥2� += E +� +F(wt) − 1 +2λ∥∇F(wt)∥2 + λ +2 ∥∆t∥2� +. +(37) +We now pay attention to the upper bound of ∥∆t∥2. Based on +Jensen’s inequality and Eqn. (34), we obtain +E∥∆t∥2 ≤ εη2 +I +� +i=1 +pt,i +� +1 − αt,i(2 − αt,i) +� +βt,i +�2 +� +�� +� +(D) +E∥∇F(wt)∥2. +(38) +By putting Eqn. (13) into (A), we have +E∥D∥ ≤ E +�������� +I +I� +i=1 +1 +(1−αt,i(2−αt,i)√ +βt,i) +2 +�������� +(c) +≤E +�������� +I +I� +i=1 +1 +1−α4 +t,iβt,i +�������� +, +(39) +where (c) always holds for αt,i ∈ [0, 1] and βt,i ∈ [0, 1]. +According to Definition 3, we have gt,i = α4 +t,iβt,i and gt = +� +i gt,i/I. Since 1/ +�� +i +1 +1−gt,i +� +is a concave function with +respect to gt,i, based on Jensen’s inequality, we obtain +E∥A∥ ≤ +I +� +i +1 +1−E(α4 +t,iβt,i) += 1 − gt. +(40) +Since the training strategies of each device and the norm of +the gradient of global data ∥∇F(wt)∥ are independent, by +putting Eqn. (40) back to Eqn. (38), we obtain +E∥∆t∥2 ≤ E +� +εη2� +1 − gt +� +∥∇F(wt)∥2� +. +(41) +Next, by putting Eqn. (41) back to Eqn. (37), we have +E +� +F(wt+1) +� +≤ E +� +F(wt) − 1 + ε +� +gt − 1 +� +2λ +∥∇F(wt)∥2� +. (42) +Subtracting F(w∗) in both sides of Eqn. (42) yields +E +� +F(wt+1 − F(w∗) +� +≤ E +� +F(wt) − 1 + ε(gt − 1) +2λ +∥∇F(wt)∥2 − F(w∗) +� +. +(43) +Based on Assumptions 2 and 3, we have [5], [45] +∥∇F(wt)∥2 ≥ 2ν +� +F(wt) − F(w∗) +� +. +(44) +Plugging Eqn. (44) into Eqn. (43), we have +E +� +F(wt+1) − F(w∗) +� +≤ ZtE +� +F(wt) − F(w∗) +� +, +(45) +where Zt = 1 − ν +λ (1 − ε(1 − gt)). +Let gmin = min{gt}∀t be the minimal global learning +gain over T global rounds. 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Krizhevsky, G. Hinton et al., “Learning multiple layers of features +from tiny images,” 2009. +[43] K. Simonyan and A. Zisserman, “Very deep convolutional networks for +large-scale image recognition,” in ICLR, 2015. +[44] W. Shi, S. Zhou, Z. Niu, M. Jiang, and L. Geng, “Joint device scheduling +and resource allocation for latency constrained wireless federated learn- +ing,” IEEE Transactions on Wireless Communications, vol. 20, no. 1, +pp. 453–467, 2021. +[45] S. Boyd, S. P. Boyd, and L. Vandenberghe, Convex optimization. +Cambridge university press, 2004. + diff --git a/29E1T4oBgHgl3EQfSAP8/content/tmp_files/load_file.txt b/29E1T4oBgHgl3EQfSAP8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..74c002f5a5fd3532154fba0380bcd1428d104860 --- /dev/null +++ b/29E1T4oBgHgl3EQfSAP8/content/tmp_files/load_file.txt @@ -0,0 +1,1250 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf,len=1249 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='03062v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='LG] 8 Jan 2023 AnycostFL: Efficient On-Demand Federated Learning over Heterogeneous Edge Devices Peichun Li∗,†, Guoliang Cheng∗, Xumin Huang∗,†, Jiawen Kang∗, Rong Yu∗, Yuan Wu†, and Miao Pan‡ ∗School of Automation, Guangdong University of Technology, Guangzhou, China †State Key Laboratory of Internet of Things for Smart City, University of Macau, Macau, China ‡Department of Electrical and Computer Engineering, University of Houston, Houston, USA Email: peichun@mail2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='gdut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='cn, guoliang cheng@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='com, huangxu min@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='com, {kavinkang, yurong}@gdut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='cn, yuanwu@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='mo, mpan2@uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='edu Abstract—In this work, we investigate the challenging prob- lem of on-demand federated learning (FL) over heterogeneous edge devices with diverse resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We propose a cost-adjustable FL framework, named AnycostFL, that enables diverse edge devices to efficiently perform local updates under a wide range of efficiency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To this end, we design the model shrinking to support local model training with elastic computation cost, and the gradient compression to allow param- eter transmission with dynamic communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' An enhanced parameter aggregation is conducted in an element-wise manner to improve the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Focusing on Any- costFL, we further propose an optimization design to minimize the global training loss with personalized latency and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' By revealing the theoretical insights of the conver- gence analysis, personalized training strategies are deduced for different devices to match their locally available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Ex- periment results indicate that, when compared to the state-of-the- art efficient FL algorithms, our learning framework can reduce up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='9 times of the training latency and energy consumption for realizing a reasonable global testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Moreover, the results also demonstrate that, our approach significantly improves the converged global accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Index Terms—Federated learning, edge intelligence, mobile computing, resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' INTRODUCTION Federated learning (FL) is an emerging distributed learning paradigm that enables multiple edge devices to train a common global model without sharing individual data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' This privacy- friendly data analytics technique over massive devices is envi- sioned as a promising solution to realize pervasive intelligence [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' However, in many real-world application areas, mobile de- vices are often equipped with different local resources, which raises the emerging challenges for locally on-demand training [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given different local resources status (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', computing capability and communication channel state) and personalized efficiency constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', latency and energy), it is crucial to customize training strategies for heterogeneous edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We perform an in-depth analysis on the time delay and the energy consumption for performing the local model updates at edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, we evaluate and record the cost of local training on three different NVIDIA Jetson family plat- forms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', Nano, NX AGX, and Xavier AGX) under different channel states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', good, medium, and poor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' On the one hand, we observe that the learning efficiency differs significantly Xavier-Good NX-Medium Nano-Poor 0 1 2 3 4 5 6 7 8 time delay (in second) of single-round local update local model training parameter transmission 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0 times 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7 times Xavier-Good NX-Medium Nano-Poor 0 2 4 6 8 10 energy consumption (in joule) of single-round local update local model training parameter transmission Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The time delay (top) and energy consumption (bottom) of single- round local update on different hardware platforms with varying communica- tion conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' with diverse learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1, the single- epoch training on Nano with poor communication condition consumes about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0 times training latency than that of Xavier AGX with good communication condition, while its energy consumption is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7 times less than the latter one’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' On the other hand, we observe that the bottlenecks of latency and energy are induced by parameter transmission and local model training, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The above observations provide insights for a proper design of the on-demand FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To handle the resource hetero- geneity, it is suggested to alleviate the energy and the latency cost of the local device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' More importantly, the computation and communication costs should be jointly reduced to achieve efficient local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In the literature, most existing studies either employ resource allocation and device scheduling to mitigate the system cost [4]–[10], or design gradient com- pression to accelerate the parameter transmission procedure [11]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The former method inherits the ideas of traditional design for mobile edge systems and takes no account of the optimization for neural networks, while the latter overlooks the computation cost of local model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In this paper, we propose “anycost” FL, named AnycostFL, to break the latency and energy bottlenecks for on-demand distributed training over heterogeneous edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Our goal is to develop a cost-adjustable FL framework that enables edge devices to perform local updates under diverse learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To this end, we first design the model shrinking and gradient compression to enable adaptive local updates with different computation and communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Meanwhile, an enhanced parameter aggregation scheme is proposed to fuse the knowledge of the local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Following that, we investigate the on-demand learning of AnycostFL by regulating the local model structure, gradient compression policy and computing frequency under personalized latency and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' However, customizing training strategy for different learning scenarios is a non-trivial task, since how the global accuracy is affected by the local model structure and compression rate is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To address this issue, we theoretically reveal the convergence insights of our framework, which are further leveraged to guide optimization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Finally, the optimal training strategy is derived for each device according to its locally available resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Our main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We propose a novel FL framework, named AnycostFL, that enables the local updates with elastic computation cost and communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We theoretically present the optimal aggregation scheme and convergence analysis for AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We investigate the on-demand training problem of Any- costFL, and the optimal training strategy is devised to adapt the locally available resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Extensive experiments indicate that the proposed Any- costFL outperforms the state-of-the-art efficient FL meth- ods in terms of resource utilization and learning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Section II describes related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In Section III, we detail the main operations of AnycostFL to fulfill the single-round training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The problem formulation, theoretical analysis and the corre- sponding solution are provided in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The experiment evaluations are presented in Section V, and we finally conclude the paper in Section VI and discuss the future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' RELATED WORK Resource Management Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Resource management methods aim to reduce the FL system cost by arranging the local and system resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Resource allocation methods employ frequency scheduling [18], transmission power control [19], and bandwidth allocation [20] to balance the cost of local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Recent device selection methods directly exclude those weak devices with poor computation or communication capabilities to accelerate the convergence time [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Besides, topology-aware management is another very effective method to mitigate the network throughput [18], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' However, these methods inherit the ideas of the efficient design for traditional mobile systems and overlook the optimization of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Neuron-aware Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Neuron-aware techniques focus on revealing the black box of neural networks to improve the training efficiency of the FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Early gradient compres- sion utilizes sparsification [11], [26], and quantization [14], [27], [28] to reduce the transmission cost of FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In addition, feature maps fusion and knowledge distillation can be carried out to improve the information aggregation [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Besides, FedMask proposes to train a personalized mask for each device to improve the test accuracy on the local dataset [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Recently, model structure pruning enables multiple de- vices with different model architectures to train a shared global model [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Such methods can reduce the cost of local training, but how to customize optimal training strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', gradient compression and model pruning policy) for different learning scenarios is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' TRAINING WITH ANYCOSTFL In this section, we first outline the overall design of Any- costFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Next, we detail the key techniques of our framework, including elastic model shrinking (EMS), flexible gradient compression (FGC), and all-in-one aggregation (AIO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Outline of AnycostFL We consider a generic application scenario of FL with a set of I edge devices I = {1, 2, · · · , I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We use Di to denote the local training data of the device i, and D = ∪I i=1Di indicates the global data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let Fi(w) = ℓ(w, Di) represent the local training loss of device i with respect to model weight w, where ℓ(·, ·) is the predetermined loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The objective of the FL system is to minimize the following global loss function F(w) ∆= I � i=1 |Di| |D| Fi(w), (1) where |Di| is the size of Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given the specified learning task, the original training workload of single sample W and the data size of uncompressed gradient S can be empirically measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2(a), to reduce the computational com- plexity of the local model training and the communication cost of gradient update transmission, we propose AnycostFL with two device-side techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', model shrinking and gradient compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' At the t-th global iteration of AnycostFL, the device i is enabled to adjust its training workload and gradient size as Wt,i = αt,iW and St,i = βt,iS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Here, αt,i ∈ (0, 1] and βt,i ∈ (0, 1] are defined as the model shrink- ing factor and the gradient compression rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The training procedure of AnycostFL is summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1) Elastic local training: At the t-th global round, the device i downloads the latest global model wt from the pa- rameter server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' With the pre-calculated model shrinking factor αt,i, the specialized sub-model wα t,i = shrink(wt, αt,i) can be efficiently derived, where function shrink(·, ·) indicates the operations for model shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, the local training is conducted with sub-model wα t,i and local data Di, and the updated local sub-model wα t+1,i is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Furthermore, the local gradient update can be acquired as ut,i = wα t,i −wα t+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2) Flexible gradient upload: To further reduce the uplink traffic, the local device i is motivated to compress the gradient update ut,i before the parameter transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' With the given compression rate βt,i, the compressed gradient update ˜ut,i = cmprs(ut,i, βt,i) is uploaded to the server, where cmprs(·, ·) is the function for gradient compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' computation capacity communication capacity device C device A device B local data compressed update comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' capacity comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' capacity the neural structure becomes larger the data size of local update becomes larger C A B param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' server global model aggregation model distribution param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' upload power the size of each hidden layer is reduced by half, and the training complexity is reduced by ¼ approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 … … … … … … … … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' sparsification binary mask quantization entropy encoding Golomb encoding compressed update an example of gradient compression (single layer) (b) optimization for the training strategy (c) model shrinking & gradient compression (a) outline of AnycostFL … … global model sub-model 16 32 64 8 16 32 size: 16x8x3x3 encoding an example of model shrinking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' left: AnycostFL over heterogeneous edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' middle: the neural structure and gradient compression strategies are customized for diverse devices according to their locally available resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' the darker color indicates the higher computing complexity for training and the larger marker size denotes the larger data size of the local update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' right: illustrations of the model shrinking for the local model and the gradient compression for the local update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 3) Parameter aggregation: The server collects the com- pressed local updates {˜ut,i}∀i with different shrinking factors {αt,i}∀i and compression rates {βt,i}∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After that, the global update is calculated by ˜ut = aioagg({˜ut,i}∀i), where aioagg(·) is the server-side all-in-one aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, the updated global model is computed as wt+1 = wt − ˜ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After the T -round training of the above three-step iterations, the final global model wT is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Before introducing how to customize the values of {αt,i}∀i and {βt,i}∀i in Section IV, we illustrate the details of model shrinking, gradient compression and update aggregation in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Elastic Model Shrinking We aim to derive the sub-model wα t,i with training complex- ity of αt,iW from global model wt by reducing the width of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The shrinking operations work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1) Server-side channel sorting: To avoid incurring extra memory cost for the edge devices, the server first sorts the channels of the latest global model before the model distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given one layer of the weight of the global model, the server sorts the output channels in the current layer in descending order according to their values of L2 norm, and meanwhile, the input channels of the next layer should be sorted accordingly in the same order to maintain the permutation invariance of the whole model [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2) Layer-wise uniform shrinking: Next, the server broad- casts the weight of each layer of the global model in a channel- by-channel manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Instead of downloading the full global model, each device only receives those important parameters from the global model to assemble the local sub-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Here, we utilize the fixed shrinking ratio for each layer in the same sub-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Empirically, given model shrinking factor αt,i, we can reduce the size of the hidden layer by √αt,i to acquire the sub-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2(c), when shrinking a global model with hidden sizes of {16, 32, 64} under αt,i = 1 4, we approximately reduce the size of each hidden layer by half as {8, 16, 32} to form the sub-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' At the beginning of the t-th global round, all device ini- tialize their local sub-models {wα t,i}∀i by choosing the most important channels from the global model wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In this way, the training complexity is significantly reduced while maintaining the performance of local sub-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After that, the local training of device k is conducted with sub-model wα t,i, which produces the local gradient ut,i with data size of αt,iS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Flexible Gradient Compression Given the local update ut,i with the desired compression rate βt,i, we aim to obtain the compressed update ˜ut,i with data size of αt,iβt,iS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let ρt,i and Lt,i denote the sparsity rate and the number of quantization levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The gradient compression scheme works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1) Kernel-wise sparsification: Without loss of generality, we take the convolution neural network (CNN) as an example to illustrate the sparsification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We aim to acquire the sparse update ˆut,i from ut,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let ut,i[k] denote the k-th kernel of ut,i, and ut,i = {ut,i[k]}∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We measure the importance of each kernel and obtain N = {∥ut,i[k]∥2}∀k, where ∥ · ∥2 denotes the L2 norm operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Next, by selecting the ⌈ρt,iK⌉- th largest value in N as the threshold Π, the kernel-wise sparsification is expressed as ˆut,i[k] = � 0 if ∥ut,i[k]∥2 < Π, ut,i[k] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (2) Meanwhile, the binary mask of ˆut,i is denoted as mt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2) Probabilistic quantization: Motivated by the studies in [35], [36], we aim to obtain the quantized update ˜ut,i with the given sparse ˆut,i and the quantization level Lt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let u ∈ ˆut,i be a scalar value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To begin with, we first calculate the magnitude range of the non-zero elements of ˆut,i, denoted as [umin, umax], where umin = min{|u|}∀u̸=0, and umax = max{|u|}∀u̸=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Next, let Q = {Ql}Lt,i l=1 denote the set of quantization points, where Ql is computed by Ql = l (umax − umin) Lt,i + umin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='model structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update of each layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 2 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='local compressed updates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='global update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='legend ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='normal update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='zero update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='not existing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='zero update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by device 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by all devices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by devices 1&2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by device 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by devices 1&3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by devices 2&3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='update by device 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='�ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1 �ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2 �ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3 �ut Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' An illustration of the all-in-one aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For any u ∈ ˆut,i and u ̸= 0, we can always find a quantization interval [Ql, Ql+1] such that Ql ≤ |u| ≤ Ql+1, and its corresponding quantized value ˜u is further computed by ˜u = � sgn(u) · Ql with probability Ql+1−|u| Ql+1−Ql , sgn(u) · Ql+1 otherwise, (4) where sgn(·) calculates the sign of the given scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fur- thermore, the set of the quantization indices of all ˜u ∈ ˜ut,i is denoted as Lt,i = {l, Ql = ˜u}∀˜u̸=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Now, ˜ut,i can be represented by a tuple of {umin, umax, Lt,i, mt,i, Lt,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 3) Lossless encoding: Due to the distribution characteristics of Lt,i that smaller indices may occur more frequently, we apply entropy coding to reduce the data size [14], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Besides, the sparse binary matrix mt,i can be compressed by Golomb encoding [11], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After determining the compression scheme, we can vary the combinations of {ρt,i, Lt,i} and record the corresponding compression rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on the results, we can build a piecewise linear function to predict the compression strategy {ρt,i, Lt,i} with the given βt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Notably, this function can be efficiently fitted by the server with a rather small amount of public training data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', 16 samples) in an offline manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' All-in-One Aggregation After all the devices upload their encoded updates, the server receives, decodes and then reconstructs the compressed local updates {˜ut,i}∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Our goal is to obtain the global update ˜ut by aggregating {˜ut,i}∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' However, the aggregation of local updates in our framework cannot be supported by conventional FedAvg [1], since the local updates are produced by different model structures with different levels of precision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', different quantization levels and sparsity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To tackle the above challenge, we propose an all-in-one aggregation scheme that fuses the local updates in an element- wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let the set {1, 2, · · · , J} index elements of the global update ˜ut, and ˜u[j] t denote the j-th element of ˜ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To accomplish the aggregation for ˜u[j] t , we first determine the subset of devices Ij ⊆ I whose local model structure also contains the j-th element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, we have ˜u[j] t = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0 if � i∈Ij m[j] t,i = 0, 1 � i∈Ij pt,im[j] t,i � i∈Ij pt,im[j] t,iu[j] t,i otherwise, (5) where pt,i is the aggregation coefficient for the j-th device at the t-th global round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The optimal values of {pt,i}∀i will be further analyzed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 3 gives an example to illus- trate the aggregation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, different elements in the global update are updated by different subsets of devices, and more important elements will “absorb” knowledge from more devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' When the j-th element is zeroed out by all the devices in Ij, we have ˜u[j] t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' THEORETICAL ANALYSIS AND OPTIMIZATION In this section, we focus on the optimization of our frame- work by customizing the training strategies for diverse devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We first formulate the on-demand training problem of Any- costFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, we derive the upper bound of the convergence rate and reveal the key insights to improve the performance of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on the analysis, the optimization problem is transformed into a tractable form, and the closed-form solution is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' AnycostFL over Wireless Networks In this subsection, we formulate the computation and com- munication models for our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After that, we build up an on-demand learning problem that minimizes the global training loss with given delay and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 1) Computation model: For the device i at the t-th global round, given the model shrinking factor αt,i and computing frequency ft,i, the time consumption of local model training can be measured by T cmp t,i = τ|Di|αt,iW ft,i , (6) where τ denotes the number of local epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Meanwhile, the corresponding energy consumption can be given by Ecmp t,i = ǫif 2 t,iτ|Di|αt,iW, (7) where ǫi is the hardware energy coefficient of the device i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2) Communication model: We consider the frequency divi- sion multiple access (FDMA) scheme for the transmission of the local gradient update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the device i at the t-th global round, the achievable transmitting rate can be estimated by rt,i = bilog2 � 1 + |ht,i|P com t,i N0bi � , (8) where P com t,i is the transmitting power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' bi is the achievable bandwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' |ht,i| denotes the path loss of wireless channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' N0 is the power spectral density of the additive white Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the device i at t-th global round, given the update ˜ut,i generated by the local model with a shrinking factor of αt,i and compression rate of βt,i, the required time T com t,i and energy consumption Ecom t,i of uplink transmission can be respectively measured by T com t,i = αt,iβt,iS rt,i , and Ecom t,i = T com t,i P com t,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (9) With the above computation and communication models, we next focus on the optimization problem of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 3) Problem formulation: To optimize AnycostFL, we study an on-demand training problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, the shared max- imal latency for each round T max is determined by the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The local energy consumption budget for each round Emax t,i is customized by the device itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given multiple devices with diverse local resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', computation, communication and data), our goal is to customize the training strategy for each device to minimize the global training loss with personalized constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', latency and energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To sum up, at the t-th global round, we aim to optimize the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (P1) min F � wt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' {αt,i}∀i, {βt,i}∀i � (10) subject to: T cmp t,i + T com t,i ≤ T max, ∀i, (10a) Ecmp t,i + Ecom t,i ≤ Emax t,i , ∀i, (10b) αmin ≤ αt,i ≤ 1, ∀i, (10c) 0 ≤ βt,i ≤ βmax, ∀i, (10d) f min i ≤ ft,i ≤ f max i , ∀i, (10e) variables: {αt,i, βt,i, ft,i}∀i, where F � wt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' {αt,i}∀i, {βt,i}∀i � denotes the global loss of the t-th round with given the global model weight wt under the training strategies of {αt,i}∀i and {βt,i}∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In the rest of this section, we analyze the relationship between training loss and training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' After that, Problem (P1) is further solved based on the theoretical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumptions and Key Lemmas Being in line with the studies in [5], [39], we make the following assumptions for the local loss function Fi, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fi is λ-Lipschitz: ∥Fi(w) − Fi(w′)∥ ≤ λ ∥w − w′∥, where λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fi is ν-strongly convex: Fi(w) ≥ Fi(w′) + (w − w′)⊤∇Fi(w′) + ν 2 ∥w − w′∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fi is twice-continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Assumptions 1 and 2, we have νI ⪯ ∇2Fi(w) ⪯ λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The ratios between the norms of ∇Fi(w) and ∇F(w) are bounded: ∥∇Fi(w)∥2 ≤ ε ∥∇F(w)∥2, where ε ≥ 0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the moderate shrinking factor α ≥ αmin, the first-shrinking-then-training can be approximated as first- training-then-shrinking: ∇Fi(wα) = [∇Fi(w)]α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Here, we use [∇Fi(w)]α to denote the shrinking operation for ∇Fi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Next, we give the following two definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Definition 1 (Local gradient divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The local gradient divergence δt,i is defined as the difference between ut,i and ˜ut,i, which is given by δt,i = ∥ut,i − ˜ut,i∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Definition 2 (Global gradient divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The global gra- dient divergence ∆t is defined as the difference between ut and ˜ut, which is measured by ∆t = ∥ut − ˜ut∥ = ��� I� i=1 pt,iut,i − I� i=1 pt,i˜ut,i ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Notably, in Definition 1, ut,i and ˜ut,i may have different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We pad the missing elements in ˜ut,i with zeros before the arithmetic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Next, we are interested in how the training strategies {αt,i, βt,i}∀i affect {δt,i}∀i and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We derive the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the local training with the model shrinking factor αt,i and compression rate βt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The square of the local gradient divergence is bounded by E∥δt,i∥2 ≤ � 1 − αt,i(2 − αt,i) � βt,i �2E∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the local update {˜ut,i, ∀i} with the corre- sponding training strategies {αt,i, βt,i}∀i and aggregation co- efficients {pt,i}∀i, the square of the global gradient divergence is bounded by E∥∆t∥2 ≤ Iεη2 I � i=1 p2 t,i � 1 − αt,i(2 − αt,i) � βt,i �2E∥∇F(wt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Optimal Aggregation Scheme and Convergence Analysis Intuitively, the local update ut,i generated with larger {αt,i, βt,i} may carry more accurate information, and thus a larger pt,i should be assigned during the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Lemma 2, we deduce the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Theorem 1 (Optimal aggregation scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given the lo- cal updates {˜ut,i}∀i with corresponding training strategies {αt,i, βt,i}∀i, the optimal aggregation coefficients are p∗ t,i = 1 � 1−αt,i(2−αt,i)√ βt,i �2 � i 1 � 1−αt,i(2−αt,i)√ βt,i �2 , ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Lemma 2, we study the following optimiza- tion problem to minimize the global gradient divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (P2) min {pt,i}∀i I � i=1 p2 t,i � 1 − αt,i(2 − αt,i) � βt,i �2 (14) subject to: pt,i ≥0, ∀i, (14a) I � i=1 pt,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (14b) It can be verified that Problem (P2) is a convex op- timization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We further solve the problem by the Karush–Kuhn–Tucker (KKT) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let {̟}∀i and θ be the Lagrange multipliers for Constraints (14a) and (14b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, we obtain ̟i ≥ 0, ̟ipt,i = 0, pt,i ≥ 0, I � i=1 pt,i = 1, 2pt,i � 1 − αt,i(2 − αt,i) � βt,i �2 − ̟i + θ = 0, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (15) Being in line with the study in [40], we can obtain pt,i = − θ 2 � 1 − αt,i(2 − αt,i) � βt,i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (16) By putting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (16) into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (14b), we obtain θ = − 2 � k 1 � 1−αt,i(2−αt,i)√ βt,i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (17) Putting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (17) into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (16) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' With the optimal aggregation scheme, we investigate the upper bound of the convergence rate of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Definition 3 (Local and global learning gains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The local and global learning gains are defined as gt,i = α4 t,iβt,i and gt = � i gt,i/I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, the local and global learning gains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', gt,i ∈ [0, 1] and gt ∈ [0, 1]) measure the amount of effective information carried in the local and global updates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Theorem 2 (Convergence rate of AnycostFL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let gmin = min{gt}∀t be the minimal global learning gain over the T - round training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The upper bound of the convergence rate of AnycostFL satisfies E � F(wT ) − F(w∗) � ≤ ZT −1E � F(w0) − F(w∗) � , (18) where Z = 1 − ν λ � 1 − ε(1 − gmin) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Recall that parameters ν, λ and ǫ are defined in Assumptions 1 to 4 before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Definition 3 and Theorem 2, we derive the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The key to minimizing the training loss of AnycostFL is to maximize the learning gain gt for each global round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' If gt = 1 ∀t, AnycostFL degrades to conventional FL without model shrinking and gradient compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Solution for Problem (P1) Based on Theorem 2 and Proposition 1, Problem (P1) can be transformed into the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (P3) max 1 I I � t=1 α4 t,iβt,i (19) subject to: Constrains (10a) to (10e), variables: {αt,i, βt,i, ft,i}∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Constraints (10a) and (10b) for the training latency and energy, we obtain the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The equality will always hold for Constraints (10a) and (10b) when confirming the optimal training strategy {α∗ t,i, β∗ t,i, f ∗ t,i}∀i, and thus T ∗ t,i = T max and Et,i = E∗ t,i ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The lemma can be proved by showing the contradic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Suppose that there exists i0 such that T ∗ t,i0 < T max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We can find a new solution {α′ t,i0, β∗ t,i0, f ′ t,i0} for device i0 and α′ t,i0 > α∗ t,i0, f ′ t,i0 < f ∗ t,i0, such that T ′ t,i0 = T max and E′ t,t0 = Emax t,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Since the global learning gain increases with the increase of αt,i0, we have g′ t > g∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Likewise, the contradiction also appears when E∗ t,i0 < Emax t,i0 , and thus we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Lemma 3, we employ two intermediate variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', φt,i and ϕt,i) for each device to reparameterize Problem (P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, φt,i ∈ [0, 1] and ϕt,i ∈ [0, 1] are the splitting factors for latency and energy, respectively, such that T cmp t,i = φt,iT max, T com t,i = (1 − φt,i)T max, Ecmp t,i = ϕt,iEmax t,i , Ecom t,i = (1 − ϕt,i)Emax t,i , ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (20) By combining Eqns (6) and (20), the local learning gain of the device i at the t-th round can be rewritten as gt,i(φt,i) = κt,i � Emax t,i − (1 − φt,i)T maxP com t,i � (φ2 t,i − φ3 t,i), (21) where κt,i = rt,i Sǫi � T max τ|Di|W �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Note that Problem (P3) can be transformed into I sub- problems because the decision-making procedure of each device is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (21), the i-th sub- problem can be expressed as a single-variable optimization problem with respect to φt,i as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (P4) max φt,i gt,i � φt,i � (22) subject to: φmin t,i ≤ φt,i ≤φmax t,i , where the lower and upper limits of φt,i can be acquired by φmin t,i = max �αminτ |Di| W f max i T max , 1 − βmaxS rt,iT max � , φmax t,i = min � τ |Di| W f min i T max , 1 − αminβminS rt,iT max � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (23) Based on the first-order optimality condition ∂gt,i/φt,i = 0, we obtain the stationary points as φs1 t,i = � ψt,i − 3Emax t,i 8P com t,i T max + 3 4, φs2 t,i = − � ψt,i + 3Emax t,i 8P com t,i T max − 3 4, (24) where ψt,i = 4(P com t,i T max)2 − 4Emax t,i P com t,i T max + 9(Emax t,i )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let St,i = {φmin t,i , φmax t,i , φs1 t,i, φs2 t,i} denote the union of the stationary points and the boundary points for Problem (P4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, S′ t,i = {φt,i|φt,i ∈ [φmin t,i , φmax t,i ], φt,i ∈ St,i} is the set of the feasible solutions of St,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The optimal solution for Problem (P4) can be acquired by φ∗ t,i = arg max φt,i∈S′ t,i gt,i(φt,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (25) Furthermore, we obtain the optimal solution for device i at the t-th global round by putting φ∗ t,i into the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' ϕ∗ t,i = 1 − (1 − φ∗ t,i)T maxP com t,i Emax t,i , α∗ t,i = 3 � (φ∗ t,iT max)2ϕ∗ t,iEmax t,i ǫi(τ |Di| W )3 , β∗ t,i = rt,i(1 − φ∗ t,i)T max α∗ t,iS , f ∗ t,i = α∗ t,iτ |Di| W φ∗ t,iT max .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (26) Notably, the decision-making process of each device does not involve the auxiliary information of the resource status from other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' At the beginning of each global round, each device can determine its training strategy locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' EXPERIMENT EVALUATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Experiment Settings 1) Setup for FL training: We consider the FL application with image classification on Fashion-MNIST and CIFAR- 10 datasets [41], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For Fashion-MNIST, we use a small convolutional neural network (CNN) with data size of model update as 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='22Mb [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the CIFAR-10 dataset, we employ VGG-9 with data size of model update as 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7Mb [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For IID and non-IID data settings, we follow the dataset partition strategy in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the learning hyper-parameters, the learning rate, batch size and local epoch are set as {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='01, 32, 1} for Fashion-MNIST and {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='08, 64, 1} for CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The maximal latency is set as T max = 10 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='89 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='89 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Test accuracy (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Time consumption (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Time consumption (min) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='STC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='QSGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='UVeQFed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='HeteroFL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='FedHQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='AnycostFL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='(a) FMNIST IID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='(b) FMNIST non-IID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='(c) CIFAR-10 IID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='(d) CIFAR-10 non-IID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Energy consumption (KJ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Energy consumption (KJ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Performance on various network architectures and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' ((a-b): global accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' time consumption with Fashion MNIST on 2-layer CNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (c-d): global accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' energy consumption with CIFAR-10 on VGG-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=') TABLE I PERFORMANCE COMPARISON BETWEEN ANYCOSTFL AND OTHER METHODS ON FASHION-MNIST AND CIFAR-10 DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' IID non-IID Dataset Method #Round Energy (KJ) Latency (min) Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (TFLOPs) Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (GB) Best Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (%) #Round Energy (KJ) Latency (min) Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (TFLOPs) Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (GB) Best Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (%) FMNIST {90%, 89%}∗ STC 305 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7×) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='94 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='42 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7×) 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='71 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='18 283 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3×) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='17 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1×) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='56 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3×) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='66 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='16 QSGD 283 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='6×) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='40 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='56 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='6×) 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='80 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='04 279 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3×) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='27 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2×) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='28 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3×) 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='79 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='07 UVeQFed 247 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='36 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='58 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 123.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='65 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='55 413 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1×) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='9×) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='78 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4×) 3990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='05 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='45 FedHQ 340 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2×) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='95 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='9×) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='67 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2×) 4148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='32 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='22 435 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2×) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='99 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='9×) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='2×) 5303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='96 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='41 AnycostFL 294 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='43 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='94 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='56 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='23 372 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='51 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='06 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='0×) 3118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='98 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='51 {x, y}: x and y denote the target global model accuracy under IID and non-IID data settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' and the energy budget is set as Emax t,i ∼ U[3, 9] joules for the CIFAR-10 dataset, and the corresponding hyper-parameters for the FMNIST dataset are halved by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Additionally, we set αmin = 1/4 and βmax = 1/15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 2) Setup for mobile system: We investigate a mobile system with I = 60 devices located within a circle cell with a radius of 550 meters, and a base station is situated at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' To simulate the mobility, the position of each device is refreshed randomly at the beginning of each round [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the computation, the energy coefficient is set as ǫi ∼ U[5 × 10−27, 1 × 1−26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For communication, the bandwidth is set as 1MHz equally for each device, and the path loss exponent is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The transmission power is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='1W, and N0 is set as −114dBm/MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Performance Comparisons We compare the proposed AnycostFL with the following efficient FL algorithms with three different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' STC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The sparse ternary compression (STC) is adapted to reduce the cost of uplink parameter transmission [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' QSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The TopK sparsification and probabilistic quanti- zation are combined to compress the local gradient [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' UVeQFed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The TopK sparsification and universal vector quantization are used to compress the local gradient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' HeteroFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Each device trains the local sub-model in different widths to match its computation capacity [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' FedHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Each device uses different quantization levels to compress the gradient according to its channel state [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 4 shows the performance of the global model over time consumption and energy consumption under the IID and the non-IID data setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' With the same training efficiency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', time and energy consumption), the proposed AnycostFL consistently outperforms the baseline schemes to improve the test accuracy of the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Meanwhile, Table I provides the best accuracy and required system cost for achieving the specified test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Particularly, when compared with HeterFL and FedHQ, AnycostFL can reduce up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='9 times the energy consumption to reach the test accuracy of 82% on CIFAR-10 dataset under the IID setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' When compared with STC, AnycostFL can reduce up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='7 times the time consumption to reach the test accuracy of 90% on FMNIST dataset under the IID setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Moreover, our framework can significantly improve the best accuracy of the global model by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='33% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='82% on CIFAR-10 dataset under the IID and the non-IID settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Impact of Key Mechanisms and Hyper-parameters Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 5(a) verifies the advantages of the main techniques of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We gradually remove the elastic model shrinking (w/o EMS), the flexible gradient compression (w/o FGC) and the all-in-one aggregation (w/o AIO), and record the required system cost to achieve 80% test accuracy with CIFAR-10 dataset under the IID setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We observe that the proposed EMS and FGC can significantly save the energy consumption and training time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Besides, AIO contributes to saving both energy and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' AnycostFL w/o EMS w/o FGC w/o AIO 20 30 40 50 60 Energy consumption Required time Energy consumption (KJ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='4x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='5x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='3x 35 40 45 50 55 60 65 70 75 Required time (min) 0 2 4 6 8 10 45 60 75 90 105 120 6 7 8 9 70 80 Average time consumption (min) Level of communication heterogeneity STC QSGD UVeQFed HeteroFL FedHQ AnycostFL 0 2 4 6 8 10 12 24 32 40 48 56 Average energy consumption (KJ) Level of computation heterogeneity STC QSGD UVeQFed HeteroFL FedHQ AnycostFL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='12 78 81 84 87 90 STC HeteroFL AnycostFL Global test accuracy (%) Computational complexity (GFLOPs) (a) Impact of key mechanisms (b) Impact of comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' heterogeneity (c) Impact of comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' heterogeneity (d) Performance of sub-models Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The main advantages of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' ((a): the impact of key mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (b-c): the impact of system heterogeneity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (d): the performance of sub-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=') We next evaluate the impact of resource heterogeneity on the training efficiency in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 5(b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We set the average energy coefficient ǫi as 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='5×10−27 and the average distance between the base station and edge devices as 400 meters, and then change their variances to simulate the computation and com- munication heterogeneity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The larger variance indicates a higher level of system heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' As we expect, the proposed AnycostFL shows more resilience than other baselines to tackle the high level of system heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We also evaluate the performance of sub-models in different widths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Specifically, We compare AnycostFL with HeteroFL (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', local training with different widths) and STC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=', the best-performing compression-only method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The sub- models are derived from the well-trained global model without further re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Surprisingly, the sub-models of the global model trained by AnycostFL can still maintain satisfactory test accuracy, which provides dynamic inference for diverse edge devices after the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed AnycostFL, a joint computation and communication efficient framework for FL, that enables edge devices with diverse resources to train a shared global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' We aimed to minimize the global training loss under given personalized latency and energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' By leverag- ing the theoretical insight of AnycostFL, we decomposed the optimization problem into multiple sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Following that, the optimal training strategy is derived for each de- vice according to its locally available resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Experiments demonstrate the advantage of our framework in improving the system efficiency and model performance compared to the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' ACKNOWLEDGMENT Rong Yu and Yuan Wu are the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' This work was supported in part by National Key R&D Program of China under Grant 2020YFB1807802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' in part by National Natural Science Foundation of China under Grants 61971148,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' 62102099,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' U22A2054 and 62001125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' in part by Science and Technology Development Fund of Macau SAR under Grant 0162/2019/A3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' in part by FDCT-MOST Joint Project under Grant 0066/2019/AMJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' in part by the Guangdong Basic and Applied Basic Research Foundation (2022A1515011287),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' and in part by US National Science Foundation under grant CNS- 2107057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For the given local gradient ˜ut,i with shrinking factor αt,i and gradient compression rate βt,i, we aim to capture the divergence between ˜ut,i and ut,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Suppose that the absolute value of the element in ut,i follows uniform distribution |u| ∼ U(0, umax), and umax = max{|u|}∀u∈ut,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' For clear notation, we sort the element-wise absolute value of ut,i in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, we obtain ut,i = [u[1] t,i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , u[j] t,i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , u[J] t,i ]⊤ and |u[j] t,i| ≤ |u[j+1] t,i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Thus, we have E∥ut,i∥2 = E J � j=1 |u[j] t,i|2 = JE|u[j] t,i|2 = Ju2 max 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (27) Based on Assumption 5, the update generated from lo- cal training with wα t,i is equal to shrink(ut,i, αt,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The operation of model shrinking on ut,i with αt,i can be viewed as removing (1 − αt,i)J elements with the least value from ut,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, we obtain shrink(ut,i, αt,i) = [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , 0, u[(1−αt,i)J+1] t,i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , u[J] t,i ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Thus, we have E∥ut,i − shrink(ut,i, αt,i)∥2 = E (1−αt,i)J � j=1 |u[j] t,i|2 = J(1 − αt,i)3u2 max/3 = (1 − αt,i)3E∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (28) We next focus on the gradient compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' The operation of gradient sparsification on ut,i with sparsity of ρt,i can be viewed as removing ρt,iJ elements with the least value from ut,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Then, the quantization is conducted on the non- zero elements of ˆut,i, and we obtain cmprs(ut,i, βt,i) = [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , 0, ˜u[ρt,iJ+1] t,i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' , ˜u[J] t,i ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Furthermore, we have E∥ut,i − cmprs(ut,i, βt,i)∥2 = E ρt,iJ � j=1 |u[j] t,i|2 � �� � (A) + E J � j=ρt,iJ+1 |u[j] t,i − ˜u[j] t,i|2 � �� � (B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (29) Likewise to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (28), we have (A) = ρ3 t,iE∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (4) and the statistical feature of ut,i, we obtain (B) = (1 − ρt,i)3E∥ut,i∥2/(2L2 t,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Given plain update ut,i in 32-bit floating point and the desired compression rate βt,i, we can set ρt,i = 1− � βt,i and Lt,i = 232√ βt,i for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' In this way, the operations of sparsification and quantization contribute equally to the gradient compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Furthermore, we have E∥ut,i − cmprs(ut,i, βt,i)∥2 ≤ (1 − βt,i)2E∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (30) Next, we focus on the local divergence δt,i with respect to αt,i and βt,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' According to the Definition 1, we have E∥δt,i∥2 = E∥ut,i − cmprs([ut,i]α, βt,i)∥2 = E∥ut,i − [ut,i]α∥2 + E∥[ut,i]α − cmprs([ut,i]α, βt,i)∥2 + 2 < ut,i − [ut,i]α, [ut,i]α − cmprs([ut,i]α, βt,i) > � �� � (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (31) It can be verified that the two vectors in term (C) are orthogonal, and we obtain (C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' According to Eqns (28) and (30), we further obtain E∥δt,i∥2 ≤ (1 − αt,i)3E∥ut,i∥2 + (1 − � βt,i)2E∥[ut,i]α∥2 (a) ≤ (1 − αt,i)3E∥ut,i∥2 + (1 − � βt,i)2αt,i(α2 t,i − 3αt,i + 3)E∥ut,i∥2 (b) ≤ � 1 − αt,i(2 − αt,i) � βt,i �2E∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (32) Likewise to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (28), inequality (a) stems from the fact that E∥[ut,i]α∥2 = αt,i(α2 t,i−3αt,i+3)E∥ut,i∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Besides, inequal- ity (b) holds for all αt,i ∈ [αmin, 1] and βt,i ∈ [0, βmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Thus, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' APPENDIX B PROOF OF LEMMA 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Definition 2 and Lemma 1, we have E∥∆t∥2 = E ��� I � i=1 pt,iut,i − I � i=1 pt,i˜ut,i ��� 2 ≤ E � I � i=1 pt,i � 1 − αt,i(2 − αt,i) � βt,i � ∥ut,i∥ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (33) We use η to denote the learning rate, and ut,i = η∇Fi(wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Assumption 4, we obtain E∥∆t∥2 ≤ εη2� I � i=1 pt,i � 1 − αt,i(2 − αt,i) � βt,i ��2 E∥∇F(wt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (34) According to Cauchy–Schwarz inequality, we obtain E∥∆t∥2 ≤ Iεη2 I � i=1 p2 t,i � 1 − αt,i(2 − αt,i) � βt,i �2E∥∇F(wt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (35) Thus, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' APPENDIX C ON THE CONVERGENCE OF ANYCOSTFL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Inspired by the studies in [5], [39], we deduce the convergence analysis of AnycostFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' According to Taylor expansion and Assumption 3, we have F(wt+1) ≤ F(wt) + (wt+1 − wt)⊤∇F(wt) + λ 2 ∥wt+1 − wt∥2 = F(wt) − ˜u⊤ t ∇F(wt) + λ 2 ��˜ut ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (36) By using learning rate η = 1 λ, we obtain E � F(wt+1) � ≤ E � F(wt) − λ (ut − ∆t)⊤ut + λ 2 ∥ut − ∆t∥2� = E � F(wt) − 1 2λ∥∇F(wt)∥2 + λ 2 ∥∆t∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (37) We now pay attention to the upper bound of ∥∆t∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Based on Jensen’s inequality and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (34), we obtain E∥∆t∥2 ≤ εη2 I � i=1 pt,i � 1 − αt,i(2 − αt,i) � βt,i �2 � �� � (D) E∥∇F(wt)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (38) By putting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (13) into (A), we have E∥D∥ ≤ E �������� I I� i=1 1 (1−αt,i(2−αt,i)√ βt,i) 2 �������� (c) ≤E �������� I I� i=1 1 1−α4 t,iβt,i �������� , (39) where (c) always holds for αt,i ∈ [0, 1] and βt,i ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' According to Definition 3, we have gt,i = α4 t,iβt,i and gt = � i gt,i/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Since 1/ �� i 1 1−gt,i � is a concave function with respect to gt,i, based on Jensen’s inequality, we obtain E∥A∥ ≤ I � i 1 1−E(α4 t,iβt,i) = 1 − gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (40) Since the training strategies of each device and the norm of the gradient of global data ∥∇F(wt)∥ are independent, by putting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (40) back to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (38), we obtain E∥∆t∥2 ≤ E � εη2� 1 − gt � ∥∇F(wt)∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (41) Next, by putting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (41) back to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (37), we have E � F(wt+1) � ≤ E � F(wt) − 1 + ε � gt − 1 � 2λ ∥∇F(wt)∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (42) Subtracting F(w∗) in both sides of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (42) yields E � F(wt+1 − F(w∗) � ≤ E � F(wt) − 1 + ε(gt − 1) 2λ ∥∇F(wt)∥2 − F(w∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (43) Based on Assumptions 2 and 3, we have [5], [45] ∥∇F(wt)∥2 ≥ 2ν � F(wt) − F(w∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (44) Plugging Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (44) into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' (43), we have E � F(wt+1) − F(w∗) � ≤ ZtE � F(wt) − F(w∗) � , (45) where Zt = 1 − ν λ (1 − ε(1 − gt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Let gmin = min{gt}∀t be the minimal global learning gain over T global rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' By recursively applying the above inequality from iteration round 0 to T , we can obtain E � F(wT ) − F(w∗) � ≤ ZT −1E � F(w0) − F(w∗) � , (46) where Z = 1 − ν λ � 1 − ε(1 − gmin) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' Thus, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' REFERENCES [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} +page_content=' B.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E1T4oBgHgl3EQfSAP8/content/2301.03062v1.pdf'} diff --git a/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf b/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3dd9fa6cd7e814205c0bd24a9a07dc57bbf1a90a --- /dev/null +++ b/2dFAT4oBgHgl3EQfkh3i/content/2301.08612v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ae926b2786f1c0908dbe584c205f3fd27ecf3526958a29449dc5f21c1488d6be +size 13089278 diff --git a/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/2301.00741v1.pdf.txt b/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/2301.00741v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e96f2b840a9164438e3bf08bd01533680dae0e22 --- /dev/null +++ b/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/2301.00741v1.pdf.txt @@ -0,0 +1,597 @@ +Astronomy & Astrophysics manuscript no. 45358arxaa +©ESO 2023 +January 3, 2023 +Letter to the Editor +Analysis of the first infrared spectrum of quasi-bound +H2 line emission in Herbig-Haro 7 +E. Roueff1, M. G. Burton2, T. R. Geballe3, and H. Abgrall1 +1 Sorbonne Université, Observatoire de Paris, PSL University, CNRS, LERMA, F-92190, Meudon, France +e-mail: evelyne.roueff@obspm.fr +2 Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DB, Northern Ireland +e-mail: Michael.Burton@Armagh.ac.uk +3 Gemini Obsevatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, HI 96720, USA +e-mail: tom.geballe@noirlab.edu +Accepted in Astronomy Astrophysics Letters on december 22, 2022 +ABSTRACT +Context. Highly excited molecular hydrogen (H2) has been observed in many regions of shocked molecular gas. A recently published +K-band spectrum of Herbig-Haro 7 (HH7) contains several vibration-rotation lines of H2 from highly excited energy levels that +have not been detected elsewhere, including a line at 2.179 µm identified as arising from the v=2 J=29 level, which lies above the +dissociation limit of H2. One emission line at 2.104 µm in this spectrum was unidentified. +Aims. We aim to complete the analysis of the spectrum of HH7 by including previously missing molecular data that have been recently +computed. +Methods. We re-analysed the K-band spectrum, emphasising the physics of quasi-bound upper levels that can produce infrared +emission lines in the K band. +Results. We confirm the identification of the 2 − 1 S (27) line at 2.1785 µm and identify the line at 2.1042 µm as due to the 1-0 S (29) +transition of H2, whose upper level energy is also higher than the dissociation limit. This latter identification, its column density, and +the energy of its upper level further substantiate the existence of a hot thermal component at 5000 K in the HH7 environment. +Conclusions. The presence of the newly identified 1 − 0 S (29) line, whose quasi-bound upper level (v=1, J=31) has a significant +spontaneous dissociation probability, shows that dissociation of H2 is occurring. The mechanism by which virtually all of the H2 in +levels with energies from 20,000 K to 53,000 K is maintained in local thermodynamic equilibrium at a single temperature of ∼5,000 +K remains to be understood. +Key words. molecular hydrogen – interstellar medium – shocks +1. Introduction +The interaction of the collimated outflow from the protostar +SSV13 (Strom et al. 1976) and the molecular cloud out of which +it formed has produced a collection of Herbig-Haro (HH) ob- +jects, HH7-HH11, in a more or less linear arrangement on the +sky. The most distant of these from SSV13, HH7, has a classic +bow shock shape. It is bright in line emission from shock-excited +vibrational states of molecular hydrogen (H2), first observed in +the v = 1 − 0 S (1) transition by Zealey et al. (1984), Harti- +gan et al. (1989), and Garden et al. (1990) and subsequently in +vibrational levels 0 − 4 and rotational levels 1 − 15 by Burton +et al. (1989) and Fernandes & Brand (1995). HH7 also emits +strongly in pure rotational lines of H2 and CO (Neufeld et al. +2006; Yuan & Neufeld 2011; Neufeld et al. 2019; Molinari et al. +2000) as well as in [OI]63µ (Sperling et al. 2020), Hα, [OI]λ6300, +and [SII]λ6716 (Hartigan et al. 2019). The vibrationally excited +H2 lines, observed mainly in the 2.0 − 2.5 µm interval, are emit- +ted predominantly in the hottest shock-heated gas, while the pure +rotational low-J transitions of H2 and the pure rotational transi- +tions of CO, observed in the mid- and far-infrared, arise in a +somewhat cooler gas downstream. +Much more highly vibrationally and rotationally excited +molecular hydrogen was found by Pike et al. (2016, hereafter +P16) in a 3′′× 3′′region near the tip of the HH7 bow shock, in +K-band spectra they obtained at a resolving power, R, of 5000. +Figure 6 of their paper shows the 2.01 − 2.45 µm spectrum of a +0′′.6×0′′.9 area in that region. Their paper demonstrated the ex- +istence in HH7 of a small percentage (1.5%) of the emitting +H2 at a temperature of ∼5,000 K. Subsequently, Geballe et al. +(2017) discovered the presence of small percentages of 5,000 K +H2 in shocked gas at several locations in the Orion Molecular +Cloud. Giannini et al. (2015) detected a similar phenomenon in +another bright HH object, HH1, at a somewhat higher tempera- +ture, ∼6300 K. +P16 identified a weak emission line at 2.179 µm in the HH7 +spectrum as the 2 − 1 S (27) transition of H2, which arises from +the upper level, v = 2, J = 29, whose energy is above the +dissociation limit of the ground state of H2; this corresponds +to 51,965.84 K, using the latest measurements (Hölsch et al. +2019) and the Committee on Data for Science and Technology +(CODATA) definition of fundamental constants (Tiesinga et al. +2021). The column density of H2 in the upper level could only be +crudely estimated by P16, as the Einstein A coefficient for that +transition was not known. P16 also reported the detection of a +faint line near 2.104 µm, which they were unable to identify. +Roueff & Abgrall (2022) have recently proposed a simple +and efficient method for computing the emission spectrum pro- +Article number, page 1 of 5 +arXiv:2301.00741v1 [astro-ph.GA] 2 Jan 2023 + +A&A proofs: manuscript no. 45358arxaa +duced by quasi-bound levels, providing accurate wavenumbers +and Einstein emission coefficients. The application to H2 al- +lowed them to calculate the Einstein coefficient of the 2-1 S (27) +transition and suggested that the line at 2.104 µm in HH7 is the +1 − 0 S (29) transition of H2, whose upper-state energy also lies +above the dissociation limit. +The present paper analyses these two high excitation lines in +the light of the new theoretical developments. Section 2 revisits +the observations of HH7, Sect. 3 summarises the recent theoret- +ical achievements, and Sect. 4 contains the resulting extended +observational analysis of the H2 line emission in HH7. We pro- +vide a discussion of our results in Sect. 5. +2. Observations +A detailed description of the observations of HH7 and a re- +duction of the spectral data have been given by P16. In brief, +the Gemini facility integral field spectrometer, the Near Infrared +Field Spectrometer (NIFS; McGregor et al. 2003), was used at +the Frederick C. Gillett Gemini North Telescope on Maunakea, +Hawai’i, to obtain spectra of a 3′′× 3′′region near the tip of the +HH7 bow shock, for program GN-2007B-Q-47. The angular res- +olution of the spectra was 0′′.35. Within this 3′′× 3′′region, the +spectra showed H2 ro-vibrational line emission from upper-state +levels covering a wide range of energies, including a dozen in +the range 40, 000 − 50, 000 K. Because some rotational energies +and associated rotational quantum numbers of the upper levels of +these lines are high (J ≳ 15), collisions rather than the absorp- +tion of ultraviolet (UV) photons are probably the main producer +of the populations in those rotational levels. Somewhat lower +values of J associated with high vibrational quantum numbers +are commonly found in dense photon-dominated regions (PDRs) +such as NGC 2023, the Orion Bar, S140, and IC63. (Burton et al. +1992; McCartney et al. 1999; Kaplan et al. 2021). H2 is excited +in PDRs by UV pumping, which is followed by electronic fluo- +rescence, but the ∆J = ±1 selection rule for electronic transitions +maintains J at values below ∼ 13.1 +P16 concentrated their analysis on the spectrum of the 0′′.6 +× 0′′.9 area shown in their Fig. 2; the spectrum is plotted in their +Fig. 6. The upper two panels of our Fig. 1 show in more de- +tail two 0.01 µm wide portions of that spectrum, each contain- +ing one of the two highly shock-excited H2 lines discussed in +the Introduction. Wavelength calibration employed the spectrum +of an argon lamp and is accurate to ∼ 0.00002 µm. The hor- +izontal scales are vacuum laboratory wavelengths and as such +can be directly compared with the theoretically calculated wave- +lengths (see Sect. 3).The uppermost panel contains the previ- +ously unidentified line at 2.1042 µm along with the nearby 4 − 3 +S (7) line. Similarly, the middle panel contains the previously +identified 2−1 S (27) line and the adjacent 5−4 S (15) line. Spec- +tral images of the four lines, extracted from the NIFS data cube, +are shown in the bottom panel of the figure and demonstrate +that, to within the limits imposed by the noise levels, the four +emission lines have identical morphologies, which also match +the morphology of the strong 1 − 0 S (1) line shown in Fig. 2 of +P16. Based on the fluctuations in the baseline, we estimate the +confidence of the detection of the 1 − 0 S (29) line to be 3.5σ. +The wavelengths of these two weak lines are slightly different +than those reported by P16 and are more accurate. +1 We contacted K. Kaplan to check if the two transitions at 2.1785 µm +and 2.1042 µm were present in his PDR spectra obtained with the Im- +mersion Grating INfrared Spectrometer (IGRINS), and they were not. +Fig. 1. Observational data showing highly excited H2 lines in HH7. Top +two panels: Spectra of a 0′′.6 × 0′′.9 area of HH7 in two narrow wave- +length intervals, each containing a line of H2 from a quasi-bound energy +level and one adjacent line, from Fig. 6 of P16. Vertical dashed lines are +the line wavelengths calculated as described in Sect. 3. Bottom: Spec- +tral images of the four lines shown above, extracted from the NIFS data +cube. The field of view is 2′′.5 × 2′′.5 and corresponds to the left part of +Fig. 2 of P16; the field centre corresponds to RA = 3:29:08.42, Dec = ++31:15:27:45 (J2000), with an estimated uncertainty of 0′′.25. +Article number, page 2 of 5 + +2HH +1 +Density +Flux +0.5 +Rel. +res. +4-3 S(7) +-0S(29) +2.098 +2.1 +2.102 +2.104 +2.106 +LabVacuumWavelength(um) +2HH +res. +Rel. +4 S(15) +2.176 +2.178 +2.18 +2.182 +2.184 +Lab Vacuum Wavelength (um) +S(15)E. Roueff et al.: Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7 +3. Theoretical aspects +Molecular quasi-bound levels correspond to states whose ener- +gies lie above the dissociation limit of the ground state of the +molecule but well below the dissociation energy of the electron- +ically excited molecule. For H2, the Schrödinger equation rele- +vant to excited rotational levels is +− ℏ2 +2µ · d2 fv,J(R) +dR2 ++ Vmod +e f f (R, J) fv,J(R) = Ev,J fv,J(R), +(1) +where Vef f (R, J) = V(R) + ℏ2J (J+1) +2µR2 +, with V(R) the ground state +electronic molecular potential of H2 and µ = Mp/2 the nuclear +reduced mass of H2. +Fig. 2. H2 molecular potentials in eV as a function of the interatomic +distance, R, expressed in atomic units. The zero value corresponds to +photo-dissociated H2. The black curve is the electronic potential of the +X 1Σ+ +g ground state from Czachorowski et al. (2018) expressed in eV. +The blue and red curves denote the effective potentials with J= 29 and +J= 31, respectively. The quasi-bound levels v = 2, J = 29 and v = +1, J = 31 are also displayed in blue and red, respectively, in the allowed +ranges of interatomic distances. +Figure 2 displays the electronic molecular potential of the +X1Σ+ +g ground state of H2 as well as the effective potentials cor- +responding to J = 29 and J = 31, the two quasi-bound lev- +els (sometimes referred to as shape resonances) previously men- +tioned. The presence of the centrifugal potential, ℏ2J (J+1) +2µR2 +, signif- +icantly modifies the shape of the electronic contribution, V(R), +by reducing the potential well, shifting the minima to larger in- +teratomic distances and exhibiting broad bump maxima above +the dissociation limit, peaking near 4.5 atomic units. +Figure 2 also displays the resonant quasi-bound eigenval- +ues, Er, which are located above the dissociation limit and are +trapped inside the centrifugal barrier. The associated wave func- +tion for each level has a non-vanishing probability in the inter- +atomic range displayed, becomes vanishingly small after the sec- +ond turning point when Er ≤ Vef f (R), and has an oscillatory be- +haviour for large R when Er becomes larger than Vef f (R). The +associated quasi-discrete stationary states have complex energy +eigenvalues, E = Er − (i Γ/2), where Er is the energy at reso- +nance and Γ characterises the width of the level and determines +its lifetime against dissociation, τ = ℏ/Γ, due to tunnelling from +the quasi-bound to the continuum oscillatory dissociating state at +large interatomic distances. Roueff & Abgrall (2022) computed +the various resonance energy level positions of H2 and the corre- +sponding emission spectrum arising from these levels by using +the recent highly accurate molecular potential of the H2 ground +state of Czachorowski et al. (2018) and extending the effective +potential by a constant value from the maximum value of the po- +tential function. This method allows one to use a standard numer- +ical integration of the Schrödinger equation applied to strictly +bound levels and has been demonstrated to be very precise for +determining the resonant energy level positions and the emission +rates. However, it does not allow a derivation of the widths or +the dissociation lifetimes. Those are obtained through different +methods based on scattering properties (Schwenke 1988; Selg +2010). +These computations predict wavelengths of 2.1785 µm for +the 2 − 1 S (27) transition and 2.1042 µm for the 1 − 0 S (29) +transition. The predicted wavelengths for the two stronger lines +in Fig. 1 are 2.10043 µm for 4−3 S (7) and 2.18179 µm for 5−4 +S (15). As can be seen in the figure, all are in excellent agreement +with the observed wavelengths. Therefore, we are confident in +the previous identification of the 2 − 1 S (27) line by P16 and in +our identification of the weak and previously unidentified feature +at 2.1042 µm as the 1 − 0 S (29) line. These two transitions are +the only lines in the 2.01 - 2.45 µm interval from quasi-bound +levels that would have been detectable in our data. (We note in +Table 1 the small Einstein A coefficient of the 2 − 0 Q(29) line +at 2.4007 µm.) +4. Column density analysis +The analysis undertaken here follows that described in P16 for +the H2 line emission from HH7 reported in that paper, with the +addition of the 1–0 S (29) and 2–1 S (27) lines presented here. A +two-component Boltzmann distribution with temperatures Thot +and Twarm was fitted to the column densities obtained from the +de-reddened line intensities, +Ni = Ni,hot + Ni,warm, +(2) +with each component described by a Boltzmann distribution at +the corresponding temperatures, as per P16. This is shown in +Fig. 3. +We obtain Twarm = 1, 783 ± 20 K and Thot = 5, 133 ± 17 K, +with 98.5% of the total column of excited H2 gas in the warm +component of the gas and 1.5% in the hot component. This com- +pares to values of Twarm = 1, 803±12 K and Thot = 5, 200±12 K +found without these two extra lines included in the analysis2. +The additional lever arm provided by the two higher excitation +energy levels has only led to a marginal decrease in the derived +temperatures; in other words, the result is essentially the same. +We conclude that the two quasi-bound H2 lines are well mod- +elled by the same hot local thermodynamic equilibrium (LTE) +component as per all lines measured in HH7 arising from energy +levels ≥15,000 K. The level populations for the two quasi-bound +lines are ∼ 10−5 times that of the v = 1, J = 3 upper level of the +brightest H2 emission line, 1 − 0 S (1). +5. Discussion +Table 1 summarises the present knowledge available for the two +quasi-bound levels of H2 v = 2, J = 29 and v = 1, J = 31, that +2 The errors quoted here are the formal errors derived from the least +squares fit. +Article number, page 3 of 5 + +J=29 +J=31 +0 +dissociation limit +-1 +-2 +-3 +-4 +g +-5 +2 +6 +8 +10 +12 +0 +4 +R (au)A&A proofs: manuscript no. 45358arxaa +Table 1. Properties of the two detected quasi-bound levels, v = 2, J = 29 and v = 1, J = 31, of H2 and their emission spectrum. +Transition +˜ν +λ +A +Ar +τd +Eqb +upper +label +cm−1 +µm +s−1 +s−1 +s +K +2 − 0 O(31) +1387.04 +7.2096 +2.704E-11 +5.482E-06 +8.130E12 +676.0 +2 − 0 Q(29) +4165.40 +2.4007 +4.592E-08 +5.482E-06 +8.130E12 +676.0 +2 − 0 S (27) +7034.45 +1.4216 +2.240E-07 +5.482E-06 +8.130E12 +676.0 +2 − 1 Q(29) +1944.67 +5.1423 +3.947E-07 +5.482E-06 +8.130E12 +676.0 +2 − 1 S (27) +4590.29 +2.1785 +2.944E-06 +5.482E-06 +8.130E12 +676.0 +2 − 2 S (27) +2399.19 +4.1681 +1.873E-06 +5.482E-06 +8.130E12 +676.0 +2 − 3 S (27) +489.60 +20.4248 +2.582E-10 +5.482E-06 +8.130E12 +676.0 +1 − 0 Q(31) +1974.02 +5.0658 +2.452E-07 +5.219E-06 +4.083E06 +1520.5 +1 − 0 S (29) +4752.38 +2.1042 +2.101E-06 +5.219E-06 +4.083E06 +1520.5 +1 − 1 S (29) +2531.65 +3.9500 +2.872E-06 +5.219E-06 +4.083E06 +1520.5 +1 − 2 S (29) +586.98 +17.0364 +4.963E-10 +5.219E-06 +4.083E06 +1520.5 +Notes. ˜ν is the computed transition frequency; λ is the corresponding vacuum wavelength. A is the sum of the electric quadrupole and the magnetic +dipole contributions to the Einstein radiative emission coefficients of the transition from Roueff & Abgrall (2022). Ar is the total radiative decay +probability, and τd is the dissociation lifetime of the upper level. Eqb +upper is the quasi-bound upper level energy expressed in K above the dissociation +limit. +Fig. 3. Level column densities, divided by their degeneracies, Ni/gi, +plotted as a function of level energy, Ti, for the H2 lines measured in +HH7. They are normalised to unity for the (v, J) = (1, 3) upper-state +level at 6,952 K, which emits the 1–0 S (1) line. The two blue points (in +the lower right) are for the newly analysed 2–1 S (27) and 1–0 S (29) +lines. The dashed red line shows the best two-temperature LTE fit, as +described in Sect. 4. +have been detected. The upper level involved in the 2 − 1 S (27) +transition at 2.1785 µm, 676 K above the dissociation energy of +the ground state, is very stable against dissociation, whereas that +of the 1 − 0 S (29) transition at 2.1042 µm, located 845 K higher, +has a dissociation probability of approximately five percent and +a dissociative lifetime, τd = ℏ/Γd, resulting from quantum tun- +nelling through the centrifugal barrier (see Fig. 2) of 4.083 × 106 +s, corresponding to less than two months. This indicates that the +shock wave in HH7 is partially dissociative. +As shown in Fig. 3, the 5,000 K component represents a +small percentage of the line-emitting H2 in HH7. As noted pre- +viously, similar small percentages have been observed in HH1 +and in the Orion molecular outflow. Figure 3 also shows that H2 +in energy levels greater than ∼ 20,000 K above the ground state +are populated only by this component. In the case of HH1, Gi- +annini et al. (2015) observed a wide range of neutral and ionised +species emitting in close proximity to the H2, many at opti- +cal wavelengths. Their analysis yields a temperature range of +8, 000 − 80, 000 K to account for the emission. They further find +that neutral and fully ionised regions coexist inside the shock. +However, for the heavily extincted H2 line emission from HH7 +(AV = 12−28 mag; P16), the species producing the optical emis- +sion lines observed by Solf & Boehm (1987), Hartigan et al. +(1989), and Hartigan et al. (2019) cannot be mixed with the H2. +In view of the detections by Giannini et al. (2015), P16, and +Geballe et al. (2017) of 5, 000 − 6, 000 K H2 in diverse envi- +ronments, it seems likely that a small percentage of H2 existing +at those temperatures is a common occurrence in collisionally +shocked molecular gas, at least in cases where collisions between +outflows and ambient molecular material occur at velocities of +many tens of km s−1, as is the case for HH1, HH7, and the Orion +Molecular Cloud. In addition, although transitions emitted from +quasi-bound levels have only been detected towards HH7, we +expect that they are present in HH1 and OMC-1 at roughly the +same intensities relative to the stronger H2 lines, as in HH7. +It is generally accepted that the maximum temperatures of +nearly all of the vibrationally excited H2 in each of the above +shocked clouds and in many others are suppressed by continu- +ous shocks, in which the collisional acceleration of the ambient +clouds and deceleration of the colliding outflows from the proto- +stars are sufficiently gradual to heat the H2 only to temperatures +of ∼2,000 K and prevent its dissociation (for more details, see +Sect. I of P16 and references therein). The existence of H2 at a +range of lower temperatures in gas cooling behind the contin- +uous shocks, which has been demonstrated by observations of +pure rotational lines (e.g. Neufeld et al. 2019), is also unsurpris- +ing. However, it seems remarkable that virtually all of the highly +ro-vibrationally excited H2 in levels with energies from 20,000 +K to 53,000 K is maintained in LTE at a single temperature of +∼5,000 K, and that there is virtually no H2 at temperatures be- +tween 2,000 K and 5,000 K. The mechanism that produces this +bimodal temperature distribution is unclear. +The location and morphology of the 5,000 K gas also is un- +clear. The gas could be located in thin (currently unresolvable) +sheets where the molecular cloud is being collisionally acceler- +ated, the wind is being collisionally decelerated, or both. Its line +emission could alternatively also be occurring in small clumps +Article number, page 4 of 5 + +100 +N[T, +1 = 98.5% +warm +N[Thot] = 1.5% +10-2 +10-4 +10-5 +10-6 +0 +1×10° +2×10° +3×10° +4×10° +5×104 +6×104 +Energy Level (K)E. Roueff et al.: Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7 +of unusually hot and/or unusually dense gas scattered along the +shock front. Comparisons of the velocity profiles of lines origi- +nating from levels whose populations are dominated by the gas +at 5,000 K with those from levels dominated by the 2,000 K +component, at higher spectral resolution than has been employed +to date, might reveal small differences and constrain the rela- +tive locations of the two components. The good fit of the v=1, +J = 31 column density to the fit to the population-energy di- +agram (Fig. 3) indicates that dissociation is taking place in the +5,000 K gas. +One can consider if the short lifetime of the v=1, J=31 quasi- +bound level indicates a significant continuous reformation of +molecular hydrogen in the gas phase at high temperatures. We +have estimated the formation rate of H2 through radiative asso- +ciation via that resonance level, i, H + H ↔ H2i → H2 + hν, +following the theory of Bain & Bardsley (1972), to be +αres +i += +� 2πℏ2 +MkT +�3/2 +(2I + 1)(2Ji + 1) +Ai +r Ad +Air + Ad +exp(−Ei/kT), +(3) +where Ad = 1/τ and M is the reduced mass of the colliding +atoms. The contribution of v = 1, J = 31 with I = 1 and similar +values of Ar and Ad is the most efficient by orders of magnitude. +However, the derived value for its contribution at 5000K is 1.37 +× 10−30 cm3 s−1, which is negligible. +Although it is difficult to assess the direct implication of the +measurable presence of these quasi-bound H2 states for shock +chemistry, their detections confirm the predictability of theoreti- +cal computations based on highly accurate potential curves. The +physical conditions associated with the astrophysical environ- +ments in which their lines are emitted may not be reproducible +in the laboratory due to their very large rotational quantum num- +bers. Thus, they probably offer the only way to probe these lev- +els. Martin et al. (1996) introduced quasi-bound levels of H2 in +their master equation studies of collisional excitation of H2 by +H and specifically mentioned the v = 2, J = 29, and v = 1, +J = 31 quasi-bound levels detected here. However, they find +that the highly excited rotational levels are not thermally popu- +lated for the range of physical conditions that they considered, +in contrast to what astronomical observations have revealed. Fi- +nally, we note that the contribution of quasi-bound levels to the +partition function of H2 and its isotopologues has been recently +computed by Zúñiga et al. (2021) using the same potential as us. +Acknowledgements. We thank K. Kaplan for having searched the two transi- +tions in his IGRINS spectra of various PDRs. We are grateful to the referee for +helpful comments. E.R. and H.A. acknowledge support by the Programme Na- +tional de Physique et de Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU +with INC/INP co-funded by CEA and CNES.This research is based in large part +on observations obtained at the international Gemini Observatory, a program of +NSF’s NOIRLab, which is managed by the Association of Universities for Re- +search in Astronomy (AURA) under a cooperative agreement with the National +Science Foundation, on behalf of the Gemini Observatory partnership: the Na- +tional Science Foundation (United States), National Research Council (Canada), +Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, +Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Ino- +vações e Comunicações (Brazil), and Korea Astronomy and Space Science In- +stitute (Republic of Korea). +References +Bain, R. A. & Bardsley, J. N. 1972, Journal of Physics B Atomic Molecular +Physics, 5, 277 +Burton, M. G., Brand, P. W. J. 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A, 9226 +Article number, page 5 of 5 + diff --git a/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/load_file.txt b/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d520b66ec92eb74b6f302cc348136657b96e0d16 --- /dev/null +++ b/3NAyT4oBgHgl3EQf1vm8/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf,len=544 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 45358arxaa ©ESO 2023 January 3, 2023 Letter to the Editor Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Roueff1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Burton2, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Geballe3, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Abgrall1 1 Sorbonne Université, Observatoire de Paris, PSL University, CNRS, LERMA, F-92190, Meudon, France e-mail: evelyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='roueff@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='fr 2 Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DB, Northern Ireland e-mail: Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='Burton@Armagh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='uk 3 Gemini Obsevatory/NSF’s NOIRLab, 670 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' A’ohoku Place, Hilo, HI 96720, USA e-mail: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='geballe@noirlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='edu Accepted in Astronomy Astrophysics Letters on december 22, 2022 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Highly excited molecular hydrogen (H2) has been observed in many regions of shocked molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' A recently published K-band spectrum of Herbig-Haro 7 (HH7) contains several vibration-rotation lines of H2 from highly excited energy levels that have not been detected elsewhere, including a line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='179 µm identified as arising from the v=2 J=29 level, which lies above the dissociation limit of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' One emission line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='104 µm in this spectrum was unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We aim to complete the analysis of the spectrum of HH7 by including previously missing molecular data that have been recently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We re-analysed the K-band spectrum, emphasising the physics of quasi-bound upper levels that can produce infrared emission lines in the K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We confirm the identification of the 2 − 1 S (27) line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1785 µm and identify the line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm as due to the 1-0 S (29) transition of H2, whose upper level energy is also higher than the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' This latter identification, its column density, and the energy of its upper level further substantiate the existence of a hot thermal component at 5000 K in the HH7 environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The presence of the newly identified 1 − 0 S (29) line, whose quasi-bound upper level (v=1, J=31) has a significant spontaneous dissociation probability, shows that dissociation of H2 is occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The mechanism by which virtually all of the H2 in levels with energies from 20,000 K to 53,000 K is maintained in local thermodynamic equilibrium at a single temperature of ∼5,000 K remains to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' molecular hydrogen – interstellar medium – shocks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Introduction The interaction of the collimated outflow from the protostar SSV13 (Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1976) and the molecular cloud out of which it formed has produced a collection of Herbig-Haro (HH) ob- jects, HH7-HH11, in a more or less linear arrangement on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The most distant of these from SSV13, HH7, has a classic bow shock shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' It is bright in line emission from shock-excited vibrational states of molecular hydrogen (H2), first observed in the v = 1 − 0 S (1) transition by Zealey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1984), Harti- gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1989), and Garden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1990) and subsequently in vibrational levels 0 − 4 and rotational levels 1 − 15 by Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1989) and Fernandes & Brand (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' HH7 also emits strongly in pure rotational lines of H2 and CO (Neufeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Yuan & Neufeld 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Neufeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2000) as well as in [OI]63µ (Sperling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2020), Hα, [OI]λ6300, and [SII]λ6716 (Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The vibrationally excited H2 lines, observed mainly in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 µm interval, are emit- ted predominantly in the hottest shock-heated gas, while the pure rotational low-J transitions of H2 and the pure rotational transi- tions of CO, observed in the mid- and far-infrared, arise in a somewhat cooler gas downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Much more highly vibrationally and rotationally excited molecular hydrogen was found by Pike et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2016, hereafter P16) in a 3′′× 3′′region near the tip of the HH7 bow shock, in K-band spectra they obtained at a resolving power, R, of 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Figure 6 of their paper shows the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='01 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='45 µm spectrum of a 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='6×0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='9 area in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Their paper demonstrated the ex- istence in HH7 of a small percentage (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5%) of the emitting H2 at a temperature of ∼5,000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Subsequently, Geballe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2017) discovered the presence of small percentages of 5,000 K H2 in shocked gas at several locations in the Orion Molecular Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Giannini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2015) detected a similar phenomenon in another bright HH object, HH1, at a somewhat higher tempera- ture, ∼6300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' P16 identified a weak emission line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='179 µm in the HH7 spectrum as the 2 − 1 S (27) transition of H2, which arises from the upper level, v = 2, J = 29, whose energy is above the dissociation limit of the ground state of H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' this corresponds to 51,965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='84 K, using the latest measurements (Hölsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2019) and the Committee on Data for Science and Technology (CODATA) definition of fundamental constants (Tiesinga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The column density of H2 in the upper level could only be crudely estimated by P16, as the Einstein A coefficient for that transition was not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' P16 also reported the detection of a faint line near 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='104 µm, which they were unable to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Roueff & Abgrall (2022) have recently proposed a simple and efficient method for computing the emission spectrum pro- Article number, page 1 of 5 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='00741v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='GA] 2 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 45358arxaa duced by quasi-bound levels, providing accurate wavenumbers and Einstein emission coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The application to H2 al- lowed them to calculate the Einstein coefficient of the 2-1 S (27) transition and suggested that the line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='104 µm in HH7 is the 1 − 0 S (29) transition of H2, whose upper-state energy also lies above the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The present paper analyses these two high excitation lines in the light of the new theoretical developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Section 2 revisits the observations of HH7, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3 summarises the recent theoret- ical achievements, and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 4 contains the resulting extended observational analysis of the H2 line emission in HH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We pro- vide a discussion of our results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Observations A detailed description of the observations of HH7 and a re- duction of the spectral data have been given by P16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' In brief, the Gemini facility integral field spectrometer, the Near Infrared Field Spectrometer (NIFS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' McGregor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2003), was used at the Frederick C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Gillett Gemini North Telescope on Maunakea, Hawai’i, to obtain spectra of a 3′′× 3′′region near the tip of the HH7 bow shock, for program GN-2007B-Q-47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The angular res- olution of the spectra was 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Within this 3′′× 3′′region, the spectra showed H2 ro-vibrational line emission from upper-state levels covering a wide range of energies, including a dozen in the range 40, 000 − 50, 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Because some rotational energies and associated rotational quantum numbers of the upper levels of these lines are high (J ≳ 15), collisions rather than the absorp- tion of ultraviolet (UV) photons are probably the main producer of the populations in those rotational levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Somewhat lower values of J associated with high vibrational quantum numbers are commonly found in dense photon-dominated regions (PDRs) such as NGC 2023, the Orion Bar, S140, and IC63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' McCartney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' H2 is excited in PDRs by UV pumping, which is followed by electronic fluo- rescence, but the ∆J = ±1 selection rule for electronic transitions maintains J at values below ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1 P16 concentrated their analysis on the spectrum of the 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='6 × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='9 area shown in their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' the spectrum is plotted in their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The upper two panels of our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1 show in more de- tail two 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='01 µm wide portions of that spectrum, each contain- ing one of the two highly shock-excited H2 lines discussed in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Wavelength calibration employed the spectrum of an argon lamp and is accurate to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='00002 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The hor- izontal scales are vacuum laboratory wavelengths and as such can be directly compared with the theoretically calculated wave- lengths (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='The uppermost panel contains the previ- ously unidentified line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm along with the nearby 4 − 3 S (7) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Similarly, the middle panel contains the previously identified 2−1 S (27) line and the adjacent 5−4 S (15) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Spec- tral images of the four lines, extracted from the NIFS data cube, are shown in the bottom panel of the figure and demonstrate that, to within the limits imposed by the noise levels, the four emission lines have identical morphologies, which also match the morphology of the strong 1 − 0 S (1) line shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2 of P16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Based on the fluctuations in the baseline, we estimate the confidence of the detection of the 1 − 0 S (29) line to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The wavelengths of these two weak lines are slightly different than those reported by P16 and are more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1 We contacted K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Kaplan to check if the two transitions at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1785 µm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm were present in his PDR spectra obtained with the Im- mersion Grating INfrared Spectrometer (IGRINS), and they were not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Observational data showing highly excited H2 lines in HH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Top two panels: Spectra of a 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='6 × 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='9 area of HH7 in two narrow wave- length intervals, each containing a line of H2 from a quasi-bound energy level and one adjacent line, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 6 of P16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Vertical dashed lines are the line wavelengths calculated as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Bottom: Spec- tral images of the four lines shown above, extracted from the NIFS data cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The field of view is 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 × 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 and corresponds to the left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2 of P16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' the field centre corresponds to RA = 3:29:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='42, Dec = +31:15:27:45 (J2000), with an estimated uncertainty of 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Article number, page 2 of 5 2HH 1 Density Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 4-3 S(7) 0S(29) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='098 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='102 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='106 LabVacuumWavelength(um) 2HH res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 4 S(15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='176 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='178 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='182 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='184 Lab Vacuum Wavelength (um) S(15)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Roueff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' : Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Theoretical aspects Molecular quasi-bound levels correspond to states whose ener- gies lie above the dissociation limit of the ground state of the molecule but well below the dissociation energy of the electron- ically excited molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' For H2, the Schrödinger equation rele- vant to excited rotational levels is − ℏ2 2µ · d2 fv,J(R) dR2 + Vmod e f f (R, J) fv,J(R) = Ev,J fv,J(R), (1) where Vef f (R, J) = V(R) + ℏ2J (J+1) 2µR2 , with V(R) the ground state electronic molecular potential of H2 and µ = Mp/2 the nuclear reduced mass of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' H2 molecular potentials in eV as a function of the interatomic distance, R, expressed in atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The zero value corresponds to photo-dissociated H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The black curve is the electronic potential of the X 1Σ+ g ground state from Czachorowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2018) expressed in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The blue and red curves denote the effective potentials with J= 29 and J= 31, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The quasi-bound levels v = 2, J = 29 and v = 1, J = 31 are also displayed in blue and red, respectively, in the allowed ranges of interatomic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Figure 2 displays the electronic molecular potential of the X1Σ+ g ground state of H2 as well as the effective potentials cor- responding to J = 29 and J = 31, the two quasi-bound lev- els (sometimes referred to as shape resonances) previously men- tioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The presence of the centrifugal potential, ℏ2J (J+1) 2µR2 , signif- icantly modifies the shape of the electronic contribution, V(R), by reducing the potential well, shifting the minima to larger in- teratomic distances and exhibiting broad bump maxima above the dissociation limit, peaking near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Figure 2 also displays the resonant quasi-bound eigenval- ues, Er, which are located above the dissociation limit and are trapped inside the centrifugal barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The associated wave func- tion for each level has a non-vanishing probability in the inter- atomic range displayed, becomes vanishingly small after the sec- ond turning point when Er ≤ Vef f (R), and has an oscillatory be- haviour for large R when Er becomes larger than Vef f (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The associated quasi-discrete stationary states have complex energy eigenvalues, E = Er − (i Γ/2), where Er is the energy at reso- nance and Γ characterises the width of the level and determines its lifetime against dissociation, τ = ℏ/Γ, due to tunnelling from the quasi-bound to the continuum oscillatory dissociating state at large interatomic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Roueff & Abgrall (2022) computed the various resonance energy level positions of H2 and the corre- sponding emission spectrum arising from these levels by using the recent highly accurate molecular potential of the H2 ground state of Czachorowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2018) and extending the effective potential by a constant value from the maximum value of the po- tential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' This method allows one to use a standard numer- ical integration of the Schrödinger equation applied to strictly bound levels and has been demonstrated to be very precise for determining the resonant energy level positions and the emission rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' However, it does not allow a derivation of the widths or the dissociation lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Those are obtained through different methods based on scattering properties (Schwenke 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Selg 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' These computations predict wavelengths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1785 µm for the 2 − 1 S (27) transition and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm for the 1 − 0 S (29) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The predicted wavelengths for the two stronger lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 1 are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='10043 µm for 4−3 S (7) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='18179 µm for 5−4 S (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' As can be seen in the figure, all are in excellent agreement with the observed wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Therefore, we are confident in the previous identification of the 2 − 1 S (27) line by P16 and in our identification of the weak and previously unidentified feature at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm as the 1 − 0 S (29) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' These two transitions are the only lines in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='01 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='45 µm interval from quasi-bound levels that would have been detectable in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (We note in Table 1 the small Einstein A coefficient of the 2 − 0 Q(29) line at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='4007 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Column density analysis The analysis undertaken here follows that described in P16 for the H2 line emission from HH7 reported in that paper, with the addition of the 1–0 S (29) and 2–1 S (27) lines presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' A two-component Boltzmann distribution with temperatures Thot and Twarm was fitted to the column densities obtained from the de-reddened line intensities, Ni = Ni,hot + Ni,warm, (2) with each component described by a Boltzmann distribution at the corresponding temperatures, as per P16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We obtain Twarm = 1, 783 ± 20 K and Thot = 5, 133 ± 17 K, with 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5% of the total column of excited H2 gas in the warm component of the gas and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5% in the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' This com- pares to values of Twarm = 1, 803±12 K and Thot = 5, 200±12 K found without these two extra lines included in the analysis2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The additional lever arm provided by the two higher excitation energy levels has only led to a marginal decrease in the derived temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' in other words, the result is essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We conclude that the two quasi-bound H2 lines are well mod- elled by the same hot local thermodynamic equilibrium (LTE) component as per all lines measured in HH7 arising from energy levels ≥15,000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The level populations for the two quasi-bound lines are ∼ 10−5 times that of the v = 1, J = 3 upper level of the brightest H2 emission line, 1 − 0 S (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Discussion Table 1 summarises the present knowledge available for the two quasi-bound levels of H2 v = 2, J = 29 and v = 1, J = 31, that 2 The errors quoted here are the formal errors derived from the least squares fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Article number, page 3 of 5 J=29 J=31 0 dissociation limit 1 2 3 4 g 5 2 6 8 10 12 0 4 R (au)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 45358arxaa Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Properties of the two detected quasi-bound levels, v = 2, J = 29 and v = 1, J = 31, of H2 and their emission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Transition ˜ν λ A Ar τd Eqb upper label cm−1 µm s−1 s−1 s K 2 − 0 O(31) 1387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='2096 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='704E-11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 0 Q(29) 4165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='4007 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='592E-08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 0 S (27) 7034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='4216 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='240E-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 1 Q(29) 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1423 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='947E-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 1 S (27) 4590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1785 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='944E-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 2 S (27) 2399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1681 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='873E-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 2 − 3 S (27) 489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='60 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='4248 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='582E-10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='482E-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='130E12 676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0 1 − 0 Q(31) 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0658 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='452E-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='219E-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='083E06 1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 1 − 0 S (29) 4752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='101E-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='219E-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='083E06 1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 1 − 1 S (29) 2531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='9500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='872E-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='219E-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='083E06 1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 1 − 2 S (29) 586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='98 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='0364 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='963E-10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='219E-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='083E06 1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' ˜ν is the computed transition frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' λ is the corresponding vacuum wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' A is the sum of the electric quadrupole and the magnetic dipole contributions to the Einstein radiative emission coefficients of the transition from Roueff & Abgrall (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Ar is the total radiative decay probability, and τd is the dissociation lifetime of the upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Eqb upper is the quasi-bound upper level energy expressed in K above the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Level column densities, divided by their degeneracies, Ni/gi, plotted as a function of level energy, Ti, for the H2 lines measured in HH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' They are normalised to unity for the (v, J) = (1, 3) upper-state level at 6,952 K, which emits the 1–0 S (1) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The two blue points (in the lower right) are for the newly analysed 2–1 S (27) and 1–0 S (29) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The dashed red line shows the best two-temperature LTE fit, as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The upper level involved in the 2 − 1 S (27) transition at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1785 µm, 676 K above the dissociation energy of the ground state, is very stable against dissociation, whereas that of the 1 − 0 S (29) transition at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='1042 µm, located 845 K higher, has a dissociation probability of approximately five percent and a dissociative lifetime, τd = ℏ/Γd, resulting from quantum tun- nelling through the centrifugal barrier (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2) of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='083 × 106 s, corresponding to less than two months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' This indicates that the shock wave in HH7 is partially dissociative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3, the 5,000 K component represents a small percentage of the line-emitting H2 in HH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' As noted pre- viously, similar small percentages have been observed in HH1 and in the Orion molecular outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Figure 3 also shows that H2 in energy levels greater than ∼ 20,000 K above the ground state are populated only by this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' In the case of HH1, Gi- annini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2015) observed a wide range of neutral and ionised species emitting in close proximity to the H2, many at opti- cal wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Their analysis yields a temperature range of 8, 000 − 80, 000 K to account for the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' They further find that neutral and fully ionised regions coexist inside the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' However, for the heavily extincted H2 line emission from HH7 (AV = 12−28 mag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' P16), the species producing the optical emis- sion lines observed by Solf & Boehm (1987), Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1989), and Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2019) cannot be mixed with the H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' In view of the detections by Giannini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2015), P16, and Geballe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2017) of 5, 000 − 6, 000 K H2 in diverse envi- ronments, it seems likely that a small percentage of H2 existing at those temperatures is a common occurrence in collisionally shocked molecular gas, at least in cases where collisions between outflows and ambient molecular material occur at velocities of many tens of km s−1, as is the case for HH1, HH7, and the Orion Molecular Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' In addition, although transitions emitted from quasi-bound levels have only been detected towards HH7, we expect that they are present in HH1 and OMC-1 at roughly the same intensities relative to the stronger H2 lines, as in HH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' It is generally accepted that the maximum temperatures of nearly all of the vibrationally excited H2 in each of the above shocked clouds and in many others are suppressed by continu- ous shocks, in which the collisional acceleration of the ambient clouds and deceleration of the colliding outflows from the proto- stars are sufficiently gradual to heat the H2 only to temperatures of ∼2,000 K and prevent its dissociation (for more details, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' I of P16 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The existence of H2 at a range of lower temperatures in gas cooling behind the contin- uous shocks, which has been demonstrated by observations of pure rotational lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Neufeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 2019), is also unsurpris- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' However, it seems remarkable that virtually all of the highly ro-vibrationally excited H2 in levels with energies from 20,000 K to 53,000 K is maintained in LTE at a single temperature of ∼5,000 K, and that there is virtually no H2 at temperatures be- tween 2,000 K and 5,000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The mechanism that produces this bimodal temperature distribution is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The location and morphology of the 5,000 K gas also is un- clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The gas could be located in thin (currently unresolvable) sheets where the molecular cloud is being collisionally acceler- ated, the wind is being collisionally decelerated, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Its line emission could alternatively also be occurring in small clumps Article number, page 4 of 5 100 N[T, 1 = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5% warm N[Thot] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='5% 10-2 10-4 10-5 10-6 0 1×10° 2×10° 3×10° 4×10° 5×104 6×104 Energy Level (K)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Roueff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' : Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7 of unusually hot and/or unusually dense gas scattered along the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Comparisons of the velocity profiles of lines origi- nating from levels whose populations are dominated by the gas at 5,000 K with those from levels dominated by the 2,000 K component, at higher spectral resolution than has been employed to date, might reveal small differences and constrain the rela- tive locations of the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The good fit of the v=1, J = 31 column density to the fit to the population-energy di- agram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' 3) indicates that dissociation is taking place in the 5,000 K gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' One can consider if the short lifetime of the v=1, J=31 quasi- bound level indicates a significant continuous reformation of molecular hydrogen in the gas phase at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We have estimated the formation rate of H2 through radiative asso- ciation via that resonance level, i, H + H ↔ H2i → H2 + hν, following the theory of Bain & Bardsley (1972), to be αres i = � 2πℏ2 MkT �3/2 (2I + 1)(2Ji + 1) Ai r Ad Air + Ad exp(−Ei/kT), (3) where Ad = 1/τ and M is the reduced mass of the colliding atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The contribution of v = 1, J = 31 with I = 1 and similar values of Ar and Ad is the most efficient by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' However, the derived value for its contribution at 5000K is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='37 × 10−30 cm3 s−1, which is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Although it is difficult to assess the direct implication of the measurable presence of these quasi-bound H2 states for shock chemistry, their detections confirm the predictability of theoreti- cal computations based on highly accurate potential curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' The physical conditions associated with the astrophysical environ- ments in which their lines are emitted may not be reproducible in the laboratory due to their very large rotational quantum num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Thus, they probably offer the only way to probe these lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (1996) introduced quasi-bound levels of H2 in their master equation studies of collisional excitation of H2 by H and specifically mentioned the v = 2, J = 29, and v = 1, J = 31 quasi-bound levels detected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' However, they find that the highly excited rotational levels are not thermally popu- lated for the range of physical conditions that they considered, in contrast to what astronomical observations have revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Fi- nally, we note that the contribution of quasi-bound levels to the partition function of H2 and its isotopologues has been recently computed by Zúñiga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' (2021) using the same potential as us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Kaplan for having searched the two transi- tions in his IGRINS spectra of various PDRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' We are grateful to the referee for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' acknowledge support by the Programme Na- tional de Physique et de Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content='This research is based in large part on observations obtained at the international Gemini Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' a program of NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' which is managed by the Association of Universities for Re- search in Astronomy (AURA) under a cooperative agreement with the National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' on behalf of the Gemini Observatory partnership: the Na- tional Science Foundation (United States),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' National Research Council (Canada),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Agencia Nacional de Investigación y Desarrollo (Chile),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Ministerio de Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Tecnología e Innovación (Argentina),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Ministério da Ciência,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Tecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' Ino- vações e Comunicações (Brazil),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' and Korea Astronomy and Space Science In- stitute (Republic of Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' References Bain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' & Bardsley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} +page_content=' N.' metadata={'source': 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+page_content=' A, 9226 Article number, page 5 of 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQf1vm8/content/2301.00741v1.pdf'} diff --git a/4NE0T4oBgHgl3EQfeQCD/vector_store/index.faiss b/4NE0T4oBgHgl3EQfeQCD/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..dc3689bcc41e3724b7195408b0c7d00ea549ddd8 --- /dev/null +++ b/4NE0T4oBgHgl3EQfeQCD/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d2785e7874ce4b067b9798448d02dbd4c965ff8983c9dbef28750c24cd4a3bf +size 1376301 diff --git a/4NE0T4oBgHgl3EQfeQCD/vector_store/index.pkl b/4NE0T4oBgHgl3EQfeQCD/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1c4778b78ddb5bbd62f88c13dbd1b07b03825bfe --- /dev/null +++ b/4NE0T4oBgHgl3EQfeQCD/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/2301.02863v1.pdf.txt b/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/2301.02863v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55d87cf8103c29c9e811a533b09d79b01f1dcf22 --- /dev/null +++ b/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/2301.02863v1.pdf.txt @@ -0,0 +1,1952 @@ +arXiv:2301.02863v1 [math.OC] 7 Jan 2023 +Noname manuscript No. +(will be inserted by the editor) +A Regularized Limited Memory Subspace Minimization Conjugate +Gradient Method for Unconstrained Optimization +Wumei Sun1 · Hongwei Liu1 · Zexian Liu2 +Received: date / Accepted: date +Abstract In this paper, based on the limited memory techniques and subspace minimization conjugate gra- +dient (SMCG) methods, a regularized limited memory subspace minimization conjugate gradient method is +proposed, which contains two types of iterations. In SMCG iteration, we obtain the search direction by min- +imizing the approximate quadratic model or approximate regularization model. In RQN iteration, combined +with regularization technique and BFGS method, a modified regularized quasi-Newton method is used in +the subspace to improve the orthogonality. Moreover, some simple acceleration criteria and an improved +tactic for selecting the initial stepsize to enhance the efficiency of the algorithm are designed. Additionally, +an generalized nonmonotone line search is utilized and the global convergence of our proposed algorithm +is established under mild conditions. Finally, numerical results show that, the proposed algorithm has a +significant improvement over ASMCG PR and is superior to the particularly well-known limited memory +conjugate gradient software packages CG DESCENT (6.8) and CGOPT(2.0) for the CUTEr library. +Keywords Limited memory · Subspace minimization conjugate gradient method · Orthogonality · +Regularization model · Quasi-Newton method +Mathematics Subject Classification (2010) 49M37 · 65K05 · 90C30 +1 Introduction +Consider problem +min +x∈Rn f(x), +(1) +where f : Rn → R is a continuously differentiable nonlinear function. +Wumei Sun +E-mail: sunwumei1992@126.com +Hongwei Liu � +E-mail: hwliuxidian@163.com +Zexian Liu +E-mail: liuzexian2008@163.com +1 School of Mathematics and Statistics, Xidian University, Xi’an 710126, China +2 School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China + +2 +Wumei Sun1 et al. +Throughout the article, we use the following notations. sk−1 = xk − xk−1, fk = f(xk), gk = g(xk), +yk−1 = gk−gk−1, ∥·∥ represents the Euclidean norm and λmax denotes the maximum eigenvalue. Moreover, +dist{x, S} = inf{∥y − x∥, y ∈ S}, where x ∈ Rn and S ∈ Rn. +Nonlinear conjugate gradient(CG) method is a well-known method for solving the problem (1), which +main iteration is +xk+1 = xk + αkdk, k = 0, 1, 2, · · · , +(2) +where xk is the kth iteration point, αk > 0 is the stepsize and dk is the search direction obtained by +d0 = −g0, dk = −gk + βkdk−1, k ≥ 1, +(3) +where gk is the gradient of f(xk) and βk is the conjugate parameter. +It is shown in theory that the convergence and numerical performance variation of different CG meth- +ods depend on the selection of conjugate parameters. Some very classical choices of the conjugate param- +eter βk are Fletcher-Reeves(FR) [9], Polak-Ribi`ere-Polyak(PRP) [30,31], Dai-Yuan(DY) [7] and Hestenes- +Stiefel(HS) [16], and are given by +βF R +k += ∥gk+1∥2 +∥gk∥2 , +βP RP +k += gT +k+1yk +∥gk∥2 , +βDY +k += ∥gk+1∥2 +dT +k yk +, +βHS +k += gT +k+1yk +dT +k yk +. +CG algorithms have evolved considerably, and some well-known CG packages such as CG DESCENT [12, +14] and CGOPT [5] have been proposed in recent years. Other recent related studies on nonlinear CG +algorithms can be found in [4,13]. +The subspace minimization conjugate gradient (SMCG) algorithm, as a generalization of the CG algo- +rithm, has received much attention from scholars [1,37], which can be traced back to the work of Yuan and +Stoer [39]. The search direction of SMCG method is obtained by minimizing the following problem: +min +d∈Ωk +gT +k d + 1 +2dT Bkd, +(4) +where Ωk is a subspace spanned by the vectors gk and sk−1, i.e., Ωk = Span{gk, sk−1}, and Bk ∈ Rn×n is +an approximation of Hessian matrix, which is positive definite and symmetric. Then the search direction d +is given by +d = ugk + vsk−1, +(5) +where u and v are both real parameters. Substituting (5) to (4) and combined with the standard secant +equation Bksk−1 = yk−1, formula (4) is reorganized as follows: +min +u,v∈R + + ∥gk∥2 +gT +k sk−1 + + +T  + u +v + + + 1 +2 + + u +v + + +T  + +ρk +gT +k yk−1 +gT +k yk−1 sk−1yk−1 + + + + u +v + + . +(6) +where ρk ≈ gT +k Bkgk. +On the basis of the Barzilai-Borwein(BB) method [2], Dai and Kou [6] proposed an effective BBCG3 +method for strictly convex quadratic minimization problem. Afterwards, based on BBCG3 method, Liu and +Liu [26] proposed SMCG BB method for solving general unconstrained optimization problems. Motivated +by SMCG BB method, some efficient SMCG methods [20,21,36,42] were later proposed, among which + +Title Suppressed Due to Excessive Length +3 +the method based on the regularization model presented by Zhao et al. [42] is the best in the numerical +performance. +The nonlinear CG method is very effective for unconstrained optimization problems. However, the +convergence of the algorithm can be very slow for some ill-posed problems and even for quadratic problems +with very small dimensions, which may be due to the loss of orthogonality [15]. Hager and Zhang [15] pointed +out theoretically that the generated successive gradients either in the CG method or the L-BFGS method +for the quadratic test problem should be orthogonal. Yet, Hager and Zhang [15] observed that, when solving +the quadratic strictly convex minimization problem PALMER1C in the CUTEr library [10], the CG method +loses orthogonality due to the rounding errors, while L-BFGS method preserves the orthogonality. In view +of this, they developed the limited memory CG method (CG DESCENT(6.8)) to correct the possible loss +of orthogonality in ill conditioned optimization problems. For the test problems in the CUTEr library [10], +their performance results indicated that CG DESCENT(6.8) has an significant improvement over their +previously proposed package CG DESCENT(5.3). +Although CG DESCENT(6.8) [15] is an efficient method for unconstrained optimization, it still suffers +from the following shortcomings: +(i) In the numerical implementation, the AWolfe line search [14] utilized in the algorithm CG DESCENT(6.8) +does not guarantee global convergence. +(ii) CG DESCENT(6.8) contains the following three pre-conditioners, corresponding to three different it- +erations: +Pk = I, Pk = Zk ˆB−1 +k+1ZT +k , Pk = Zk ˆB−1 +k+1ZT +k + σk ¯Zk ¯ZT +k , +(7) +where σk is determined by (4.2) of [15], ˆBk+1, Zk and ¯Zk are given by the matrices in literature [15]. These +three pre-conditioners make the algorithm CG DESCENT(6.8) look complex. +(iii) In the convergence analysis, the algorithm CG DESCENT(6.8) needs to impose the following assump- +tions on the pre-conditioners: +∥Pk∥ ≤ γ0, gT +k+1Pkgk+1 ≥ γ1∥gk+1∥2, dT +k P −1 +k +dk ≥ γ2∥dk∥2, +(8) +where γ0 > 0, γ1 > 0 and γ2 > 0. These assumptions are comparatively strict and difficult to be verified in +actual practice. +To address the above-mentioned shortcomings, Liu et al. [27] presented an improved Dai¨CKou CG +algorithm called CGOPT(2.0), which combines limited memory technology and Dai-Kou CG method. +In CGOPT(2.0) [27], they utilized a modified quasi-Newton method to restore the lost orthogonality, and +established the convergence of CGOPT(2.0) with fewer assumptions. Some numerical experiments indicated +that CGOPT(2.0) is better than the famous CG software package CG DESCENT(6.8) [15]. +In view of the above discussion, a regularized limited memory subspace minimization conjugate gradient +method on the basis of SMCG method and limited memory technique is studied in this paper. To recover +orthogonality, we propose a modified regularized quasi-Newton method. The major contributions of this +paper are the following. +1. A regularized limited memory subspace minimization conjugate gradient algorithm is proposed, which +combines limited memory technology and SMCG method. + +4 +Wumei Sun1 et al. +2. Based on the idea of regularization and BFGS method, an improved regularized quasi-Newton method +is exploited to improve orthogonality. +3. Some simple acceleration criteria and an improved initial stepsize selection strategy are designed to +enhance the efficiency of the algorithm. Additionally, an generalized nonmonotone line search condition +is presented, which may be regarded as an extension of the Zhang-Hager’s [41] nonmonotone line search. +4. The convergence of the method is built under mild conditions and the corresponding numerical perfor- +mance shows that the new method is much more effective than the existing methods. +The structure of the paper is as follows. In Section 2, we describe the detail of the regularized limited +memory subspace minimization conjugate gradient algorithm, including the direction selection of SMCG +iteration and regularized Quasi-Newton iteration and an effective acceleration technique. Moreover, the +decision of the initial step size and the generalized nonmonotone Wolfe line search are also given in this +section. In Section 3, some important properties of the search direction are analyzed and the global con- +vergence of the proposed algorithm is established. Numerical experiments for algorithm comparison are +showed in Section 4. Conclusions are given in the last section. +2 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm +In the section, combining the idea of subspace minimization and regularization quasi-Newton method, +we present a regularized limited memory subspace minimization conjugate gradient algorithm. Firstly, +we give the choices of search direction under different iterations. Subsequently, we develop a very effec- +tive acceleration technique, a modified initial step selection strategy and generalized nonmonotonic line +search technology to optimize the performance of the proposed algorithm. Finally, the details of algorithm +RL SMCG are described. +2.1 Direction Selection of SMCG Iteration and Regularized Quasi-Newton Iteration +The regularized limited memory subspace minimization conjugate gradient method mainly contains two +kinds of iterations which are SMCG iteration and regularized quasi-Newton(RQN) iteration, respectively. +Furthermore, the search direction derivation of the two iterations is also different. +2.1.1 SMCG iteration +The search direction selection of SMCG iteration is closely related to the properties of the objective function +f(x) at the iteration point xk. By reference [3,38], defined +tk = +���2 +� +fk−1 − fk + gT +k sk−1 +� +/ +� +sT +k−1yk−1 +� +− 1 +��� , +(9) +to describe how f(x) approaches a quadratic function on a line segment between xk−1 and xk. Literature +[24] indicates that if the condition +tk ≤ ¯ξ4 or +� +tk ≤ ¯ξ5 and tk−1 ≤ ¯ξ5 +� +, +(10) + +Title Suppressed Due to Excessive Length +5 +is satisfied, where ¯ξ4 and ¯ξ5 are the smaller positive constants and ¯ξ4 < ¯ξ5, f(x) may be near to a quadratic +function on a line between xk−1 and xk. Moreover, According to [32], we know that if the following condition +¯ξ1 ≤ sT +k−1yk−1 +∥sk−1∥2 ≤ ∥yk−1∥2 +sT +k−1yk−1 +≤ ¯ξ2, +(11) +is satisfied, then the condition number of the Hessian matrix of the normal function may be not very large, +here ¯ξ1 and ¯ξ2 are positive constants. +Similar to [42], based on some certain properties of the function f(x) at the current point xk, we derive +different search direction by dividing it into the following four cases. +(i) If the condition (11) is satisfied while the condition (10) are not, this implies that the quadratic +model may not be able to approach the objective function f(x) well at the present iteration point xk. Then, +search direction dk will be obtained by minimizing the following cubic regular subproblem, i.e. +min +dk∈Ωk mk (dk) = dT +k gk + 1 +2dT +k Bkdk + 1 +3σk ∥dk∥3 +Bk , +(12) +where Ωk is a subspace spanned by the vectors gk and sk−1, Bk ∈ Rn×n is an approximation of Hessian +matrix, which is positive definite and symmetric and satisfying the secant condition Bksk−1 = yk−1, σk ≥ 0 +is an adaptive regularization parameter obtained from interpolation condition and dk is determined by +dk = ukgk + vksk−1, +(13) +where vk and uk are parameters to be established. Obviously, we could obtain (12) by giving (4) a weighted +regularization term 1 +3σk ∥dk∥3 +Bk. Substituting (13) to (12), it is easy to obtain that (12) is equivalent to +min +uk,vk∈R + + ∥gk∥2 +gT +k sk−1 + + +T  + uk +vk + + + 1 +2 + + uk +vk + + +T +¯Bk + + uk +vk + + + σk +3 +������ + + uk +vk + + +������ +3 +¯ +Bk +. +(14) +where ¯Bk = + + +ρk +gT +k yk−1 +gT +k yk−1 sk−1yk−1 + + is a positive definite and symmetric matrix, ρk is an estimate of +gT +k Bkgk. Similar to BBCG3 [6], we also use +3 +2 +∥yk−1∥2 +sT +k−1yk−1 I to estimate Bk in the term ρk, which means +ρk = 3 +2 +∥yk−1∥2 +sT +k−1yk−1 ∥gk∥2. Then, by solving problem (14) we obtain the following solutions about uk and vk: + + uk +vk + + = + + +1 +(1+σk(̟∗))∆k +� +gT +k yk−1gT +k sk−1 − sT +k−1yk−1∥gk∥2� +1 +(1+σk(̟∗))∆k +� +gT +k yk−1∥gk∥2 − ρkgT +k sk−1 +� + + , +(15) +among them, +∆k = +������ +ρk +gT +k yk−1 +gT +k yk−1 sk−1yk−1 +������ += ρksk−1yk−1 − (gT +k yk−1)2 > 0, +(16) +σk and ̟∗ are the same as those in literature [42], which will not be repeated here. +(ii) If both conditions (11) and (10) hold, this indicates that the objective function f(x) may approach +the quadratic model at the current iteration point xk. Since that is the case, let σk = 0, i.e. we consider deriv- +ing the search direction by solving the minimization problem (6). Like (i), we choose ρk = 3 +2 +∥yk−1∥2 +sT +k−1yk−1 ∥gk∥2 +and ∆k is determined by (16), then we obtain the following unique solution of quadratic approximate + +6 +Wumei Sun1 et al. +problem (6): + + ¯uk +¯vk + + = + + +1 +∆k (gT +k yk−1gT +k sk−1 − sT +k−1yk−1∥gk∥2) +1 +∆k (gT +k yk−1∥gk∥2 − ρkgT +k sk−1) + + , +(17) +here the search direction is calculated by dk = ¯ukgk + ¯vksk−1, where ¯uk and ¯vk are determined by (17). +(iii) If condition (11) is not satisfied and the conditions +���gT +k yk−1gT +k sk−1 +��� ≤ ¯ξ3sT +k−1yk−1∥gk∥2 and sT +k−1yk−1 ≥ ¯ξ1∥sk−1∥2, +(18) +are satisfied, where 0 ≤ ¯ξ3 ≤ 1, the condition number of the Hessian matrix may be lager, hence the search +direction obtained in cases (i) and (ii) may not be better. However, the condition (18) can ensure sufficient +descent and linear growth in HS conjugate gradient method. Moreover, because of the finite termination +nature of the HS conjugate gradient method for solving exact convex quadratic minimization problems, this +choice of direction allows for faster convergence of the algorithm. Then, in this case, the search direction is +determined by (3) and βk = βHS +k +. +(iv) If neither condition (11) nor (18) holds, then we pick the following direction, i.e. : +dk = −gk. +(19) +In summary, the search direction in the SMCG iteration can be described as in the following: +dk = + + + + + + + + + + + + + + + +ukgk + vksk−1, +if (11) holds and (10) does not hold, +¯ukgk + ¯vksk−1, +if (11) holds and (10) holds, +−gk + βHS +k +dk−1, +if (11) does not hold and (18) holds, +−gk, +if neither (11) nor (18) holds, +(20) +where uk and vk are determined by (15); ¯uk and ¯vk are determined by (17). +If the successive gradients have orthogonality or the lost orthogonality is restored, the algorithm performs +SMCG iteration. On the contrary, if the orthogonality is lost, the iteration will turn to the following +regularized quasi-Newton iteration to improve the orthogonality. +2.1.2 Regularized Quasi-Newton(RQN) iteration +When the successive gradients lose their orthogonality, the iteration switches from SMCG iteration to RQN +iteration. In other words, a modified regularized BFGS algorithm in subspace Sk is proposed to restore the +orthogonality, where Sk is a subspace generated by the following limited memory m search directions +Sk = span {dk−1, dk−2, · · · , dk−m} , +where m > 0 and m is the number of limited memory. In this article, the limited memory m selected in our +algorithm does not exceed 11. Then, as soon as orthogonality is corrected, the RQN iteration is terminated +and the SMCG iteration is triggered immediately. +First, we introduce some preparations for turning to RQN iteration. Let Sk ∈ Rn×m be a matrix which +has columns consisting of dk−1, dk−2, · · · , dk−m. In similar fashion to limited memory CG method [15], we +also assume that columns of Sk are line-independent. Let the QR factorization of Sk be Sk = Zk ¯Rk, where + +Title Suppressed Due to Excessive Length +7 +the columns of Zk ∈ Rn×m form the normal orthogonal bases for subspace Sk and ¯Rk ∈ Rm×m is the +upper triangular matrix with positive diagonal terms. +If gk is included almost in subspace Sk, then we think that the orthogonality property of the algorithm +may be lost. In this case, we interrupt the SMCG iteration and move to minimize the objective function in +the subspace Sk: +min +z∈Sk f(xk + z). +(21) +The solution to the subspace problem (21) will improve the orthogonality and guide us to a suitable search +direction that will lead us out of the subspace Sk. Similar to [15], we utilize the distance from gk to subspace +Sk to judge whether orthogonality is lost. If the condition +dist {gk, Sk} ≤ ˜η0∥gk∥ +(22) +is satisfied, where 0 < ˜η0 < 1 and ˜η0 is small, we think gk is almost contained in Sk, it means that the +orthogonality of the successive gradients has lost. Then, we switch to RQN iteration to solve the subspace +problem (21) until the gradient is nearly orthogonal enough to the subspace to meet the condition +dist {gk, Sk} ≥ ˜η1∥gk∥, +(23) +where 0 < ˜η0 < ˜η1 < 1. At this time, the algorithm iteration will go away subspace Sk and turn to the +SMCG iteration. Because the column of Zk is the orthonormal basis of Sk, it’s not hard to know from the +definition of dist {gk, Sk} that (22) and (23) can be expressed as +� +1 − ˜η2 +0 +� +∥gk∥2 ≤ +���ZT +k gk +��� +2 +, +(24) +and +� +1 − ˜η2 +1 +� +∥gk∥2 ≥ +���ZT +k gk +��� +2 +. +(25) +In [15], Hager and Zhang utilized the limited memory BFGS (L-BFGS) [22,28] method to solve the subspace +problem (21) for restoring the orthogonality, and achieved better numerical results. However, it should +be noted that the convergence analysis of the limited memory CG method [15] requires imposing strict +assumptions (8) on the preprocessors (7). Because the dimension m of the chosen subspace Sk is usually +small and when orthogonality is lost, the properties of the function at the iteration point maybe not +very good. Based on these, we consider a regularized L-BFGS method in the subspace Sk for solving the +subproblem (21). +The search direction of general quasi-Newton method [40] for unconstrained optimization (1) is the +form of dk = −B−1 +k gk, where Bk is a positive definite and symmetric approximation to the Hessian matrix. +As one of the most popular methods of quasi-Newton method, L-BFGS method stores the approximate +Hessian matrix of the objective function using small memory and computes the search direction dk using +the nearest m vector pairs of (sk−i, yk−i), i = 0, 1, . . . , m − 1. + +8 +Wumei Sun1 et al. +Ueda and Yamashita [35] presented a regularized Newton method for nonconvex unconstrained opti- +mization, whose search direction dk is obtained by solving the following linear equations: +� +∇2f(xk) + µI +� +dk = −∇f(xk), +(26) +where µ > 0 is referred to as the regularized parameter. The regularized Newton method [35] generally +defaults to a step size of 1, and global convergence is guaranteed by controlling the parameter µk. However, +as a type of Newton method, the regularized Newton method in [35] must solve the Hessian matrix of +f which is particularly computationally complex. To address this drawback, some scholars proposed the +regularized limited memory BFGS-type method [33,23] for solving unconstrained optimization problems, +i.e. the search direction dk is the solution of the following equations +(Bk + µI) dk = −∇f(xk), +(27) +where matrix Bk is an approximate Hessian determined by a particular quasi-Newton method. Regular- +ization technology can effectively improve the efficiency of quasi-Newton method in solving ill-conditioned +problems. Nevertheless, when computing Bk by the L-BFGS method, it is very hard to calculate (Bk + µI)−1. +Hence, motivated by [34], we present a regularized quasi-Newton method which combines the BFGS method +with the regularized technique to improve orthogonality in the m-dimensional subspace Sk. In this paper, +we consider Bk + µI as an approximation of ∇2f(xk) + µI. Because the matrix Bk is the approximate +Hessian of f(xk) and Bk + µI can be used as an approximate Hessian of f(xk) + µ +2 ∥x∥2. At this point, we +utilize (sk, yk(µ)) instead of (sk, yk), where +yk(µ) = (∇f(xk+1) + µxk+1) − (∇f(xk) + µxk) = yk + µsk. +Note that the regularized BFGS method stores as many vector pairs as the traditional BFGS method and +hence it does not require additional memory. +In [19], a effective BFGS quasi-Newton method for solving nonconvex unconstrained minimization was +proposed by Li and Fukushima [19], in which the matrix Bk+1 is updated by +Bk+1 = + + + +Bk − BksksT +k Bk +sT +k Bksk ++ ykyT +k +sT +k yk , +if +sT +k yk +∥sk∥2 > υ∥gk∥α, +Bk, +otherwise , +where υ > 0 and α > 0. Some recent advances about modified BFGS method can be found in [18,11,34]. +Inspired by the quasi-Newton methods described above, we propose an improved regularized BFGS +method to solve the subproblem (21) in subspace Sk. +Remark 1. In what follows, the variables with hats belong to subspace Sk , distinguished from the ones +found in the full space Rn. +Let ˆx = (ˆx1, ˆx2, · · · , ˆxm, )T ∈ Rm. The subproblem (21) can be expressed as +min +ˆx∈Rm f(xk + ˆx1dk−1 + ˆx2dk−2 + · · · + ˆxmdk−m). +(28) + +Title Suppressed Due to Excessive Length +9 +Similar to [27], because the regularized quasi-Newton directions in the subspace Sk always transform to +the full space Rn and QR decomposition of matrix Sk, we can obtain dk = Zk ˆdk, ˆgk = ZT +k gk, ˆyk = ZT +k yk, +ˆsT +k ˆyk = sT +k yk, ∥ˆsk∥2 = ∥sk∥2 and ˆfk = fk. +Let Bk(µ) = Bk + µI, then inspired by Li and Fukushima [19], we develop an improved regularized +BFGS method to solve the above subproblem (28) with a search direction of the form +ˆdk+1 = − ˆB−1 +k+1(µ)ˆgk+1, +(29) +where ˆBk+1(µ) is given by +ˆBk+1(µ) = + + + +ˆBk(µ) − +ˆ +Bk(µ)ˆskˆsT +k ˆ +Bk(µ) +ˆsT +k ˆ +Bk(µ)ˆsk ++ ˆyk(µ)ˆyT +k (µ) +ˆsT +k ˆyk(µ) , +if mod(k, l) ̸= 0 and ˆsT +k ˆyk(µ) +ˆsT +k ˆsk +≥ υ, +ˆI, +otherwise , +(30) +where υ > 0, mod(k, l) ̸= 0 represents the remainder for k modulo l, ˆyk(µ) = ˆyk + µˆsk and µ > 0 is an +important regularized parameter. The condition mod(k, l) ̸= 0 means the matrix ˆBk(µ) will be reset to +the identity matrix ˆI after updating l times, which ensures the good convergence of the algorithm. In the +paper, we set l = max(m2, 20). Obviously, ˆsT +k ˆyk(µ) > 0, and as soon as the matrix ˆBk(µ) is symmetric and +positive definitive, it is not hard to prove that the matrix ˆBk+1(µ) is symmetric and positive definitive. +As a very important regularization parameter, µ is closely related to the convergence analysis of the +regularized BFGS method. In this paper, the idea of the trust-region radius is used to find the suitable +search direction by controlling µ, in other words, The ratio of objective function value reduction to model +function value reduction is utilized. Then, give the definition of a ratio function rk( ˆdk, µ) as follows +rk( ˆdk, µ) = +ˆf(xk) − ˆf(xk + αk ˆdk) +ˆf(xk) − ˆqk( ˆdk, µ) +, +(31) +where ˆqk : Rm × R → R is a function of the form +ˆqk( ˆdk, µ) = ˆf(xk) + αkˆgT +k ˆdk + 1 +2α2 +k ˆdT +k ˆBk(µ) ˆdk. +(32) +Then, if the ratio function rk( ˆdk, µ) is relatively large, this means that compared with the reduction of the +model function, the reduction of the objective function is large enough, we choose to reduce the parameter +µ. On the flip side, if the ratio function rk( ˆdk, µ) is relatively small, i.e., ˆf(xk) − ˆf(xk + αk ˆdk) is small, +we will increase µ. In addition, to ensure that the algorithms converge well, we limit µ to an interval, i.e. +0 < µmin < µ < µmax. In general, if the next iteration point is closer to the current iteration point, the +reduction of the function value may not be obvious. At this time, we hope to get a new iteration point by +modifying the search direction, then the search direction improved by regular parameter µ may be a good +choice. Therefore, if ∥ˆsk∥2 ≤ ˆτ (ˆτ > 0), our choice and update of µ are as follows: +µk+1 = + + + +max {µmin, σ1µk} , +if rk( ˆdk, µ) ≥ σ3, +min {µmax, σ2µk} , +otherwise, +(33) +where 0 < σ1 ≤ 1, σ2 > 1 and 0 < σ3 ≤ 1. Otherwise, we choose µ = 0, i.e., the regularized BFGS method +is reduced to a general BFGS method. + +10 +Wumei Sun1 et al. +Remark 2. In order to simplify the symbol and facilitate writing, we still record the updated symbol +µk+1 as µ. +In the process of algorithm implementation, the search direction (29) in subspace Sk always converts +to the full space Rn at each RQN iteration, i.e., +dk+1 = −Pkgk+1, +(34) +where +Pk = Zk ˆB−1 +k+1(µ)ZT +k +(35) +and ˆBk+1(µ) is given by (30). +In Section 3, we will show that matrices ˆBk+1(µ) and Pk have some good properties in the RQN +iteration, which is critical for the convergence analysis. +2.2 An Effective Acceleration Technique +In order to optimize the performance of the algorithm, Sun et al. [32] proposed an acceleration technique, +which replaces (2) with the following new iterative form +xk+1 = xk + ¯ηkαkdk, +(36) +where ¯ηk ≥ 0 is an acceleration parameter obtained from an interpolation function. In view of the numerical +effect of the acceleration technique, our algorithm also takes it into account. Similar to reference [32], we +minimize the following interpolation function to get the acceleration parameter ¯ηk: +¯ηk = arg min q(ϕk(¯η)), +(37) +where ¯η ≥ 0, ϕk(¯η) = f(xk + ¯ηαkdk), and q(ϕk(¯η)) represents the interpolation function defined by ϕk(¯η). +In the paper, we consider minimizing the quadratic interpolation function [29] q(ϕk(0), ϕ′ +k(0),ϕ′ +k(1)), then, +¯ηk = arg min q(ϕk(0), ϕ′ +k(0), ϕ′ +k(1)), +(38) +By minimizing (38) we have +¯ηk = −¯ak +¯bk +, ¯bk ≥ ¯ǫ, +(39) +where ¯ak = αkgT +k dk, ¯bk = αk(g¯z − gk)Tdk, g¯z = ∇f(¯z), ¯z = xk + αkdk and ¯ǫ > 0 is a small constant. +We propose the following acceleration criterion, which is simpler than the rule in reference [32], that is +¯bk ≥ ¯ǫ, ∥s¯z∥2 ≤ ¯τ, ∥gk∥2 ≤ ˆτ, |¯tk+1| < ¯c, and |sT +k g¯z| ≥ Max(ς, ¯ς · ¯bk) +(40) +where ¯ǫ, ¯τ, ˆτ, ¯c, ς and ¯ς are all small positive constants, ¯bk = αk(g¯z − gk)T dk, s¯z = ¯z − xk, ¯z = xk + αkdk, +|¯tk+1| = | 2(fk−f¯z+gT +¯z s¯z) +sT +¯z g¯z +− 1|, f¯z = f(¯z) and g¯z = ∇f(¯z). When the condition (40) holds, we accelerate +the algorithm and update the relevant variables. In addition, one of the necessary conditions for successful +acceleration is that the trial iteration point must satisfy the line search condition. Therefore, if the algorithm + +Title Suppressed Due to Excessive Length +11 +accelerates successfully, update the iteration point xk+1 by using (36). Otherwise the algorithm acceleration +fails and returns to the original algorithm, at which point ¯ηk = 1, update the iteration point xk+1 with (2). +In reference [32], the acceleration criterion is divided into three cases, which seems to be more complex, +while our acceleration criterion has only one case and the form is simpler. +2.3 Choices of the Initial Stepsize and the Generalized Nonmonotone Wolfe Line Search +It is well known that the design of the search direction and the conditions of the line search are two +critical factors which affect the efficiency of the line search algorithm. In this subsection, we will develop +an improved nonmonotone Wolfe line search which can be regarded as an extension of the Zhang-Hager’s +[41] nonmonotone line search. In addition, an improved initial step selection strategy is designed. +For the sake of convenience, we express the one-dimensional line search function as +φk(α) = f(xk + αdk), α ≥ 0. +The choice of the initial stepsize α0 +k is of great importance for a line search in an optimization method. For +the Newton-like methods, choosing the initial step α0 +k = 1 is important to speed up convergence. For the +conjugate gradient methods, it is essential to use information from the current iteration of the problem to +make initial guesses [29]. In the conjugate gradient method, there have been various ways to choose the +initial stepsize, for example, see [5,12,15,29]. However, it did not have an agreement on which is the best. +In particular, Hager and Zhang [15] select the initial step in CG DESCENT as below: +α0 +k = + + + +arg min ¯q +� +φk (0) , φ′ +k (0) , φk (¯τ1αk−1) +� +, if φk (¯τ1αk−1) ≤ φk (0) , +¯τ2αk−1, +otherwise, +(41) +where ¯q +� +φk (0) , φ′ +k (0) , φk (τ1αk−1) +� +represents the interpolation function given by the three values φk (0) , +φ′ +k (0) and φk (τ1αk−1) , ¯τ1 and ¯τ2 are positive parameters. In CGOPT, Dai and Kou [5] determined the +initial stepsize in the following way: +α0 +k = + + + +α +if |φk (α) − φk (0)| / (τ3 + φk (0)) > τ4, +arg min ¯q +� +φk (0) , φ′ +k (0) , φk (α) +� +, otherwise, +(42) +where α = max +� +τ5αk−1, −2 |fk − fk−1| /gT +k dk +� +, τ3 > 0, τ4 > 0 and τ5 > 0. Most recently, Liu and Liu +[26] discussed the development a very effective initial stepsize selection strategy for SMCG method by +combining the BB methods and the interpolation technique. +Based on the above research, we devise an improved strategy to obtain the initial stepsize. We first +consider the initial stepsize for the search direction in the RQN iteration. +(i) Initial stepsize of the search direction (34) with Bk+1(µ) ̸= I. +Since the search direction ˆd is a quasi-Newton direction in the subspace Sk, then the initial stepsize +α0 +k = 1 may be a good choice. Therefore, the trial initial stepsize can be stated as +α0 +k = + + + +ˆαk, +if +((10) or ̟ ≤ τ2) holds and ¯αk > 0, +1, +otherwise, +(43) + +12 +Wumei Sun1 et al. +where +ˆαk = min{max{¯αk, αmin}, αmax}, +¯αk = min ¯q(φk(0), φk +′(0),φk(1)), +̟ = |φk (1) − φk (0)| / (τ1 + φk (0)) , τ1 > 0, τ2 > 0 and αmax > αmin > 0. +Here, ¯q +� +φk (0) , φ′ +k (0) , φk (1) +� +is a quadratic interpolation function for φk (0) , φ′ +k (0) , and φk (1) , and +αmax and αmin represent two positive constants. +(ii) Initial stepsize of the search direction (34) with Bk+1(µ) = I. +α0 +k = + + + +ˆαk, +if +((10) or ̟ ≤ τ2) holds and ¯αk > 0, +¯¯αk, +otherwise, +(44) +where +¯¯αk = + + + +max{min{αBB2 +k +, αmax}, αmin}, if gT +k sk−1 > 0, +max{min{αBB1 +k +, αmax}, αmin}, if gT +k sk−1 ≤ 0, +(45) +For the initial stepsize of the search direction in the SMCG iteration. If the search direction dk is +calculated by (20) with dk ̸= −gk, the initial stepsize is chosen in the same way as the RQN iteration, +which is determined by (43). If the search direction dk is given by (19), the initial stepsize is determined +by +α0 +k = + + + +min{max{˜˜αk, αmin}, αmax}, if (10) holds, ∥gk∥2 ≤ 1, dk−1 ̸= −gk−1 and ˜˜αk > 0, +¯¯αk, +otherwise, +(46) +where ¯¯αk is determined by (45) and ˜˜αk = min q(φk(0), φk′(0),φk(¯¯αk)). +Next, we introduce a generalized line search condition, which can be regarded as a development of the +Zhang-Hager’s nonmonotone line search. We recall the nonmonotone line search introduced by Zhang and +Hager [41] +f(xk + αkdk) ≤ Ck + δαkgT +k dk, +(47) +where +Ck+1 = ηkQkCk + f k+1 +Qk+1 +, Qk+1 = ηkQk + 1, +(48) +0 < δ < 1, and ηk ∈ [0, 1]. From (48), it is easy to see that Ck+1 is a convex combination of fk+1 and +Ck. If C0 = f(x0), it is thus clear that Ck can be regard as a convex combination of the function values +f(x0), f(x1), · · · , f(xk). It means that Ck can employ information about the known function values from +the previous iteration. The Zhang-Hager’s nonmonotone line search (47) is reduced to the standard Armijo +line search condition when ηk = 0 for each k. +As it was reported in [41], the nonmonotone line search proposed by Zhang and Hager plays a crucial +role in generating an appropriate stepsize compared to the monotone line search method. Based on (47) +and (48), Huang et al. [17] presented a very effective nonmonotone line search technique, which can be + +Title Suppressed Due to Excessive Length +13 +regard as an extension of Zhang-Hager’s nonmonotone line search, that is +Ck+1 = ηkQkCk + fk+1 +Qk+1 +≤ Ck + δkαkgT +k dk, +(49) +where ηk ∈ [ηmin, ηmax], δmax < 1, 0 < δmin < (1 − ηmax)δmax, δmin ≤ δk ≤ +δmax +Qk+1 and Qk+1 is computed +by (48). +Inspired by the previous discussion, we will study a generalized nonmonotone Wolfe line search technique +based on (48) and (49). Considering the acceleration technique, the generalized nonmonotone Wolfe line +search conditions are as follows: +Ck+1 ≤ Ck + δk¯ηkαkgT +k dk, +(50) +gT +k+1dk ≥ σgT +k dk, +(51) +where 0 < δmin < δk < δmax < 1, σ ∈ (0, 1), Q0 = 1, C0 = f0, ¯ηk is an acceleration parameter determined +by (39), Ck and Qk are updated as follows +Ck+1 = ηkQkCk + f(xk+1) +Qk+1 +, Qk+1 = ηkQk + 1, f(xk+1) = f(xk + ¯ηkαkdk), +(52) +where ηk ∈ [0, 1]. Specially, +Q1 = 2.0, C1 = min{C0, f1 + 1.0}, +(53) +when k ≥ 1, Ck+1 and Qk+1 are updated by (52), and ηk is given as +ηk = + + + +1, +if Ck − fk+1 > 0.95|Ck| and k > 100, +0.9, otherwise. +(54) +Here ηk is a parameter that controls the degree of non-monotonicity, referred to [25]. +Furthermore, we demonstrate that the generalized nonmonotone Wolfe line search is an extension of +the Zhang-Hager’s nonmonotone Wolfe line search method. It follows from (50) that we get +f(xk + ¯ηkαkdk) ≤ (Qk+1 − ηkQk)Ck + Qk+1δk¯ηkαkgT +k dk. +(55) +Since Qk+1 − ηkQk = 1, (50) is equivalent to +f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT +k dk, +(56) +It is easy to see that if δk = +δ +Qk+1 , nonmonotone line search condition (56) reduces to the Zhang-Hager’s +nonmonotone Wolfe line search condition (47). This means that the Zhang-Hager’s nonmonotone Wolfe +line search condition in [41] can be considered as a particular version of (50). +2.4 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm(RL SMCG) +In this subsection, we describe the regularized limited memory subspace minimization conjugate gradient +algorithm in detail. As mentioned above, the regularized limited memory subspace minimization conjugate +gradient algorithm is made of two kinds of iterations. The “state” in Algorithm 1 represents for the type of + +14 +Wumei Sun1 et al. +iteration, i.e., state= “SMCG” means that SMCG iteration will be carried out, and state= “RQN” means +that RQN iteration will be performed. +Algorithm 1 RL SMCG +Step 0. Chosen x0 ∈ Rn, ε > 0, ˜η0, ˜η1, υ, m, ξ1, ξ2, ξ3, ξ4, ξ5, σ1, σ2, σ3, µmin, µmax, τ, ¯τ, ¯c, ς, ¯ς, ¯ǫ, τ1, +τ2, δk, σ, IterRestart := 0, IterQuad := 0 and MinQuad. Set state = “SMCG” and k := 0. +Step 1. If ∥gk∥∞ ≤ ε, stop. +Step 2. Compute the search direction. +If (state = “SMCG”), then +If k = 0, then d0 = −g0. +elseif (IterQuad = MinQuad and IterQuad ̸= IterRestart), set +dk = −gk, IterQuad = 0, and IterRestart = 0. +else +Determine the search direction dk by (20). +end +elseif (state = “RQN”), then +Compute Pk by (35), and compute the search direction dk by (34). +end +Step 3. Determine the corresponding initial step size α0 +k from (43), (44) and (46) according to the different +iteration directions in the Step 2. +Step 4. Determine a stepsize αk satisfying the generalized nonmonotone Wolfe line search (50) and (51) +with initial stepsize α0 +k. +Step 5.Compute the trial iteration ¯z = xk + αkdk and g¯z = ∇f(¯z). If ∥g¯z∥∞ ≤ ε, then stop; otherwise, go +to Step 6. +Step 6. Acceleration procedure. +If the condition (40) holds, then go to 6.1. +6.1. Compute ¯ak = αkgT +k dk, ¯bk = αk(g¯z − gk)T dk and ¯ηk by (39). +6.2. Update the iteration point as xk+1 = xk + ¯ηkαkdk and compute fk+1 and gk+1. +6.3. If fk+1 satisfies (50) and gk+1 satisfies (51), go to Steps 8. Otherwise, go to Steps 7. +else +go to Steps 7. +end +Step 7. Update the variable as xk+1 = xk + αkdk. Compute fk+1 and gk+1. +Step 8. Update restart conditions. +Step 9. Update Qk+1 and Ck+1 with (52). +Step 10. Update iteration type. +If (state = “SMCG”), then +If (24) holds, then state = “RQN”. +elseif (state = “RQN”), then +If (25) holds, then state = “SMCG”. +end +Step 11. Set k := k + 1 and go to Step 1. +Remark 3. Notably, when the lost orthogonality is corrected, our algorithm terminates the RQN +iteration and immediately calls the SMCG iteration. However, the limited memory CG method [15] first + +Title Suppressed Due to Excessive Length +15 +carries out the complex preprocessing CG iteration after the orthogonality is improved. This means that +algorithm RL SMCG is more simple compared to the limited memory CG method [15]. +3 Convergence Analysis +In the section, we establish the global convergence of the algorithm RL SMCG under the following assump- +tions and properties. +Define N to be an open neighborhood of the level set L (x0) = {x ∈ Rn : f (x) ≤ f (x0)} , where x0 is +an initial point. +Assumption 1 (i) The objective function f is continuously differentiable in N and the level set is bounded +from below. (ii) The gradient g of the objective function is Lipschitz continuous in N, i.e., there exists a +constant L > 0 such that ∥g(x) − g(y)∥ ≤ L ∥x − y∥ , ∀x, y ∈ N. +Under these assumptions, we have the following several properties. +Lemma 1 Suppose that Assumption 1 holds. Then, for ˆBk+1(µ) in (30), there exist three constants ˆξ1 > +0, ˆξ2 > 0 and ˆξ3 > 0 such that +λmax +� +ˆBk+1(µ) +� +≤ ˆξ1, λmax +� +ˆB−1 +k+1(µ) +� +≤ ˆξ2, +��� ˆB−1 +k+1(µ) +��� ≤ ˆξ3. +Proof We know that Zk is a normal orthogonal basis of Sk and the dimension m < +∞, hence we have +ξ0 > 0 such that ∥Zk∥ ≤ ξ0. According to (30) and the property of the matrix norm in finite dimensional +spaces, we can get that λmax +� +ˆBk(µ) +� += 1 or +λmax +� +ˆBk+1(µ) +� +≤ λmax +� +ˆBk(µ) +� ++ λmax +� +− +ˆBk(µ)ˆskˆsT +k ˆBk(µ) +ˆsT +k ˆBk(µ)ˆsk +� ++ λmax +� ˆyk(µ)ˆyT +k (µ) +ˆsT +k ˆyk(µ) +� +(57) +≤ λmax +� +ˆBk(µ) +� ++ ˆyT +k (µ)ˆyk(µ) +ˆsT +k ˆyk(µ) +. +Further, by ˆyk(µ) = ˆyk + µˆsk, µ > 0, we get +ˆyT +k (µ)ˆyk(µ) +ˆsT +k ˆyk(µ) += ∥ˆyk∥2 + µ2 +k∥ˆsk∥2 + 2µˆsT +k ˆyk +ˆsT +k ˆyk + µ∥ˆsk∥2 += ∥ˆyk∥2 + µˆsT +k ˆyk +ˆsT +k ˆyk + µ∥ˆsk∥2 + µˆsT +k ˆyk + µ2 +k∥ˆsk∥2 +ˆsT +k ˆyk + µ∥ˆsk∥2 +≤ ∥ˆyk∥2 + µˆsT +k ˆyk +ˆsT +k ˆyk ++ µ +≤ L2ξ2 +0∥ˆsk∥2 +ˆsT +k ˆyk ++ 2µ +≤ L2ξ2 +0 +υ ++ 2µmax. +The fourth inequality above is obtained from ˆyk = ZT +k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii). Because +ˆBk(µ) will be set to ˆI after a maximum of l updates, combining with (57) easy to get λmax +� +ˆBk+1(µ) +� +≤ +1 + lL2ξ2 +0 +υ ++ 2lµmax ≜ ˆξ1. + +16 +Wumei Sun1 et al. +Let ˆPk(µ) = ˆB−1 +k+1(µ). According to (30) and some simple matrix operations, we have that ˆPk(µ) = ˆI +or +ˆPk(µ) = +� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�T +ˆPk−1(µ) +� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +� ++ +ˆskˆsT +k +ˆsT +k ˆyk(µ). +(58) +It is not difficult to that λmax +�� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�T � +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�� += ∥ˆyk(µ)∥2∥ˆsk∥2 +(ˆsT +k ˆyk(µ)) +2 +. For any ˆz ̸= 0 ∈ Rm and +ˆPk(µ) in (58), we have +ˆzT ˆPk(µ)ˆz = ˆzT +� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�T +ˆPk−1(µ) +� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +� +ˆz + +� +ˆsT +k ˆz +�2 +ˆsT +k ˆyk(µ) +≤ λmax +� +ˆPk−1(µ) +� +ˆzT +� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�T � +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +� +ˆz + +� +ˆsT +k ˆz +�2 +ˆsT +k ˆyk(µ) +≤ λmax +� +ˆPk−1(µ) +� +λmax +�� +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�T � +ˆI − ˆyk(µ)ˆsT +k +ˆsT +k ˆyk(µ) +�� +∥ˆz∥2 + +� +ˆsT +k ˆz +�2 +ˆsT +k ˆyk(µ) +≤ λmax +� +ˆPk−1(µ) +� ∥ˆyk(µ)∥2∥ˆsk∥2 +� +ˆsT +k ˆyk(µ) +�2 +∥ˆz∥2 + +∥ˆsk∥2 +ˆsT +k ˆyk(µ)∥ˆz∥2. +The above inequality is divided by ∥ˆz∥2, and the resulting inequality is maximized, then we have +λmax +� +ˆPk(µ) +� +≤ λmax +� +ˆPk−1(µ) +� ∥ˆyk(µ)∥2∥ˆsk∥2 +� +ˆsT +k ˆyk(µ) +�2 ++ +∥ˆsk∥2 +ˆsT +k ˆyk(µ) +≤ λmax +� +ˆPk−1(µ) +� + + +∥ˆyk(µ)∥2 +ˆsT +k ˆyk(µ) +∥ˆsk∥2 +ˆsT +k ˆyk(µ) + + + ∥ˆsk∥2 +ˆsT +k ˆyk +≤ λmax +� +ˆPk−1(µ) +� �L2ξ2 +0 +υ ++ 2µmax +� ∥ˆsk∥2 +ˆsT +k ˆyk ++ ∥ˆsk∥2 +ˆsT +k ˆyk +≤ +�L2ξ2 +0 +υ2 ++ 2µmax +υ +� +λmax +� +ˆPk−1(µ) +� ++ 1 +υ . +The third inequality above is obtained from ˆyk = ZT +k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii). Because ˆPk(µ) +will be set to ˆI after a maximum of l updates, it is easy to know that there exists a constant ˆξ2 > 0 such +that λmax +� +ˆB−1 +k+1(µ) +� += λmax +� +ˆPk(µ) +� +≤ ˆξ2. +Since ˆB−1 +k+1(µ) is a positive definite and symmetric matrix, we have +��� ˆB−1 +k+1(µ) +��� +2 = λmax +� +ˆB−1 +k+1(µ) +� +≤ +ˆξ2. As a result, using the equivalence property of matrix norm in a finite dimensional space, it follows that +there exists a constant ˆξ3 > 0 such that +��� ˆB−1 +k+1(µ) +��� ≤ ˆξ3. The proof is completed. +⊓⊔ +Lemma 2 Suppose that Assumption 1 holds. Then, for Pk in (35), there exist three constants γ0 > 0, γ1 > 0 +and γ2 > 0 such that +∥Pk∥ ≤ γ0, gT +k+1Pkgk+1 ≥ γ1 ∥gk+1∥2 , dT +k P −1 +k +dk ≥ γ2 ∥dk∥2 , +(59) +where P −1 +k +denotes the pseudoinverse of Pk. +Proof By (25), (35) and Lemma 1, we obtain that +∥Pk∥ = +���Zk ˆB−1 +k+1(µ)ZT +k +��� = +��� ˆB−1 +k+1(µ) +��� ≤ ˆξ3 ≜ γ0, +gT +k+1Pkgk+1 = gT +k+1Zk ˆB−1 +k+1(µ)ZT +k gk+1 + +Title Suppressed Due to Excessive Length +17 += ˆgT +k+1 ˆB−1 +k+1(µ)ˆgk+1 +≥ λmin +� +ˆB−1 +k+1(µ) +� +∥ˆgk+1∥2 +≥ 1 +ˆξ1 +� +1 − ˜η2 +1 +� +∥gk+1∥2 ≜ γ1 ∥gk+1∥2 , +dT +k P −1 +k +dk = dT +k Zk ˆB−1 +k+1(µ)ZT +k dk = ˆdT +k ˆB−1 +k+1(µ) ˆdk ≥ 1 +ˆξ2 +��� ˆdk +��� +2 += 1 +ˆξ2 +∥dk∥2 ≜ γ2 ∥dk∥2 . +Therefore, we can get the conclusions. The proof is completed. +⊓⊔ +Subsequently, we provide some properties of the search directions produced by the algorithm RL SMCG, +which are crucial for the following convergence analysis. +Lemma 3 Suppose that Assumption 1 holds. Then, there exists a constant c1 > 0 such that the search +directions (20) and (34) are calculated by algorithm RL SMCG satisfy the sufficient descent condition: +gT +k dk ≤ −¯c1∥gk∥2. +(60) +Proof We divide the proof into the following two cases. +(i) SMCG iteration. Similar to the proof of Lemma 4.1 of [42], it is easy to have +gT +k dk ≤ −c1∥gk∥2, +where c1 = min +� +1 +2, 1 − ¯ξ3, +2 +3¯ξ2 , +1 +3¯ξ2 , +2 +5¯ξ2 +� +. +(ii) RQN iteration. According to Lemma 2, we have +gT +k dk = −gT +k Pk−1gk ≤ −γ1 ∥gk∥2 . +By setting ¯c1 = min {c1, γ1}, we can obtain (60). The proof is completed. +⊓⊔ +Lemma 4 Suppose that Assumption 1 holds. Then, there exists a constant c1 > 0 such that the search +directions (20) and (34) are calculated by algorithm RL SMCG satisfy +∥dk∥ ≤ ¯c2∥gk∥. +(61) +Proof We divide the proof into the following two cases. +(i) SMCG iteration. Referring to the proof procedure of Lemma 4.2 of [42], it is easy to get +∥dk∥ ≤ c2∥gk∥, +where c2 = max +� +1, 1 + L +¯ξ1 , 20 +¯ξ1 +� +. +(ii) RQN iteration. According to Lemma 2, we obtain ∥dk∥ = ∥−Pk−1gk∥ ≤ γ0 ∥gk∥. +By setting ¯c2 = min {c2, γ0}, we can obtain (61). The proof is completed. +⊓⊔ +The following lemmas are very critical for the convergence analysis of algorithm RL SMCG. + +18 +Wumei Sun1 et al. +Lemma 5 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG. +Then, +If acceleration succeeds: +¯ηkαk ≥ +�1 − σ +L +� ��gT +k dk +�� +∥dk∥2 . +(62) +If acceleration fails: +αk ≥ +�1 − σ +L +� ��gT +k dk +�� +∥dk∥2 . +(63) +Where σ are given by (51). +Proof We divide the proof into the following two cases. +(i) If acceleration succeeds: +From (51) and Assumptions 1 (ii), we obtain that +(σ − 1)gT +k dk ≤ g(xk + ¯ηkαkdk)T dk − gT +k dk = (g(xk + ¯ηkαkdk) − gk)T dk ≤ L¯ηkαk∥dk∥2, +which yields +¯ηkαk ≥ +�σ − 1 +L +� gT +k dk +∥dk∥2 . +This means that (62) holds. +(ii) If acceleration fails: +Let ¯ηk = 1, and the rest of the proof procedure is the same as before. +⊓⊔ +Lemma 6 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG. +Then, there holds that fk ≤ Ck for each k. +Proof We divide the proof into the following two cases. +(i) If acceleration succeeds: +The new iterative update format is xk+1 = xk + ¯ηkαkdk, where ¯ηk = − ¯ak +¯bk . Through (56), we have +fk+1 = f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT +k dk. Combining (52), δk > 0, lemma 5 and the sufficiently +descent property of the direction dk+1, we have fk+1 < Ck. The remaining proof process refers to Lemma +5.1 in [42], we can obtain fk+1 ≤ Ck+1, hence fk ≤ Ck is established for each k. +(ii) If acceleration fails: +Let ¯ηk = 1, and the rest of the proof procedure is the same as before. +⊓⊔ +Theorem 1 Suppose that Assumption 1 holds, the sequence {xk} is generated by the algorithm RL SMCG. +Then, +lim +k→∞ ∥gk∥ = 0. +(64) +Proof We divide the proof into the following two cases. +(i) If acceleration succeeds: +By Assumptions 1, lemmas 3 - 5 and the generalized nonmonotone Wolfe line search conditions (50) +and (51), we get that +Ck+1 ≤ Ck + δk¯ηkαkgT +k dk +(65) + +Title Suppressed Due to Excessive Length +19 +≤ Ck + δmin¯ηkαkgT +k dk +≤ Ck + δmin 1 − σ +L +(gT +k dk)2 +∥dk∥2 +≤ Ck + δmin(1 − σ)¯c2 +1 +L¯c2 +2 +∥gk∥2 += Ck + β∥gk∥2. +Where β = +δmin(1−σ)¯c2 +1 +L¯c2 +2 +. Combined with (53), we have C1 ≤ C0 that means that Ck is monotonically +decreasing. According to lemma 6 and Assumption 1 (i), we know Ck is bounded from below. Then +∞ +� +k=0 +β∥gk∥2 < ∞, +therefore, +lim +k→∞ ∥g(xk)∥ = 0. +(ii) If acceleration fails: +Let ¯ηk = 1, and the rest of the proof procedure is the same as before. +⊓⊔ +4 Numerical Experiments +In this section, we compare the numerical performance of RL SMCG with ASMCG PR [32], CG DESCENT(6.8) +[15] and CGOPT(2.0) [27] for the 145 test problems from CUTEr library [10]. The codes of CG DESCENT(6.8) +[15] and CGOPT(2.0) [27] can be downloaded from http://users.clas.ufl.edu/hager/papers/Software and +https://web.xidian.edu.cn/xdliuhongwei/en/paper.html or http://lsec.cc.ac.cn/ dyh/software.html, respec- +tively. +In the numerical experiments, we set the parameters of RL SMCG as: ¯ξ1 = 10−10, ¯ξ2 = 1.2 × 104, +¯ξ3 = 5 × 10−5, ¯ξ4 = 10−4, ¯ξ5 = 0.08, ˜η0 = 10−9, ˜η1 = 0.5, υ = 5 × 10−7, m = min{n, 11}, σ1 = 0.1, +σ2 = 5, σ3 = 0.85, ˆτ = 1, ¯τ = 0.225, ¯c = 0.1, ς = 5 × 10−5(n ≤ 11), ς = 5 × 10−6(n > 11), ¯ς = 5 × 10−3, +τ1 = 0.1, τ2 = 135, δk = 0.0005 and σ = 0.9999. CG DESCENT(6.8) and CGOPT(2.0) take the default +parameters in their codes but the stopping conditions. Note that the number of memory m for RL SMCG is +min{n, 11} while the number of memory for CG DESCENT(6.8) is 11. All test methods in the experiment +are terminated if ∥gk∥∞ ≤ 10−6 is satisfied, and we set the number of iterations for all test algorithms to +be no more than 200,000. In addition, all algorithms are running in Ubuntu 10.04 LTS. +We will show the performances of the test methods using the performance profiles introduced by Dolan +and Mor´e [8]. In the following Figs. 1-12, “Niter”,“Nf”,“Ng” and “Tcpu” represent the number of iterations, +the number of function evaluations, the number of gradient evaluations and CPU time(s), respectively. +We divided the numerical experiments in three teams. +In the first set of numerical experiments, figures 1-4 illustrate the performance profiles of RL SMCG +and ASMCG PR [32]. From Figs. 1, 2, 3 and 4, we can observe that RL SMCG has a quite significant +improvement over ASMCG PR in terms of the number of iterations, the number of function evaluations, + +20 +Wumei Sun1 et al. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +ASMCG_PR +Fig. 1: Niter +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +ASMCG_PR +Fig. 2: Nf +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +ASMCG_PR +Fig. 3: Ng +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +ASMCG_PR +Fig. 4: Tcpu +the number of gradient evaluations and CPU time. It indicates that the limited memory technique equipped +in RL SMCG indeed brings quite significant numerical improvements. +In the second set of numerical experiments, we give a comparison of the performance profiles of +RL SMCG with CG DESCENT(6.8) [15]. Regarding the number of iterations and the number of func- +tion evaluations in Fig. 5 and Fig. 6 respectively, we observe that RL SMCG is a little better than +CG DESCENT(6.8) for the number of iterations and the number of function evaluations. As shown in +Fig. 7, we can see that RL SMCG is much better than CG DESCENT(6.8) in terms of the number of +gradient evaluations, because RL SMCG outperforms for about 71.5% of the CUTEr test problems, while +the percentage of software CG DESCENT(6.8) is below 40%. It can be observe from Fig. 8 that RL SMCG +is faster than CG DESCENT(6.8) in terms of CPU time. By Theorem 1, RL SMCG is globally conver- +gent with the generalized nonmonotone Wolfe line search, while CG DESCENT (6.8) does not guarantee +global convergence when using the rather efficient approximate Wolfe (AWolfe) line search. This means that +RL SMCG is superior to CG DESCENT(6.8) for CUTEr library in theory and numerical performance. +In the third set of the numerical experiments, comparing the performance of RL SMCG with CGOPT(2.0) +[27]. As shown in Figs. 9 and 10, we can take a look at RL SMCG performs almost always better than +CGOPT(2.0) in terms of the number of iterations and the number of function evaluations. Figures. 11 and +12 indicates that RL SMCG outperforms CGOPT(2.0) in terms of the number of gradient evaluations and +CPU time for the CUTEr library. + +Title Suppressed Due to Excessive Length +21 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CG_DESCENT(6.8) +Fig. 5: Niter +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CG_DESCENT(6.8) +Fig. 6: Nf +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CG_DESCENT(6.8) +Fig. 7: Ng +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CG_DESCENT(6.8) +Fig. 8: Tcpu +From the results of the above three numerical experiments, it is clear that the proposed algorithm +RL SMCG is quite effective. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CGOPT(2.0) +Fig. 9: Niter +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CGOPT(2.0) +Fig. 10: Nf + +22 +Wumei Sun1 et al. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CGOPT(2.0) +Fig. 11: Ng +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +τ +0.5 +0.6 +0.7 +0.8 +0.9 +1 +P(τ) +ARL_SMCG +CGOPT(2.0) +Fig. 12: Tcpu +5 Conclusions +In this paper, combined subspace minimization conjugate gradient method with limited memory technique, +we presented a regularized limited memory subspace minimization conjugate gradient method, which con- +tains two types of iteration. In the proposed algorithm, a modified regularized quasi-Newton method is +given in small dimensional subspace to correct the orthogonality, and an improved initial step size selection +strategy and some simple acceleration criteria are designed. Moreover, we establish the global convergence +of the proposed algorithm by utilizing generalized nonmonotone Wolfe line search under some mild as- +sumptions. Some numerical results suggest that our algorithm yields a tremendous improvement over the +ASMCG PR and outperforms the most up-to-date limited memory CG software packages CG DESCENT +(6.8) and CGOPT(2.0). +6 Declarations +6.1 Ethical Approval +Not Applicable +6.2 Availability of supporting data +Data sharing not applicable to this article as no datasets were generated or analyzed during the current +study. +6.3 Competing interests +The authors declare no competing interests. + +Title Suppressed Due to Excessive Length +23 +6.4 Funding +This research was supported by the National Natural Science Foundation of China (No. 11901561), the +Natural Science Foundation of Guizhou (No. ZK[2022]084) and the Natural Science Basic Research Program +of Shaanxi (No. 2021JM-396). +6.5 Authors’ contributions +Wumei Sun wrote the main manuscript text. Hongwei Liu and Zexian Liu reviewed and revised the +manuscript. +6.6 Acknowledgments +The authors would like to thank the editor and the anonymous referees for their valuable suggestions and +comments which have greatly improved the presentation of this paper. +References +1. Andrei, N.: An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained opti- +mization. Numer. Algor. 65, 859-874 (2014) +2. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer Anal. 8, 141-148 (1988) +3. Dai, Y.H., Yuan, J.Y., Yuan, Y.X.: Modified two-point stepsize gradient methods for unconstrained optimization +problems. Comput. Optim. Appl. 22(1), 103-109 (2002) +4. Dai, Y.H.: Nonlinear Conjugate Gradient Methods. Wiley Encyclopedia of Operations Research and Management +Science(2011). https://doi.org/10.1002/9780470400531.eorms0183 +5. 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Algor. 87, 1501-1534 (2021) + diff --git a/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/load_file.txt b/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b79c4a504b539aebe6a46c42c7d48f98b46fd7a8 --- /dev/null +++ b/9tE1T4oBgHgl3EQfCQLF/content/tmp_files/load_file.txt @@ -0,0 +1,1005 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf,len=1004 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='02863v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='OC] 7 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (will be inserted by the editor) A Regularized Limited Memory Subspace Minimization Conjugate Gradient Method for Unconstrained Optimization Wumei Sun1 · Hongwei Liu1 · Zexian Liu2 Received: date / Accepted: date Abstract In this paper, based on the limited memory techniques and subspace minimization conjugate gra- dient (SMCG) methods, a regularized limited memory subspace minimization conjugate gradient method is proposed, which contains two types of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In SMCG iteration, we obtain the search direction by min- imizing the approximate quadratic model or approximate regularization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In RQN iteration, combined with regularization technique and BFGS method, a modified regularized quasi-Newton method is used in the subspace to improve the orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, some simple acceleration criteria and an improved tactic for selecting the initial stepsize to enhance the efficiency of the algorithm are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Additionally, an generalized nonmonotone line search is utilized and the global convergence of our proposed algorithm is established under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Finally, numerical results show that, the proposed algorithm has a significant improvement over ASMCG PR and is superior to the particularly well-known limited memory conjugate gradient software packages CG DESCENT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) and CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) for the CUTEr library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Keywords Limited memory · Subspace minimization conjugate gradient method · Orthogonality · Regularization model · Quasi-Newton method Mathematics Subject Classification (2010) 49M37 · 65K05 · 90C30 1 Introduction Consider problem min x∈Rn f(x), (1) where f : Rn → R is a continuously differentiable nonlinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Wumei Sun E-mail: sunwumei1992@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='com Hongwei Liu � E-mail: hwliuxidian@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='com Zexian Liu E-mail: liuzexian2008@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='com 1 School of Mathematics and Statistics, Xidian University, Xi’an 710126, China 2 School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China 2 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Throughout the article, we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' sk−1 = xk − xk−1, fk = f(xk), gk = g(xk), yk−1 = gk−gk−1, ∥·∥ represents the Euclidean norm and λmax denotes the maximum eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, dist{x, S} = inf{∥y − x∥, y ∈ S}, where x ∈ Rn and S ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Nonlinear conjugate gradient(CG) method is a well-known method for solving the problem (1), which main iteration is xk+1 = xk + αkdk, k = 0, 1, 2, · · · , (2) where xk is the kth iteration point, αk > 0 is the stepsize and dk is the search direction obtained by d0 = −g0, dk = −gk + βkdk−1, k ≥ 1, (3) where gk is the gradient of f(xk) and βk is the conjugate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' It is shown in theory that the convergence and numerical performance variation of different CG meth- ods depend on the selection of conjugate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Some very classical choices of the conjugate param- eter βk are Fletcher-Reeves(FR) [9], Polak-Ribi`ere-Polyak(PRP) [30,31], Dai-Yuan(DY) [7] and Hestenes- Stiefel(HS) [16], and are given by βF R k = ∥gk+1∥2 ∥gk∥2 , βP RP k = gT k+1yk ∥gk∥2 , βDY k = ∥gk+1∥2 dT k yk , βHS k = gT k+1yk dT k yk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' CG algorithms have evolved considerably, and some well-known CG packages such as CG DESCENT [12, 14] and CGOPT [5] have been proposed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Other recent related studies on nonlinear CG algorithms can be found in [4,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The subspace minimization conjugate gradient (SMCG) algorithm, as a generalization of the CG algo- rithm, has received much attention from scholars [1,37], which can be traced back to the work of Yuan and Stoer [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The search direction of SMCG method is obtained by minimizing the following problem: min d∈Ωk gT k d + 1 2dT Bkd, (4) where Ωk is a subspace spanned by the vectors gk and sk−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', Ωk = Span{gk, sk−1}, and Bk ∈ Rn×n is an approximation of Hessian matrix, which is positive definite and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then the search direction d is given by d = ugk + vsk−1, (5) where u and v are both real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Substituting (5) to (4) and combined with the standard secant equation Bksk−1 = yk−1, formula (4) is reorganized as follows: min u,v∈R \uf8eb \uf8ed ∥gk∥2 gT k sk−1 \uf8f6 \uf8f8 T \uf8eb \uf8ed u v \uf8f6 \uf8f8 + 1 2 \uf8eb \uf8ed u v \uf8f6 \uf8f8 T \uf8eb \uf8ed ρk gT k yk−1 gT k yk−1 sk−1yk−1 \uf8f6 \uf8f8 \uf8eb \uf8ed u v \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (6) where ρk ≈ gT k Bkgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' On the basis of the Barzilai-Borwein(BB) method [2], Dai and Kou [6] proposed an effective BBCG3 method for strictly convex quadratic minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Afterwards, based on BBCG3 method, Liu and Liu [26] proposed SMCG BB method for solving general unconstrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Motivated by SMCG BB method, some efficient SMCG methods [20,21,36,42] were later proposed, among which Title Suppressed Due to Excessive Length 3 the method based on the regularization model presented by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' [42] is the best in the numerical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The nonlinear CG method is very effective for unconstrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, the convergence of the algorithm can be very slow for some ill-posed problems and even for quadratic problems with very small dimensions, which may be due to the loss of orthogonality [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Hager and Zhang [15] pointed out theoretically that the generated successive gradients either in the CG method or the L-BFGS method for the quadratic test problem should be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Yet, Hager and Zhang [15] observed that, when solving the quadratic strictly convex minimization problem PALMER1C in the CUTEr library [10], the CG method loses orthogonality due to the rounding errors, while L-BFGS method preserves the orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In view of this, they developed the limited memory CG method (CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8)) to correct the possible loss of orthogonality in ill conditioned optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' For the test problems in the CUTEr library [10], their performance results indicated that CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) has an significant improvement over their previously proposed package CG DESCENT(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Although CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) [15] is an efficient method for unconstrained optimization, it still suffers from the following shortcomings: (i) In the numerical implementation, the AWolfe line search [14] utilized in the algorithm CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) does not guarantee global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) contains the following three pre-conditioners, corresponding to three different it- erations: Pk = I, Pk = Zk ˆB−1 k+1ZT k , Pk = Zk ˆB−1 k+1ZT k + σk ¯Zk ¯ZT k , (7) where σk is determined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2) of [15], ˆBk+1, Zk and ¯Zk are given by the matrices in literature [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' These three pre-conditioners make the algorithm CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) look complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (iii) In the convergence analysis, the algorithm CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) needs to impose the following assump- tions on the pre-conditioners: ∥Pk∥ ≤ γ0, gT k+1Pkgk+1 ≥ γ1∥gk+1∥2, dT k P −1 k dk ≥ γ2∥dk∥2, (8) where γ0 > 0, γ1 > 0 and γ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' These assumptions are comparatively strict and difficult to be verified in actual practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' To address the above-mentioned shortcomings, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' [27] presented an improved Dai¨CKou CG algorithm called CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0), which combines limited memory technology and Dai-Kou CG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) [27], they utilized a modified quasi-Newton method to restore the lost orthogonality, and established the convergence of CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) with fewer assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Some numerical experiments indicated that CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) is better than the famous CG software package CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In view of the above discussion, a regularized limited memory subspace minimization conjugate gradient method on the basis of SMCG method and limited memory technique is studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' To recover orthogonality, we propose a modified regularized quasi-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The major contributions of this paper are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' A regularized limited memory subspace minimization conjugate gradient algorithm is proposed, which combines limited memory technology and SMCG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 4 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Based on the idea of regularization and BFGS method, an improved regularized quasi-Newton method is exploited to improve orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Some simple acceleration criteria and an improved initial stepsize selection strategy are designed to enhance the efficiency of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Additionally, an generalized nonmonotone line search condition is presented, which may be regarded as an extension of the Zhang-Hager’s [41] nonmonotone line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The convergence of the method is built under mild conditions and the corresponding numerical perfor- mance shows that the new method is much more effective than the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In Section 2, we describe the detail of the regularized limited memory subspace minimization conjugate gradient algorithm, including the direction selection of SMCG iteration and regularized Quasi-Newton iteration and an effective acceleration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, the decision of the initial step size and the generalized nonmonotone Wolfe line search are also given in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In Section 3, some important properties of the search direction are analyzed and the global con- vergence of the proposed algorithm is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Numerical experiments for algorithm comparison are showed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Conclusions are given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm In the section, combining the idea of subspace minimization and regularization quasi-Newton method, we present a regularized limited memory subspace minimization conjugate gradient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Firstly, we give the choices of search direction under different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Subsequently, we develop a very effec- tive acceleration technique, a modified initial step selection strategy and generalized nonmonotonic line search technology to optimize the performance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Finally, the details of algorithm RL SMCG are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1 Direction Selection of SMCG Iteration and Regularized Quasi-Newton Iteration The regularized limited memory subspace minimization conjugate gradient method mainly contains two kinds of iterations which are SMCG iteration and regularized quasi-Newton(RQN) iteration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Furthermore, the search direction derivation of the two iterations is also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1 SMCG iteration The search direction selection of SMCG iteration is closely related to the properties of the objective function f(x) at the iteration point xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' By reference [3,38], defined tk = ���2 � fk−1 − fk + gT k sk−1 � / � sT k−1yk−1 � − 1 ��� , (9) to describe how f(x) approaches a quadratic function on a line segment between xk−1 and xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Literature [24] indicates that if the condition tk ≤ ¯ξ4 or � tk ≤ ¯ξ5 and tk−1 ≤ ¯ξ5 � , (10) Title Suppressed Due to Excessive Length 5 is satisfied, where ¯ξ4 and ¯ξ5 are the smaller positive constants and ¯ξ4 < ¯ξ5, f(x) may be near to a quadratic function on a line between xk−1 and xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, According to [32], we know that if the following condition ¯ξ1 ≤ sT k−1yk−1 ∥sk−1∥2 ≤ ∥yk−1∥2 sT k−1yk−1 ≤ ¯ξ2, (11) is satisfied, then the condition number of the Hessian matrix of the normal function may be not very large, here ¯ξ1 and ¯ξ2 are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Similar to [42], based on some certain properties of the function f(x) at the current point xk, we derive different search direction by dividing it into the following four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) If the condition (11) is satisfied while the condition (10) are not, this implies that the quadratic model may not be able to approach the objective function f(x) well at the present iteration point xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, search direction dk will be obtained by minimizing the following cubic regular subproblem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' min dk∈Ωk mk (dk) = dT k gk + 1 2dT k Bkdk + 1 3σk ∥dk∥3 Bk , (12) where Ωk is a subspace spanned by the vectors gk and sk−1, Bk ∈ Rn×n is an approximation of Hessian matrix, which is positive definite and symmetric and satisfying the secant condition Bksk−1 = yk−1, σk ≥ 0 is an adaptive regularization parameter obtained from interpolation condition and dk is determined by dk = ukgk + vksk−1, (13) where vk and uk are parameters to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Obviously, we could obtain (12) by giving (4) a weighted regularization term 1 3σk ∥dk∥3 Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Substituting (13) to (12), it is easy to obtain that (12) is equivalent to min uk,vk∈R \uf8eb \uf8ed ∥gk∥2 gT k sk−1 \uf8f6 \uf8f8 T \uf8eb \uf8ed uk vk \uf8f6 \uf8f8 + 1 2 \uf8eb \uf8ed uk vk \uf8f6 \uf8f8 T ¯Bk \uf8eb \uf8ed uk vk \uf8f6 \uf8f8 + σk 3 ������ \uf8eb \uf8ed uk vk \uf8f6 \uf8f8 ������ 3 ¯ Bk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (14) where ¯Bk = \uf8eb \uf8ed ρk gT k yk−1 gT k yk−1 sk−1yk−1 \uf8f6 \uf8f8 is a positive definite and symmetric matrix, ρk is an estimate of gT k Bkgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Similar to BBCG3 [6], we also use 3 2 ∥yk−1∥2 sT k−1yk−1 I to estimate Bk in the term ρk, which means ρk = 3 2 ∥yk−1∥2 sT k−1yk−1 ∥gk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, by solving problem (14) we obtain the following solutions about uk and vk: \uf8eb \uf8ed uk vk \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 (1+σk(̟∗))∆k � gT k yk−1gT k sk−1 − sT k−1yk−1∥gk∥2� 1 (1+σk(̟∗))∆k � gT k yk−1∥gk∥2 − ρkgT k sk−1 � \uf8f6 \uf8f8 , (15) among them, ∆k = ������ ρk gT k yk−1 gT k yk−1 sk−1yk−1 ������ = ρksk−1yk−1 − (gT k yk−1)2 > 0, (16) σk and ̟∗ are the same as those in literature [42], which will not be repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) If both conditions (11) and (10) hold, this indicates that the objective function f(x) may approach the quadratic model at the current iteration point xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Since that is the case, let σk = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' we consider deriv- ing the search direction by solving the minimization problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Like (i), we choose ρk = 3 2 ∥yk−1∥2 sT k−1yk−1 ∥gk∥2 and ∆k is determined by (16), then we obtain the following unique solution of quadratic approximate 6 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' problem (6): \uf8eb \uf8ed ¯uk ¯vk \uf8f6 \uf8f8 = \uf8eb \uf8ed 1 ∆k (gT k yk−1gT k sk−1 − sT k−1yk−1∥gk∥2) 1 ∆k (gT k yk−1∥gk∥2 − ρkgT k sk−1) \uf8f6 \uf8f8 , (17) here the search direction is calculated by dk = ¯ukgk + ¯vksk−1, where ¯uk and ¯vk are determined by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (iii) If condition (11) is not satisfied and the conditions ���gT k yk−1gT k sk−1 ��� ≤ ¯ξ3sT k−1yk−1∥gk∥2 and sT k−1yk−1 ≥ ¯ξ1∥sk−1∥2, (18) are satisfied, where 0 ≤ ¯ξ3 ≤ 1, the condition number of the Hessian matrix may be lager, hence the search direction obtained in cases (i) and (ii) may not be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, the condition (18) can ensure sufficient descent and linear growth in HS conjugate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, because of the finite termination nature of the HS conjugate gradient method for solving exact convex quadratic minimization problems, this choice of direction allows for faster convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, in this case, the search direction is determined by (3) and βk = βHS k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (iv) If neither condition (11) nor (18) holds, then we pick the following direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' : dk = −gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (19) In summary, the search direction in the SMCG iteration can be described as in the following: dk = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ukgk + vksk−1, if (11) holds and (10) does not hold, ¯ukgk + ¯vksk−1, if (11) holds and (10) holds, −gk + βHS k dk−1, if (11) does not hold and (18) holds, −gk, if neither (11) nor (18) holds, (20) where uk and vk are determined by (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ¯uk and ¯vk are determined by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If the successive gradients have orthogonality or the lost orthogonality is restored, the algorithm performs SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' On the contrary, if the orthogonality is lost, the iteration will turn to the following regularized quasi-Newton iteration to improve the orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2 Regularized Quasi-Newton(RQN) iteration When the successive gradients lose their orthogonality, the iteration switches from SMCG iteration to RQN iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In other words, a modified regularized BFGS algorithm in subspace Sk is proposed to restore the orthogonality, where Sk is a subspace generated by the following limited memory m search directions Sk = span {dk−1, dk−2, · · · , dk−m} , where m > 0 and m is the number of limited memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In this article, the limited memory m selected in our algorithm does not exceed 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, as soon as orthogonality is corrected, the RQN iteration is terminated and the SMCG iteration is triggered immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' First, we introduce some preparations for turning to RQN iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Let Sk ∈ Rn×m be a matrix which has columns consisting of dk−1, dk−2, · · · , dk−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In similar fashion to limited memory CG method [15], we also assume that columns of Sk are line-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Let the QR factorization of Sk be Sk = Zk ¯Rk, where Title Suppressed Due to Excessive Length 7 the columns of Zk ∈ Rn×m form the normal orthogonal bases for subspace Sk and ¯Rk ∈ Rm×m is the upper triangular matrix with positive diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If gk is included almost in subspace Sk, then we think that the orthogonality property of the algorithm may be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In this case, we interrupt the SMCG iteration and move to minimize the objective function in the subspace Sk: min z∈Sk f(xk + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (21) The solution to the subspace problem (21) will improve the orthogonality and guide us to a suitable search direction that will lead us out of the subspace Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Similar to [15], we utilize the distance from gk to subspace Sk to judge whether orthogonality is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If the condition dist {gk, Sk} ≤ ˜η0∥gk∥ (22) is satisfied, where 0 < ˜η0 < 1 and ˜η0 is small, we think gk is almost contained in Sk, it means that the orthogonality of the successive gradients has lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, we switch to RQN iteration to solve the subspace problem (21) until the gradient is nearly orthogonal enough to the subspace to meet the condition dist {gk, Sk} ≥ ˜η1∥gk∥, (23) where 0 < ˜η0 < ˜η1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' At this time, the algorithm iteration will go away subspace Sk and turn to the SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Because the column of Zk is the orthonormal basis of Sk, it’s not hard to know from the definition of dist {gk, Sk} that (22) and (23) can be expressed as � 1 − ˜η2 0 � ∥gk∥2 ≤ ���ZT k gk ��� 2 , (24) and � 1 − ˜η2 1 � ∥gk∥2 ≥ ���ZT k gk ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (25) In [15], Hager and Zhang utilized the limited memory BFGS (L-BFGS) [22,28] method to solve the subspace problem (21) for restoring the orthogonality, and achieved better numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, it should be noted that the convergence analysis of the limited memory CG method [15] requires imposing strict assumptions (8) on the preprocessors (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Because the dimension m of the chosen subspace Sk is usually small and when orthogonality is lost, the properties of the function at the iteration point maybe not very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Based on these, we consider a regularized L-BFGS method in the subspace Sk for solving the subproblem (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The search direction of general quasi-Newton method [40] for unconstrained optimization (1) is the form of dk = −B−1 k gk, where Bk is a positive definite and symmetric approximation to the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As one of the most popular methods of quasi-Newton method, L-BFGS method stores the approximate Hessian matrix of the objective function using small memory and computes the search direction dk using the nearest m vector pairs of (sk−i, yk−i), i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 8 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Ueda and Yamashita [35] presented a regularized Newton method for nonconvex unconstrained opti- mization, whose search direction dk is obtained by solving the following linear equations: � ∇2f(xk) + µI � dk = −∇f(xk), (26) where µ > 0 is referred to as the regularized parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The regularized Newton method [35] generally defaults to a step size of 1, and global convergence is guaranteed by controlling the parameter µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, as a type of Newton method, the regularized Newton method in [35] must solve the Hessian matrix of f which is particularly computationally complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' To address this drawback, some scholars proposed the regularized limited memory BFGS-type method [33,23] for solving unconstrained optimization problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' the search direction dk is the solution of the following equations (Bk + µI) dk = −∇f(xk), (27) where matrix Bk is an approximate Hessian determined by a particular quasi-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Regular- ization technology can effectively improve the efficiency of quasi-Newton method in solving ill-conditioned problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Nevertheless, when computing Bk by the L-BFGS method, it is very hard to calculate (Bk + µI)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Hence, motivated by [34], we present a regularized quasi-Newton method which combines the BFGS method with the regularized technique to improve orthogonality in the m-dimensional subspace Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In this paper, we consider Bk + µI as an approximation of ∇2f(xk) + µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Because the matrix Bk is the approximate Hessian of f(xk) and Bk + µI can be used as an approximate Hessian of f(xk) + µ 2 ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' At this point, we utilize (sk, yk(µ)) instead of (sk, yk), where yk(µ) = (∇f(xk+1) + µxk+1) − (∇f(xk) + µxk) = yk + µsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Note that the regularized BFGS method stores as many vector pairs as the traditional BFGS method and hence it does not require additional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In [19], a effective BFGS quasi-Newton method for solving nonconvex unconstrained minimization was proposed by Li and Fukushima [19], in which the matrix Bk+1 is updated by Bk+1 = \uf8f1 \uf8f2 \uf8f3 Bk − BksksT k Bk sT k Bksk + ykyT k sT k yk , if sT k yk ∥sk∥2 > υ∥gk∥α, Bk, otherwise , where υ > 0 and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Some recent advances about modified BFGS method can be found in [18,11,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Inspired by the quasi-Newton methods described above, we propose an improved regularized BFGS method to solve the subproblem (21) in subspace Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In what follows, the variables with hats belong to subspace Sk , distinguished from the ones found in the full space Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Let ˆx = (ˆx1, ˆx2, · · · , ˆxm, )T ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The subproblem (21) can be expressed as min ˆx∈Rm f(xk + ˆx1dk−1 + ˆx2dk−2 + · · · + ˆxmdk−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (28) Title Suppressed Due to Excessive Length 9 Similar to [27], because the regularized quasi-Newton directions in the subspace Sk always transform to the full space Rn and QR decomposition of matrix Sk, we can obtain dk = Zk ˆdk, ˆgk = ZT k gk, ˆyk = ZT k yk, ˆsT k ˆyk = sT k yk, ∥ˆsk∥2 = ∥sk∥2 and ˆfk = fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Let Bk(µ) = Bk + µI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' then inspired by Li and Fukushima [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' we develop an improved regularized BFGS method to solve the above subproblem (28) with a search direction of the form ˆdk+1 = − ˆB−1 k+1(µ)ˆgk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (29) where ˆBk+1(µ) is given by ˆBk+1(µ) = \uf8f1 \uf8f2 \uf8f3 ˆBk(µ) − ˆ Bk(µ)ˆskˆsT k ˆ Bk(µ) ˆsT k ˆ Bk(µ)ˆsk + ˆyk(µ)ˆyT k (µ) ˆsT k ˆyk(µ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' if mod(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' l) ̸= 0 and ˆsT k ˆyk(µ) ˆsT k ˆsk ≥ υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ˆI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' otherwise ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (30) where υ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' mod(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' l) ̸= 0 represents the remainder for k modulo l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ˆyk(µ) = ˆyk + µˆsk and µ > 0 is an important regularized parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The condition mod(k, l) ̸= 0 means the matrix ˆBk(µ) will be reset to the identity matrix ˆI after updating l times, which ensures the good convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the paper, we set l = max(m2, 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Obviously, ˆsT k ˆyk(µ) > 0, and as soon as the matrix ˆBk(µ) is symmetric and positive definitive, it is not hard to prove that the matrix ˆBk+1(µ) is symmetric and positive definitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As a very important regularization parameter, µ is closely related to the convergence analysis of the regularized BFGS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In this paper, the idea of the trust-region radius is used to find the suitable search direction by controlling µ, in other words, The ratio of objective function value reduction to model function value reduction is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, give the definition of a ratio function rk( ˆdk, µ) as follows rk( ˆdk, µ) = ˆf(xk) − ˆf(xk + αk ˆdk) ˆf(xk) − ˆqk( ˆdk, µ) , (31) where ˆqk : Rm × R → R is a function of the form ˆqk( ˆdk, µ) = ˆf(xk) + αkˆgT k ˆdk + 1 2α2 k ˆdT k ˆBk(µ) ˆdk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (32) Then, if the ratio function rk( ˆdk, µ) is relatively large, this means that compared with the reduction of the model function, the reduction of the objective function is large enough, we choose to reduce the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' On the flip side, if the ratio function rk( ˆdk, µ) is relatively small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', ˆf(xk) − ˆf(xk + αk ˆdk) is small, we will increase µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In addition, to ensure that the algorithms converge well, we limit µ to an interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 0 < µmin < µ < µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In general, if the next iteration point is closer to the current iteration point, the reduction of the function value may not be obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' At this time, we hope to get a new iteration point by modifying the search direction, then the search direction improved by regular parameter µ may be a good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Therefore, if ∥ˆsk∥2 ≤ ˆτ (ˆτ > 0), our choice and update of µ are as follows: µk+1 = \uf8f1 \uf8f2 \uf8f3 max {µmin, σ1µk} , if rk( ˆdk, µ) ≥ σ3, min {µmax, σ2µk} , otherwise, (33) where 0 < σ1 ≤ 1, σ2 > 1 and 0 < σ3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Otherwise, we choose µ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', the regularized BFGS method is reduced to a general BFGS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 10 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In order to simplify the symbol and facilitate writing, we still record the updated symbol µk+1 as µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the process of algorithm implementation, the search direction (29) in subspace Sk always converts to the full space Rn at each RQN iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', dk+1 = −Pkgk+1, (34) where Pk = Zk ˆB−1 k+1(µ)ZT k (35) and ˆBk+1(µ) is given by (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In Section 3, we will show that matrices ˆBk+1(µ) and Pk have some good properties in the RQN iteration, which is critical for the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2 An Effective Acceleration Technique In order to optimize the performance of the algorithm, Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' [32] proposed an acceleration technique, which replaces (2) with the following new iterative form xk+1 = xk + ¯ηkαkdk, (36) where ¯ηk ≥ 0 is an acceleration parameter obtained from an interpolation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In view of the numerical effect of the acceleration technique, our algorithm also takes it into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Similar to reference [32], we minimize the following interpolation function to get the acceleration parameter ¯ηk: ¯ηk = arg min q(ϕk(¯η)), (37) where ¯η ≥ 0, ϕk(¯η) = f(xk + ¯ηαkdk), and q(ϕk(¯η)) represents the interpolation function defined by ϕk(¯η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the paper, we consider minimizing the quadratic interpolation function [29] q(ϕk(0), ϕ′ k(0),ϕ′ k(1)), then, ¯ηk = arg min q(ϕk(0), ϕ′ k(0), ϕ′ k(1)), (38) By minimizing (38) we have ¯ηk = −¯ak ¯bk , ¯bk ≥ ¯ǫ, (39) where ¯ak = αkgT k dk, ¯bk = αk(g¯z − gk)Tdk, g¯z = ∇f(¯z), ¯z = xk + αkdk and ¯ǫ > 0 is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' We propose the following acceleration criterion, which is simpler than the rule in reference [32], that is ¯bk ≥ ¯ǫ, ∥s¯z∥2 ≤ ¯τ, ∥gk∥2 ≤ ˆτ, |¯tk+1| < ¯c, and |sT k g¯z| ≥ Max(ς, ¯ς · ¯bk) (40) where ¯ǫ, ¯τ, ˆτ, ¯c, ς and ¯ς are all small positive constants, ¯bk = αk(g¯z − gk)T dk, s¯z = ¯z − xk, ¯z = xk + αkdk, |¯tk+1| = | 2(fk−f¯z+gT ¯z s¯z) sT ¯z g¯z − 1|, f¯z = f(¯z) and g¯z = ∇f(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' When the condition (40) holds, we accelerate the algorithm and update the relevant variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In addition, one of the necessary conditions for successful acceleration is that the trial iteration point must satisfy the line search condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Therefore, if the algorithm Title Suppressed Due to Excessive Length 11 accelerates successfully, update the iteration point xk+1 by using (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Otherwise the algorithm acceleration fails and returns to the original algorithm, at which point ¯ηk = 1, update the iteration point xk+1 with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In reference [32], the acceleration criterion is divided into three cases, which seems to be more complex, while our acceleration criterion has only one case and the form is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='3 Choices of the Initial Stepsize and the Generalized Nonmonotone Wolfe Line Search It is well known that the design of the search direction and the conditions of the line search are two critical factors which affect the efficiency of the line search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In this subsection, we will develop an improved nonmonotone Wolfe line search which can be regarded as an extension of the Zhang-Hager’s [41] nonmonotone line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In addition, an improved initial step selection strategy is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' For the sake of convenience, we express the one-dimensional line search function as φk(α) = f(xk + αdk), α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The choice of the initial stepsize α0 k is of great importance for a line search in an optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' For the Newton-like methods, choosing the initial step α0 k = 1 is important to speed up convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' For the conjugate gradient methods, it is essential to use information from the current iteration of the problem to make initial guesses [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the conjugate gradient method, there have been various ways to choose the initial stepsize, for example, see [5,12,15,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, it did not have an agreement on which is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In particular, Hager and Zhang [15] select the initial step in CG DESCENT as below: α0 k = \uf8f1 \uf8f2 \uf8f3 arg min ¯q � φk (0) , φ′ k (0) , φk (¯τ1αk−1) � , if φk (¯τ1αk−1) ≤ φk (0) , ¯τ2αk−1, otherwise, (41) where ¯q � φk (0) , φ′ k (0) , φk (τ1αk−1) � represents the interpolation function given by the three values φk (0) , φ′ k (0) and φk (τ1αk−1) , ¯τ1 and ¯τ2 are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In CGOPT, Dai and Kou [5] determined the initial stepsize in the following way: α0 k = \uf8f1 \uf8f2 \uf8f3 α if |φk (α) − φk (0)| / (τ3 + φk (0)) > τ4, arg min ¯q � φk (0) , φ′ k (0) , φk (α) � , otherwise, (42) where α = max � τ5αk−1, −2 |fk − fk−1| /gT k dk � , τ3 > 0, τ4 > 0 and τ5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Most recently, Liu and Liu [26] discussed the development a very effective initial stepsize selection strategy for SMCG method by combining the BB methods and the interpolation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Based on the above research, we devise an improved strategy to obtain the initial stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' We first consider the initial stepsize for the search direction in the RQN iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) Initial stepsize of the search direction (34) with Bk+1(µ) ̸= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Since the search direction ˆd is a quasi-Newton direction in the subspace Sk, then the initial stepsize α0 k = 1 may be a good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Therefore, the trial initial stepsize can be stated as α0 k = \uf8f1 \uf8f2 \uf8f3 ˆαk, if ((10) or ̟ ≤ τ2) holds and ¯αk > 0, 1, otherwise, (43) 12 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' where ˆαk = min{max{¯αk, αmin}, αmax}, ¯αk = min ¯q(φk(0), φk ′(0),φk(1)), ̟ = |φk (1) − φk (0)| / (τ1 + φk (0)) , τ1 > 0, τ2 > 0 and αmax > αmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Here, ¯q � φk (0) , φ′ k (0) , φk (1) � is a quadratic interpolation function for φk (0) , φ′ k (0) , and φk (1) , and αmax and αmin represent two positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) Initial stepsize of the search direction (34) with Bk+1(µ) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' α0 k = \uf8f1 \uf8f2 \uf8f3 ˆαk, if ((10) or ̟ ≤ τ2) holds and ¯αk > 0, ¯¯αk, otherwise, (44) where ¯¯αk = \uf8f1 \uf8f2 \uf8f3 max{min{αBB2 k , αmax}, αmin}, if gT k sk−1 > 0, max{min{αBB1 k , αmax}, αmin}, if gT k sk−1 ≤ 0, (45) For the initial stepsize of the search direction in the SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If the search direction dk is calculated by (20) with dk ̸= −gk, the initial stepsize is chosen in the same way as the RQN iteration, which is determined by (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If the search direction dk is given by (19), the initial stepsize is determined by α0 k = \uf8f1 \uf8f2 \uf8f3 min{max{˜˜αk, αmin}, αmax}, if (10) holds, ∥gk∥2 ≤ 1, dk−1 ̸= −gk−1 and ˜˜αk > 0, ¯¯αk, otherwise, (46) where ¯¯αk is determined by (45) and ˜˜αk = min q(φk(0), φk′(0),φk(¯¯αk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Next, we introduce a generalized line search condition, which can be regarded as a development of the Zhang-Hager’s nonmonotone line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' We recall the nonmonotone line search introduced by Zhang and Hager [41] f(xk + αkdk) ≤ Ck + δαkgT k dk, (47) where Ck+1 = ηkQkCk + f k+1 Qk+1 , Qk+1 = ηkQk + 1, (48) 0 < δ < 1, and ηk ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' From (48), it is easy to see that Ck+1 is a convex combination of fk+1 and Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If C0 = f(x0), it is thus clear that Ck can be regard as a convex combination of the function values f(x0), f(x1), · · · , f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' It means that Ck can employ information about the known function values from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The Zhang-Hager’s nonmonotone line search (47) is reduced to the standard Armijo line search condition when ηk = 0 for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As it was reported in [41], the nonmonotone line search proposed by Zhang and Hager plays a crucial role in generating an appropriate stepsize compared to the monotone line search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Based on (47) and (48), Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' [17] presented a very effective nonmonotone line search technique, which can be Title Suppressed Due to Excessive Length 13 regard as an extension of Zhang-Hager’s nonmonotone line search, that is Ck+1 = ηkQkCk + fk+1 Qk+1 ≤ Ck + δkαkgT k dk, (49) where ηk ∈ [ηmin, ηmax], δmax < 1, 0 < δmin < (1 − ηmax)δmax, δmin ≤ δk ≤ δmax Qk+1 and Qk+1 is computed by (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Inspired by the previous discussion, we will study a generalized nonmonotone Wolfe line search technique based on (48) and (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Considering the acceleration technique, the generalized nonmonotone Wolfe line search conditions are as follows: Ck+1 ≤ Ck + δk¯ηkαkgT k dk, (50) gT k+1dk ≥ σgT k dk, (51) where 0 < δmin < δk < δmax < 1, σ ∈ (0, 1), Q0 = 1, C0 = f0, ¯ηk is an acceleration parameter determined by (39), Ck and Qk are updated as follows Ck+1 = ηkQkCk + f(xk+1) Qk+1 , Qk+1 = ηkQk + 1, f(xk+1) = f(xk + ¯ηkαkdk), (52) where ηk ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Specially, Q1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0, C1 = min{C0, f1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0}, (53) when k ≥ 1, Ck+1 and Qk+1 are updated by (52), and ηk is given as ηk = \uf8f1 \uf8f2 \uf8f3 1, if Ck − fk+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='95|Ck| and k > 100, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (54) Here ηk is a parameter that controls the degree of non-monotonicity, referred to [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Furthermore, we demonstrate that the generalized nonmonotone Wolfe line search is an extension of the Zhang-Hager’s nonmonotone Wolfe line search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' It follows from (50) that we get f(xk + ¯ηkαkdk) ≤ (Qk+1 − ηkQk)Ck + Qk+1δk¯ηkαkgT k dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (55) Since Qk+1 − ηkQk = 1, (50) is equivalent to f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT k dk, (56) It is easy to see that if δk = δ Qk+1 , nonmonotone line search condition (56) reduces to the Zhang-Hager’s nonmonotone Wolfe line search condition (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' This means that the Zhang-Hager’s nonmonotone Wolfe line search condition in [41] can be considered as a particular version of (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm(RL SMCG) In this subsection, we describe the regularized limited memory subspace minimization conjugate gradient algorithm in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As mentioned above, the regularized limited memory subspace minimization conjugate gradient algorithm is made of two kinds of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The “state” in Algorithm 1 represents for the type of 14 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', state= “SMCG” means that SMCG iteration will be carried out, and state= “RQN” means that RQN iteration will be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Algorithm 1 RL SMCG Step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Chosen x0 ∈ Rn, ε > 0, ˜η0, ˜η1, υ, m, ξ1, ξ2, ξ3, ξ4, ξ5, σ1, σ2, σ3, µmin, µmax, τ, ¯τ, ¯c, ς, ¯ς, ¯ǫ, τ1, τ2, δk, σ, IterRestart := 0, IterQuad := 0 and MinQuad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Set state = “SMCG” and k := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If ∥gk∥∞ ≤ ε, stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Compute the search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If (state = “SMCG”), then If k = 0, then d0 = −g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' elseif (IterQuad = MinQuad and IterQuad ̸= IterRestart), set dk = −gk, IterQuad = 0, and IterRestart = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' else Determine the search direction dk by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' end elseif (state = “RQN”), then Compute Pk by (35), and compute the search direction dk by (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' end Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Determine the corresponding initial step size α0 k from (43), (44) and (46) according to the different iteration directions in the Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Determine a stepsize αk satisfying the generalized nonmonotone Wolfe line search (50) and (51) with initial stepsize α0 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='Compute the trial iteration ¯z = xk + αkdk and g¯z = ∇f(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If ∥g¯z∥∞ ≤ ε, then stop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' otherwise, go to Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Acceleration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If the condition (40) holds, then go to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Compute ¯ak = αkgT k dk, ¯bk = αk(g¯z − gk)T dk and ¯ηk by (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Update the iteration point as xk+1 = xk + ¯ηkαkdk and compute fk+1 and gk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If fk+1 satisfies (50) and gk+1 satisfies (51), go to Steps 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Otherwise, go to Steps 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' else go to Steps 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' end Step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Update the variable as xk+1 = xk + αkdk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Compute fk+1 and gk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Update restart conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Update Qk+1 and Ck+1 with (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Step 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Update iteration type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' If (state = “SMCG”), then If (24) holds, then state = “RQN”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' elseif (state = “RQN”), then If (25) holds, then state = “SMCG”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' end Step 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Set k := k + 1 and go to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Notably, when the lost orthogonality is corrected, our algorithm terminates the RQN iteration and immediately calls the SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' However, the limited memory CG method [15] first Title Suppressed Due to Excessive Length 15 carries out the complex preprocessing CG iteration after the orthogonality is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' This means that algorithm RL SMCG is more simple compared to the limited memory CG method [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 3 Convergence Analysis In the section, we establish the global convergence of the algorithm RL SMCG under the following assump- tions and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Define N to be an open neighborhood of the level set L (x0) = {x ∈ Rn : f (x) ≤ f (x0)} , where x0 is an initial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Assumption 1 (i) The objective function f is continuously differentiable in N and the level set is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) The gradient g of the objective function is Lipschitz continuous in N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=', there exists a constant L > 0 such that ∥g(x) − g(y)∥ ≤ L ∥x − y∥ , ∀x, y ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Under these assumptions, we have the following several properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Lemma 1 Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, for ˆBk+1(µ) in (30), there exist three constants ˆξ1 > 0, ˆξ2 > 0 and ˆξ3 > 0 such that λmax � ˆBk+1(µ) � ≤ ˆξ1, λmax � ˆB−1 k+1(µ) � ≤ ˆξ2, ��� ˆB−1 k+1(µ) ��� ≤ ˆξ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Proof We know that Zk is a normal orthogonal basis of Sk and the dimension m < +∞, hence we have ξ0 > 0 such that ∥Zk∥ ≤ ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' According to (30) and the property of the matrix norm in finite dimensional spaces, we can get that λmax � ˆBk(µ) � = 1 or λmax � ˆBk+1(µ) � ≤ λmax � ˆBk(µ) � + λmax � − ˆBk(µ)ˆskˆsT k ˆBk(µ) ˆsT k ˆBk(µ)ˆsk � + λmax � ˆyk(µ)ˆyT k (µ) ˆsT k ˆyk(µ) � (57) ≤ λmax � ˆBk(µ) � + ˆyT k (µ)ˆyk(µ) ˆsT k ˆyk(µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Further, by ˆyk(µ) = ˆyk + µˆsk, µ > 0, we get ˆyT k (µ)ˆyk(µ) ˆsT k ˆyk(µ) = ∥ˆyk∥2 + µ2 k∥ˆsk∥2 + 2µˆsT k ˆyk ˆsT k ˆyk + µ∥ˆsk∥2 = ∥ˆyk∥2 + µˆsT k ˆyk ˆsT k ˆyk + µ∥ˆsk∥2 + µˆsT k ˆyk + µ2 k∥ˆsk∥2 ˆsT k ˆyk + µ∥ˆsk∥2 ≤ ∥ˆyk∥2 + µˆsT k ˆyk ˆsT k ˆyk + µ ≤ L2ξ2 0∥ˆsk∥2 ˆsT k ˆyk + 2µ ≤ L2ξ2 0 υ + 2µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The fourth inequality above is obtained from ˆyk = ZT k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Because ˆBk(µ) will be set to ˆI after a maximum of l updates, combining with (57) easy to get λmax � ˆBk+1(µ) � ≤ 1 + lL2ξ2 0 υ + 2lµmax ≜ ˆξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 16 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Let ˆPk(µ) = ˆB−1 k+1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' According to (30) and some simple matrix operations, we have that ˆPk(µ) = ˆI or ˆPk(µ) = � ˆI − ˆyk(µ)ˆsT k ˆsT k ˆyk(µ) �T ˆPk−1(µ) � ˆI − ˆyk(µ)ˆsT k ˆsT k ˆyk(µ) � + ˆskˆsT k ˆsT k ˆyk(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (58) It is not difficult to that λmax �� ˆI − ˆyk(µ)ˆsT k ˆsT k ˆyk(µ) �T � ˆI − ˆyk(µ)ˆsT k ˆsT k ˆyk(µ) �� = ∥ˆyk(µ)∥2∥ˆsk∥2 (ˆsT k ˆyk(µ)) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' For any ˆz ̸= 0 ∈ Rm and ˆPk(µ) in (58),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆzT ˆPk(µ)ˆz = ˆzT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆPk−1(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆz + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='≤ λmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆPk−1(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆzT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�T � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆz + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='≤ λmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆPk−1(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='λmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�T � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆI − ˆyk(µ)ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='∥ˆz∥2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='≤ λmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆPk−1(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ∥ˆyk(µ)∥2∥ˆsk∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='∥ˆz∥2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='∥ˆsk∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ˆsT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='k ˆyk(µ)∥ˆz∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The above inequality is divided by ∥ˆz∥2, and the resulting inequality is maximized, then we have λmax � ˆPk(µ) � ≤ λmax � ˆPk−1(µ) � ∥ˆyk(µ)∥2∥ˆsk∥2 � ˆsT k ˆyk(µ) �2 + ∥ˆsk∥2 ˆsT k ˆyk(µ) ≤ λmax � ˆPk−1(µ) � \uf8eb \uf8ed ∥ˆyk(µ)∥2 ˆsT k ˆyk(µ) ∥ˆsk∥2 ˆsT k ˆyk(µ) \uf8f6 \uf8f8 + ∥ˆsk∥2 ˆsT k ˆyk ≤ λmax � ˆPk−1(µ) � �L2ξ2 0 υ + 2µmax � ∥ˆsk∥2 ˆsT k ˆyk + ∥ˆsk∥2 ˆsT k ˆyk ≤ �L2ξ2 0 υ2 + 2µmax υ � λmax � ˆPk−1(µ) � + 1 υ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The third inequality above is obtained from ˆyk = ZT k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Because ˆPk(µ) will be set to ˆI after a maximum of l updates, it is easy to know that there exists a constant ˆξ2 > 0 such that λmax � ˆB−1 k+1(µ) � = λmax � ˆPk(µ) � ≤ ˆξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Since ˆB−1 k+1(µ) is a positive definite and symmetric matrix, we have ��� ˆB−1 k+1(µ) ��� 2 = λmax � ˆB−1 k+1(µ) � ≤ ˆξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As a result, using the equivalence property of matrix norm in a finite dimensional space, it follows that there exists a constant ˆξ3 > 0 such that ��� ˆB−1 k+1(µ) ��� ≤ ˆξ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ Lemma 2 Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, for Pk in (35), there exist three constants γ0 > 0, γ1 > 0 and γ2 > 0 such that ∥Pk∥ ≤ γ0, gT k+1Pkgk+1 ≥ γ1 ∥gk+1∥2 , dT k P −1 k dk ≥ γ2 ∥dk∥2 , (59) where P −1 k denotes the pseudoinverse of Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Proof By (25), (35) and Lemma 1, we obtain that ∥Pk∥ = ���Zk ˆB−1 k+1(µ)ZT k ��� = ��� ˆB−1 k+1(µ) ��� ≤ ˆξ3 ≜ γ0, gT k+1Pkgk+1 = gT k+1Zk ˆB−1 k+1(µ)ZT k gk+1 Title Suppressed Due to Excessive Length 17 = ˆgT k+1 ˆB−1 k+1(µ)ˆgk+1 ≥ λmin � ˆB−1 k+1(µ) � ∥ˆgk+1∥2 ≥ 1 ˆξ1 � 1 − ˜η2 1 � ∥gk+1∥2 ≜ γ1 ∥gk+1∥2 , dT k P −1 k dk = dT k Zk ˆB−1 k+1(µ)ZT k dk = ˆdT k ˆB−1 k+1(µ) ˆdk ≥ 1 ˆξ2 ��� ˆdk ��� 2 = 1 ˆξ2 ∥dk∥2 ≜ γ2 ∥dk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Therefore, we can get the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ Subsequently, we provide some properties of the search directions produced by the algorithm RL SMCG, which are crucial for the following convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Lemma 3 Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, there exists a constant c1 > 0 such that the search directions (20) and (34) are calculated by algorithm RL SMCG satisfy the sufficient descent condition: gT k dk ≤ −¯c1∥gk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (60) Proof We divide the proof into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1 of [42], it is easy to have gT k dk ≤ −c1∥gk∥2, where c1 = min � 1 2, 1 − ¯ξ3, 2 3¯ξ2 , 1 3¯ξ2 , 2 5¯ξ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) RQN iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' According to Lemma 2, we have gT k dk = −gT k Pk−1gk ≤ −γ1 ∥gk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' By setting ¯c1 = min {c1, γ1}, we can obtain (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ Lemma 4 Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, there exists a constant c1 > 0 such that the search directions (20) and (34) are calculated by algorithm RL SMCG satisfy ∥dk∥ ≤ ¯c2∥gk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (61) Proof We divide the proof into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) SMCG iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Referring to the proof procedure of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2 of [42], it is easy to get ∥dk∥ ≤ c2∥gk∥, where c2 = max � 1, 1 + L ¯ξ1 , 20 ¯ξ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) RQN iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' According to Lemma 2, we obtain ∥dk∥ = ∥−Pk−1gk∥ ≤ γ0 ∥gk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' By setting ¯c2 = min {c2, γ0}, we can obtain (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ The following lemmas are very critical for the convergence analysis of algorithm RL SMCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 18 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Lemma 5 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, If acceleration succeeds: ¯ηkαk ≥ �1 − σ L � ��gT k dk �� ∥dk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (62) If acceleration fails: αk ≥ �1 − σ L � ��gT k dk �� ∥dk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (63) Where σ are given by (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Proof We divide the proof into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) If acceleration succeeds: From (51) and Assumptions 1 (ii), we obtain that (σ − 1)gT k dk ≤ g(xk + ¯ηkαkdk)T dk − gT k dk = (g(xk + ¯ηkαkdk) − gk)T dk ≤ L¯ηkαk∥dk∥2, which yields ¯ηkαk ≥ �σ − 1 L � gT k dk ∥dk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' This means that (62) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) If acceleration fails: Let ¯ηk = 1, and the rest of the proof procedure is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ Lemma 6 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, there holds that fk ≤ Ck for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Proof We divide the proof into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) If acceleration succeeds: The new iterative update format is xk+1 = xk + ¯ηkαkdk, where ¯ηk = − ¯ak ¯bk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Through (56), we have fk+1 = f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT k dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Combining (52), δk > 0, lemma 5 and the sufficiently descent property of the direction dk+1, we have fk+1 < Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The remaining proof process refers to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1 in [42], we can obtain fk+1 ≤ Ck+1, hence fk ≤ Ck is established for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) If acceleration fails: Let ¯ηk = 1, and the rest of the proof procedure is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ Theorem 1 Suppose that Assumption 1 holds, the sequence {xk} is generated by the algorithm RL SMCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then, lim k→∞ ∥gk∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (64) Proof We divide the proof into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (i) If acceleration succeeds: By Assumptions 1, lemmas 3 - 5 and the generalized nonmonotone Wolfe line search conditions (50) and (51), we get that Ck+1 ≤ Ck + δk¯ηkαkgT k dk (65) Title Suppressed Due to Excessive Length 19 ≤ Ck + δmin¯ηkαkgT k dk ≤ Ck + δmin 1 − σ L (gT k dk)2 ∥dk∥2 ≤ Ck + δmin(1 − σ)¯c2 1 L¯c2 2 ∥gk∥2 = Ck + β∥gk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Where β = δmin(1−σ)¯c2 1 L¯c2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Combined with (53), we have C1 ≤ C0 that means that Ck is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' According to lemma 6 and Assumption 1 (i), we know Ck is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Then ∞ � k=0 β∥gk∥2 < ∞, therefore, lim k→∞ ∥g(xk)∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' (ii) If acceleration fails: Let ¯ηk = 1, and the rest of the proof procedure is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ⊓⊔ 4 Numerical Experiments In this section, we compare the numerical performance of RL SMCG with ASMCG PR [32], CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) [15] and CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) [27] for the 145 test problems from CUTEr library [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' The codes of CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) [15] and CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) [27] can be downloaded from http://users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='edu/hager/papers/Software and https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='cn/xdliuhongwei/en/paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='html or http://lsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='cn/ dyh/software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='html, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the numerical experiments, we set the parameters of RL SMCG as: ¯ξ1 = 10−10, ¯ξ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2 × 104, ¯ξ3 = 5 × 10−5, ¯ξ4 = 10−4, ¯ξ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='08, ˜η0 = 10−9, ˜η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5, υ = 5 × 10−7, m = min{n, 11}, σ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1, σ2 = 5, σ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='85, ˆτ = 1, ¯τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='225, ¯c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1, ς = 5 × 10−5(n ≤ 11), ς = 5 × 10−6(n > 11), ¯ς = 5 × 10−3, τ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1, τ2 = 135, δk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0005 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) and CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) take the default parameters in their codes but the stopping conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Note that the number of memory m for RL SMCG is min{n, 11} while the number of memory for CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' All test methods in the experiment are terminated if ∥gk∥∞ ≤ 10−6 is satisfied, and we set the number of iterations for all test algorithms to be no more than 200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In addition, all algorithms are running in Ubuntu 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='04 LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' We will show the performances of the test methods using the performance profiles introduced by Dolan and Mor´e [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the following Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1-12, “Niter”,“Nf”,“Ng” and “Tcpu” represent the number of iterations, the number of function evaluations, the number of gradient evaluations and CPU time(s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' We divided the numerical experiments in three teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the first set of numerical experiments, figures 1-4 illustrate the performance profiles of RL SMCG and ASMCG PR [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1, 2, 3 and 4, we can observe that RL SMCG has a quite significant improvement over ASMCG PR in terms of the number of iterations, the number of function evaluations, 20 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG ASMCG_PR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1: Niter 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG ASMCG_PR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2: Nf 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG ASMCG_PR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 3: Ng 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG ASMCG_PR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 4: Tcpu the number of gradient evaluations and CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' It indicates that the limited memory technique equipped in RL SMCG indeed brings quite significant numerical improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the second set of numerical experiments, we give a comparison of the performance profiles of RL SMCG with CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Regarding the number of iterations and the number of func- tion evaluations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6 respectively, we observe that RL SMCG is a little better than CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) for the number of iterations and the number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 7, we can see that RL SMCG is much better than CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) in terms of the number of gradient evaluations, because RL SMCG outperforms for about 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5% of the CUTEr test problems, while the percentage of software CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) is below 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' It can be observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 8 that RL SMCG is faster than CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) in terms of CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' By Theorem 1, RL SMCG is globally conver- gent with the generalized nonmonotone Wolfe line search, while CG DESCENT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) does not guarantee global convergence when using the rather efficient approximate Wolfe (AWolfe) line search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' This means that RL SMCG is superior to CG DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) for CUTEr library in theory and numerical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the third set of the numerical experiments, comparing the performance of RL SMCG with CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 9 and 10, we can take a look at RL SMCG performs almost always better than CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) in terms of the number of iterations and the number of function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 11 and 12 indicates that RL SMCG outperforms CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) in terms of the number of gradient evaluations and CPU time for the CUTEr library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 21 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CG_DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 5: Niter 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CG_DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6: Nf 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CG_DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 7: Ng 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CG_DESCENT(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 8: Tcpu From the results of the above three numerical experiments, it is clear that the proposed algorithm RL SMCG is quite effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 9: Niter 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 10: Nf 22 Wumei Sun1 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 11: Ng 1 2 3 4 5 6 7 8 9 10 11 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='9 1 P(τ) ARL_SMCG CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 12: Tcpu 5 Conclusions In this paper, combined subspace minimization conjugate gradient method with limited memory technique, we presented a regularized limited memory subspace minimization conjugate gradient method, which con- tains two types of iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' In the proposed algorithm, a modified regularized quasi-Newton method is given in small dimensional subspace to correct the orthogonality, and an improved initial step size selection strategy and some simple acceleration criteria are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Moreover, we establish the global convergence of the proposed algorithm by utilizing generalized nonmonotone Wolfe line search under some mild as- sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Some numerical results suggest that our algorithm yields a tremendous improvement over the ASMCG PR and outperforms the most up-to-date limited memory CG software packages CG DESCENT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='8) and CGOPT(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6 Declarations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='1 Ethical Approval Not Applicable 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='2 Availability of supporting data Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='3 Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='4 Funding This research was supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 11901561), the Natural Science Foundation of Guizhou (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' ZK[2022]084) and the Natural Science Basic Research Program of Shaanxi (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 2021JM-396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='5 Authors’ contributions Wumei Sun wrote the main manuscript text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' Hongwei Liu and Zexian Liu reviewed and revised the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content='6 Acknowledgments The authors would like to thank the editor and the anonymous referees for their valuable suggestions and comments which have greatly improved the presentation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfCQLF/content/2301.02863v1.pdf'} +page_content=' References 1.' 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Paris-Saclay, CNRS, ENS Paris-Saclay, +CentraleSupelec, LuMIn, F-91190 Gif-sur-Yvette, France +2CEA DAM DIF, F-91297 Arpajon, France +3Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes, 91680 Bruy`eres-le-Chˆatel, France +(Dated: January 13, 2023) +Engineering a layer of nitrogen-vacancy (NV) centers on the tip of a diamond anvil creates a +multipurpose quantum sensors array for high pressure measurements, especially for probing magnetic +and superconducting properties of materials. Expanding this concept above 100 GPa appears to be +a substantial challenge. We observe that deviatoric stress on the anvil tip sets a limit at 40-50 GPa +for practical magnetic measurements based on optically detected magnetic resonance (ODMR) of +NV centers under pressure. We show that this limit can be circumvented up to at least 130 GPa +by machining a micropillar on the anvil tip to create a quasi-hydrostatic stress environment for the +NV centers. This is quantified using the pressure dependence of the diamond Raman shift, the NV +ODMR dependence on applied magnetic field, and NV photoluminescence spectral shift. This paves +the way for direct and reliable detection of the Meissner effect in superconductors above 100 GPa, +such as super-hydrides. +Introduction. The diamond anvil cell (DAC) is rou- +tinely used to synthesize compounds under megabar +(100 GPa) pressures, exhibiting novel phenomena and +remarkable properties. Recent examples such as the ob- +servation of metal hydrogen [1], superconductivity close +to ambient temperature in superhydrides [2–4], or su- +perionic water ice [5] are lacking detailed magnetic or +transport measurements for their definite proof and clear +understanding. In particular, magnetic measurements re- +main challenging at megabar pressures because they are +mainly based on flux detection by inductive coils and +must thus extract the signal of the few-micrometers sam- +ples from the much larger magnetic background signal of +the bulky DAC apparatus. This constraint can be cir- +cumvented by implementing in the DAC sensing methods +that exploit the magnetic sensitivity of nitrogen-vacancy +(NV) centers in diamond [6–9]. +This method offers a +tabletop optical microscopy instrumentation, the map- +ping of the magnetic field in the sample chamber with +micrometer spatial resolution and the absence of any sen- +sitivity decrease with the sample size down to the mi- +crometer scale. Another key feature is the easy combi- +nation with synchrotron X-ray characterizations to cor- +relate the magnetic or superconducting properties with a +well-defined crystallographic structure [10]. Yet, the ex- +tension of this technique to extreme pressures remains a +challenge [11]. We investigate here how the existence of a +deviatoric stress in the diamond anvil sets effective limits +to the magnetic response of NV centers localized at the +anvil tip to maximize sample proximity [6, 7]. We then +propose and implement a method that overcomes that +limit and keeps the full NV quantum sensing capabilities +at pressures above 100 GPa. +Experimental configuration. +The negatively charged +NV center is a point defect of diamond that emits visible +photoluminescence (PL) by absorbing green photons +and re-emitting red photons (at ambient pressure), with +an electronic spin s = 1 in the ground and excited +states. In the absence of external magnetic and stress +fields, the ms = ±1 spin sublevels of the ground state +are degenerate and separated by D = 2.87 GHz from +the ms = 0 sublevel (Fig. 1a). Spin-dependent PL arises +from a spin-selective difference in the non-radiative +coupling to metastable singlet states, which also induces +optical pumping into the ms = 0 state under green +illumination [12]. +The energy difference between the +sublevels of the ground state can then be read out +from the change of the NV luminescence intensity upon +scanning the frequency of an additional microwave +excitation. +Dips in the PL intensity indicate that +the excitation microwave frequency is resonant with a +transition between two sublevels, leading to optically +detected magnetic resonance (ODMR) that can be easily +implemented by optically addressing the NV centers +through the diamond anvil [6]. +Here we use the same experimental configuration as in +Ref. [6], keeping two crucial characteristics: 1) the NV +centers are integrated in the DAC device by mounting a +IIas ultra-pure Almax-Boehler design [100]-cut diamond +anvil with a dense ensemble of NV centers (typically +104 +NV/µm2) implanted at about 10 nm beneath +the anvil surface using a nitrogen Focused Ion Beam +(FIB) [13] (Fig. 1b); 2) the microwave excitation is +applied using an external single-turn coil above the +rhenium gasket of the DAC. The metallic gasket is +machined with a slit, filled with an epoxy-glue mixture +ensuring +sample +confinement +and +DAC +mechanical +stability, that re-distributes the induced currents in the +metal, leading to a focusing and amplification of the +microwave flux in the sample chamber similarly to a +arXiv:2301.05094v1 [quant-ph] 12 Jan 2023 + +2 +(a) +(b) +(d) +(c) +NV layer +Anvil 1 +Anvil 2 +Gasket +Pressure +ms=0 +ms=±1 +D +D+훿 +FIG. 1. +(a) Energy diagram of the NV center ground state +and evolution under stress. (b) Schematic cross-section of the +location of NV centers implanted as a layer below the anvil +culet surface. (c) Design of the machined gasket compatible +with the MW excitation of the NV centers. Red arrows show +initial MW excitation current in the wire loop, blue arrows +are currents induced into the gasket. The areas shaded in red +indicate the intensity of the MW field. (d) ODMR spectra +of NV centers implanted in the tip of a standard diamond +anvil at different pressures, as a function of a magnetic field +applied along the [100] diamond axis. Green dashed lines are +fits of the eigenfrequencies computed with the NV ground +state Hamiltonian given by eq. 1. +Lenz lens [14] (Fig. 1c). +Upon pressure increase, the +PL excitation wavelength was decreased to match the +blueshift of the NV absorption spectrum [11] by using +continuous-wave (cw) lasers at successive wavelengths +532, 488, 457 and 405 nm. A customized confocal optical +microscope was used to collect the PL. A static vector +magnetic field was applied on the DAC using three +Helmholtz coil pairs with an amplitude ranging between +0 and 10 mT. The magnetic field was aligned along +the DAC axis with accuracy ±0.5◦. +This orientation +corresponds to the diamond [100] crystal axis for which +all NV centers have equivalent responses to stress and +magnetic field. Pressure in the DAC was measured using +the calibrated diamond Raman phonon mode at the +anvil tip [15]. +Stress effect on the NV magnetic response. We per- +formed cw-ODMR experiments on the NV centers un- +der pressures ranging from 10 GPa to 70 GPa. At each +pressure point, we collected the ODMR spectrum for the +ensemble of NV centers under varying amplitude of the +applied magnetic field. +1300 +1400 +1500 +1600 +1700 +Raman shift ν (cm−1) +Intensity (a.u.) +P = 92 GPa +Culet +Micropillar +40 µm +(a) +(b) +(c) +(d) +* +25 +50 +75 +Pressure (GPa) +3.0 +3.2 +3.4 +Vmol (cm3.mol−1) +1.95 +2.00 +2.05 +2.10 +2.15 +2.20 +2.25 +2.30 +NV− ZPL energy (eV) +Linear fits +Micropillar +Standard anvil +600 +670 +λ (nm) +PL (a.u.) +Anvil 1 +Anvil 2 +Gasket +PTM +NV layer +−0.20 −0.15 −0.10 −0.05 0.00 +ln(V/V0) +0.00 +0.05 +0.10 +0.15 +0.20 +ln(ν/ν0) +Occelli et al. [16] +Micropillar +FIG. 2. +(a) Scanning electron microscope image of a FIB- +machined micropillar on a diamond anvil culet of 100 µm +diameter. The bottom panel shows a schematic cross-section +with the distortion under pressure of the culet. (b) Energy of +the NV center zero-phonon line (ZPL) as a function of pres- +sure and diamond volume, recorded for NV centers implanted +in and out of the micropillar. Inset: typical PL spectra of the +NV centers recorded at 0, 37 and 78 GPa (bottom to top). +The arrows indicate the ZPL position. (c) Diamond Raman +spectra recorded on a pressurized microstructured diamond +anvil at 92 GPa, on and outside the micropillar. In the spec- +trum taken on the micropillar, the peak indicated by the star +reveals hydrostatic compression. (d) Raman frequency shift +measured on the micropillar as a function of relative diamond +volume. Data from [16] is a reference of the Raman shift of +diamond under hydrostatic pressure. +The data are shown in Fig. 1d. Four effects of stress +on the ODMR signals are observed. First, the zero-field +center frequency D = 2.87 GHz increases almost linearly +with a slope of 9.6 MHz/GPa to a value D + δ, where δ +is the pressure induced variation. Second, a splitting ∆σ +appears between the transition lines in the absence of an +external magnetic field. This splitting increases almost +linearly with pressure with a slope of 3.9 MHz/GPa and +originates in deviatoric stress at the anvil culet. Conse- +quently, at a given pressure, the quasi-linear evolution of +the Zeeman splitting due to the applied magnetic field +can only be recovered above a compensating amplitude +of the magnetic field that increases with pressure. This +detrimental influence of stress hence weakens the NV + +3.8 +51 GPa +1.025 +α =0.56 +41 GPa +3.6 +1.000 +MW frequency (GHz) +α =0.57 +30 GPa +PL intensity (a.u.) +α =0.56 +0.975 +20 GPa +3.4 +α =0.56 +0.950 +10 GPa +3.2 +α =0.67 +0.925 +0.900 +3.0 +0.875 +2.8 +0.850 +0 +5 +10 +5 +10 +5 +10 +5 +10 +5 +10 +α model fit +B applied in [100] (mT)3 +sensing magnetic sensitivity. Furthermore, the required +larger applied bias magnetic field isn’t aligned with a +given NV axis here, to overlap responses from all NV +orientations, and thus mixes the sublevels of the ground +state. This mixing perturbs the optically induced spin +polarization and quenches the PL [17]. Third, the shape +of the ODMR spectra differs from the conventional +symmetrical pair of peaks. +The contrast of the low +frequency branch becomes gradually smaller than the +high frequency branch. +After vanishing at a pressure +around 40 GPa, a slightly positive contrast reappears +(increase of PL at resonance) above 50 GPa under high +enough magnetic field. +Finally, the overall observed +ODMR contrast decreases severely under pressure. +In the diamond lattice under mechanical stress (or +equivalently strain), the Hamiltonian describing the NV +center ground state is modified by a spin-mechanical in- +teraction [18, 19] related to the stress tensor +↔σ. +The +stress tensor must exhibit the cylindrical symmetry of +the anvil. At the anvil tip, the stress components parallel +(σ∥) and perpendicular (σ⊥) to the surface differ. Due to +continuity of the normal stress component, σ⊥ is equal to +the experimental pressure P in the DAC chamber. The +tangential component, σ∥, is reduced by a factor α com- +pared to σ⊥. Using a simplified model of a semi-infinite +anvil with a flat face and a circularly symmetric distribu- +tion of pressure applied to this face, the α parameter was +estimated about 0.6 [20]. Neglecting off-diagonal shear +stress components, the stress tensor then reads as: +↔σ= +� +� +αP +0 +0 +0 +αP +0 +0 +0 +P +� +� . +(1) +Using this stress tensor, the diagonalization of the NV +ground state Hamiltonian yields modified spin resonance +frequencies which can be approximated to first order as: +ν± = D + δ ± ∆/2 +(2) +where δ is the spectral shift due to compression, and +∆ = +� +∆2σ + ∆2 +B is the quadratic sum of the splittings +respectively induced by the stress and by the magnetic +field (see Supplementary Material for the full expression). +Since eq. (2) is exact only for low off-axis magnetic field, +a full numerical diagonalization was used to accurately +fit the measured resonance frequencies, as shown by the +green dashed lines in fig. 1d. Only two parameters, α +and P, are hence needed to predict the magnetic field re- +sponse under stress. We obtained a value α = 0.56 that is +essentially constant with pressure, quantifying deviatoric +stress close to the 0.6 value given in Ref. [20]. +Deviatoric stress thus introduces major modifications +to the NV behavior as the anisotropic compression of +the diamond host lattice distorts the C3v symmetry of +the NV center. Here we quantified changes within the +NV ground triplet states, but the stress dependence of +the singlet states and the excited triplet states remains +unexplored and is difficult to assess. As a hypothesis, we +attribute the observed modification and ultimate loss of +ODMR contrast to the effect of deviatoric stress on these +levels involved in the contrast mechanism [21]. +This +hypothesis is corroborated by recent results obtained +on +microdiamonds +compressed +quasi-hydrostatically +inside the sample chamber of a DAC, for which the +ODMR signal could be conserved up to 140 GPa [22]. +These results converge toward a possible circumventing +strategy by ensuring hydrostatic compression of the NV +centers. +Restoring hydrostaticity with diamond microstructura- +tion. A strategy to try to mitigate deviatoric stress can +be implemented by microstructuring the diamond anvil +culet. A successful geometry is presented in Fig. 2a. A +pillar, 7 µm in diameter and with a 2 µm deep trench +around it was FIB-machined on an NV-implanted dia- +mond anvil culet. The pillar surface is thus disconnected +from the anvil surface submitted to deviatoric stress in- +duced by anvil cupping tension [23, 24]. +This also al- +lows the pressure-transmitting medium (PTM) to fill the +trench to immerse the pillar in a stress field close to hy- +drostatic conditions. The pillar is then equivalent to a di- +amond microdisk that would be integrated in the sample +chamber of the DAC but ensures perfect reproducibility +and removes any interface with the diamond culet to op- +timize PL measurements. As seen below, this design is +also very robust and can withstand extreme pressures. +The hydrostaticity of the stress exerted on diamond +under pressure can be tested by measuring the Raman +frequency of the diamond optical phonon. +Under hy- +drostatic conditions, the dependence of the frequency of +the Raman scattering with diamond volume follows a +Gruneisen relation of parameter γ = 0.97(1) whereas the +frequency shift is smaller under deviatoric stress [16]. As +seen in Fig. 2c, the Raman spectra measured at the dia- +mond anvil culet on the micropillar and away from it dif- +fer. In both cases, the broad asymmetric peak is associ- +ated to the stress distribution within the thickness of the +anvil that is optically probed and the high frequency edge +is used to estimate the pressure [15]. At the micropil- +lar, a well separated peak appears with higher frequency +shift. The pressure evolution of its center wavenumber +perfectly matches the value obtained for diamond under +hydrostatic pressure [16] as shown in Fig. 2d. This indi- +cates that the tip of the micropillar hosting part of the +NV center layer is then close to hydrostatic pressure. +Accordingly the PL spectrum of the NV layer in +the micropillar shows a pressure induced blue shift +(Fig. 2b) that can be quantified with the zero-phonon line +(ZPL) [11]. While the NV ZPL dependence with pressure +is not linear, its evolution becomes linear when plotted +versus the compressed diamond volume estimated using + +4 +0 +5 +10 +3.0 +3.5 +4.0 +4.5 +5.0 +MW frequency (GHz) +20 GPa +α =0.95 +α model fit +5 +10 +50 GPa +α =0.95 +5 +10 +B applied in [100] (mT) +74 GPa +α =0.95 +5 +10 +103 GPa +α =0.95 +5 +10 +131 GPa +α =0.97 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +0.99 +1.00 +PL intensity (a.u.) +(a) +4.0 +4.5 +Microwave frequency (GHz) +PL intensity (a.u.) +|B| = 6 mT +P = 73 GPa +P = 103 GPa +P = 131 GPa +(b) +FIG. 3. +(a) ODMR spectra obtained from NV centers implanted in a micropillar at varying pressures, as a function of magnetic +field applied along the diamond [100] axis. Fitted values of the stress anisotropy parameter α ≃ 0.95 indicate quasi-hydrostatic +conditions. (b) ODMR spectra recorded for NV centers in the micropillar for a magnetic field of 6 mT amplitude. The signals +at 73 GPa, 103 GPa, and 131 GPa are normalized for clarity, with contrast values of 5%, 3% and 1.5% respectively. +the diamond equation of state [25]. +Linear fit gives a +slope of −769±4 meV/(cm3·mol−1). A similar measure- +ment performed on a non modified diamond anvil yields +a weaker slope of −434 ± 2 meV/(cm3·mol−1). This sig- +nificant difference in the pressure dependence of the ZPL +is another indication of the deviatoric stress reduction +caused by the microstructuration. +ODMR measurements were also performed for the NV +centers hosted in the micropillar. +As shown in Fig. 3 +corresponding to the pressure evolution up to 130 GPa, +most of the detrimental effects previously observed and +attributed to deviatoric stress are now suppressed. The +spectra consistently show a negative contrast remaining +almost constant up to at least 100 GPa. Increasing fur- +ther the pressure up to 130 GPa (where the experiment +was stopped by one of the anvils breaking), a slight +decrease of the contrast was observed and is attributed +to a degraded efficiency of the microwave excitation +for frequencies higher than 4 GHz. The magnetic field +response remains also unchanged across the whole tested +pressure range. +The ODMR spectra exhibit a very +low zero-field splitting ∆σ of 0.29 ± 0.03 MHz/GPa +with increasing pressure, and a shift of the zero-field +center frequency D + δ of 13.42 ± 0.14 MHz/GPa. As +shown in Fig. 4 these values differ significantly from +those measured for NV centers in standard anvils, and +were consistent across four experimental runs performed +on different anvils, with pillars machined either using +a FIB or a femtosecond laser. +Applying the model +described above for the spin-mechanical interaction, +the evolution of the ODMR eigenfrequencies versus +the applied magnetic field were well-fitted using an +anisotropy parameter α ≃ 0.95 that stays constant +within the pressure range tested (Fig. 3a). Since α ≃ 1 +(a) +(b) +0 +25 +50 +75 +100 +125 +150 +Pressure (GPa) +3.0 +3.5 +4.0 +4.5 +D + δ (GHz) +Standard anvil +Micropillar run 1, 2, 3, 4 +Doherty et al. [11] +Dai et al. [22] +0 +25 +50 +75 +100 +125 +150 +Pressure (GPa) +0 +50 +100 +150 +200 +250 +∆σ (MHz) +Standard anvil +Micropillar run 1, 2, 3, 4 +FIG. 4. +(a) Pressure dependence of ODMR center frequency +D + δ, showing a quasi-linear shift of 13.42 ± 0.14 MHz/GPa +on the micropillar compared to 9.68 ± 0.8 MHz/GPa on the +standard anvil. The extrapolation of the values measured up +to 60 GPa in [11] and the fit up to 140 GPa from [22] are +given for comparison. +(b) Pressure dependence of ODMR +frequency splitting ∆σ at zero magnetic field. +At the mi- +cropillar, ∆σ increases by 0.29 ± 0.03 MHz/GPa instead of +3.89±0.06 MHz/GPa with the standard geometry of the anvil. + +5 +would indicate perfect hydrostaticity, this result gives +an independent confirmation of the almost hydrostatic +pressure applied on the NV centers in the micropillar. +Consequently, the microstructuration strategy enables +efficient magnetic field sensing at pressures higher +than 100 GPa with a sensitivity improved by orders of +magnitude compared to the use of a standard anvil with +a flat tip (see Supplementary Material). +Conclusion. +Microstructuration of diamond anvils, +implemented here by machining a micropillar on the +culet, +provides quasi-hydrostatic conditions for NV +centers implanted in the anvil up to 100 GPa and +above. +With this design NV magnetic sensing can be +implemented under such extreme pressures as if at +ambient pressure. This work opens the way to sensitive +and spatially resolved magnetic measurements in the +constrained environment of the DAC which should now +be used for a convincing observation of the Meissner +effect in super-hydrides. +We are grateful to Olivier Marie and Gr´egoire Le +Caruyer for machining of the diamond culets, to Flo- +rent Occelli for assistance in DACs preparation and to +Doroth´ee Colson and Anne Forget for annealing the dia- +mond anvils after nitrogen implantation. This work has +received funding from the EMPIR program co-financed +by the Participating States and the European Union’s +Horizon 2020 research and innovation program (20IND05 +QADeT), from the Agence Nationale de la Recherche +under the project SADAHPT and the ESR/EquipEx+ +program (grant number ANR-21-ESRE-0031), and from +the Paris ˆIle-de-France R´egion in the framework of DIM +SIRTEQ. JFR acknowledges support from Institut Uni- +versitaire de France. +∗ jean-francois.roch@ens-paris-saclay.fr +[1] P. 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M. +Wentzcovitch, and other members of the IPPS task +group, High Pressure Research 40, 299 (2020). + diff --git a/9tE4T4oBgHgl3EQfdwyk/content/tmp_files/load_file.txt b/9tE4T4oBgHgl3EQfdwyk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fa49adccf95fa6ec8eb1c503f1902fa114d4e57 --- /dev/null +++ b/9tE4T4oBgHgl3EQfdwyk/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf,len=592 +page_content='NV center magnetometry up to 130 GPa as if at ambient pressure Antoine Hilberer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='1 Lo¨ıc Toraille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3 Cassandra Dailledouze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='1 Marie-Pierre Adam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='1 Liam Hanlon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='1 Gunnar Weck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3 Martin Schmidt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='1 Paul Loubeyre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3 and Jean-Fran¸cois Roch1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' ∗ 1Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' ENS Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' CentraleSupelec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' LuMIn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' F-91190 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' France 2CEA DAM DIF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' F-91297 Arpajon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' France 3Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Laboratoire Mati`ere en Conditions Extrˆemes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 91680 Bruy`eres-le-Chˆatel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' France (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2023) Engineering a layer of nitrogen-vacancy (NV) centers on the tip of a diamond anvil creates a multipurpose quantum sensors array for high pressure measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' especially for probing magnetic and superconducting properties of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Expanding this concept above 100 GPa appears to be a substantial challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We observe that deviatoric stress on the anvil tip sets a limit at 40-50 GPa for practical magnetic measurements based on optically detected magnetic resonance (ODMR) of NV centers under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We show that this limit can be circumvented up to at least 130 GPa by machining a micropillar on the anvil tip to create a quasi-hydrostatic stress environment for the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This is quantified using the pressure dependence of the diamond Raman shift, the NV ODMR dependence on applied magnetic field, and NV photoluminescence spectral shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This paves the way for direct and reliable detection of the Meissner effect in superconductors above 100 GPa, such as super-hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The diamond anvil cell (DAC) is rou- tinely used to synthesize compounds under megabar (100 GPa) pressures, exhibiting novel phenomena and remarkable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Recent examples such as the ob- servation of metal hydrogen [1], superconductivity close to ambient temperature in superhydrides [2–4], or su- perionic water ice [5] are lacking detailed magnetic or transport measurements for their definite proof and clear understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' In particular, magnetic measurements re- main challenging at megabar pressures because they are mainly based on flux detection by inductive coils and must thus extract the signal of the few-micrometers sam- ples from the much larger magnetic background signal of the bulky DAC apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This constraint can be cir- cumvented by implementing in the DAC sensing methods that exploit the magnetic sensitivity of nitrogen-vacancy (NV) centers in diamond [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This method offers a tabletop optical microscopy instrumentation, the map- ping of the magnetic field in the sample chamber with micrometer spatial resolution and the absence of any sen- sitivity decrease with the sample size down to the mi- crometer scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Another key feature is the easy combi- nation with synchrotron X-ray characterizations to cor- relate the magnetic or superconducting properties with a well-defined crystallographic structure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Yet, the ex- tension of this technique to extreme pressures remains a challenge [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We investigate here how the existence of a deviatoric stress in the diamond anvil sets effective limits to the magnetic response of NV centers localized at the anvil tip to maximize sample proximity [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We then propose and implement a method that overcomes that limit and keeps the full NV quantum sensing capabilities at pressures above 100 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Experimental configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The negatively charged NV center is a point defect of diamond that emits visible photoluminescence (PL) by absorbing green photons and re-emitting red photons (at ambient pressure), with an electronic spin s = 1 in the ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' In the absence of external magnetic and stress fields, the ms = ±1 spin sublevels of the ground state are degenerate and separated by D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='87 GHz from the ms = 0 sublevel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Spin-dependent PL arises from a spin-selective difference in the non-radiative coupling to metastable singlet states, which also induces optical pumping into the ms = 0 state under green illumination [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The energy difference between the sublevels of the ground state can then be read out from the change of the NV luminescence intensity upon scanning the frequency of an additional microwave excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Dips in the PL intensity indicate that the excitation microwave frequency is resonant with a transition between two sublevels, leading to optically detected magnetic resonance (ODMR) that can be easily implemented by optically addressing the NV centers through the diamond anvil [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Here we use the same experimental configuration as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [6], keeping two crucial characteristics: 1) the NV centers are integrated in the DAC device by mounting a IIas ultra-pure Almax-Boehler design [100]-cut diamond anvil with a dense ensemble of NV centers (typically 104 NV/µm2) implanted at about 10 nm beneath the anvil surface using a nitrogen Focused Ion Beam (FIB) [13] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2) the microwave excitation is applied using an external single-turn coil above the rhenium gasket of the DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The metallic gasket is machined with a slit, filled with an epoxy-glue mixture ensuring sample confinement and DAC mechanical stability, that re-distributes the induced currents in the metal, leading to a focusing and amplification of the microwave flux in the sample chamber similarly to a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='05094v1 [quant-ph] 12 Jan 2023 2 (a) (b) (d) (c) NV layer Anvil 1 Anvil 2 Gasket Pressure ms=0 ms=±1 D D+훿 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (a) Energy diagram of the NV center ground state and evolution under stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (b) Schematic cross-section of the location of NV centers implanted as a layer below the anvil culet surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (c) Design of the machined gasket compatible with the MW excitation of the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Red arrows show initial MW excitation current in the wire loop, blue arrows are currents induced into the gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The areas shaded in red indicate the intensity of the MW field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (d) ODMR spectra of NV centers implanted in the tip of a standard diamond anvil at different pressures, as a function of a magnetic field applied along the [100] diamond axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Green dashed lines are fits of the eigenfrequencies computed with the NV ground state Hamiltonian given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Lenz lens [14] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Upon pressure increase, the PL excitation wavelength was decreased to match the blueshift of the NV absorption spectrum [11] by using continuous-wave (cw) lasers at successive wavelengths 532, 488, 457 and 405 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A customized confocal optical microscope was used to collect the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A static vector magnetic field was applied on the DAC using three Helmholtz coil pairs with an amplitude ranging between 0 and 10 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The magnetic field was aligned along the DAC axis with accuracy ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This orientation corresponds to the diamond [100] crystal axis for which all NV centers have equivalent responses to stress and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Pressure in the DAC was measured using the calibrated diamond Raman phonon mode at the anvil tip [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Stress effect on the NV magnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We per- formed cw-ODMR experiments on the NV centers un- der pressures ranging from 10 GPa to 70 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' At each pressure point, we collected the ODMR spectrum for the ensemble of NV centers under varying amplitude of the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1300 1400 1500 1600 1700 Raman shift ν (cm−1) Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=') P = 92 GPa Culet Micropillar 40 µm (a) (b) (c) (d) 25 50 75 Pressure (GPa) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='4 Vmol (cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='mol−1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='30 NV− ZPL energy (eV) Linear fits Micropillar Standard anvil 600 670 λ (nm) PL (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=') Anvil 1 Anvil 2 Gasket PTM NV layer −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='00 ln(V/V0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='20 ln(ν/ν0) Occelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [16] Micropillar FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (a) Scanning electron microscope image of a FIB- machined micropillar on a diamond anvil culet of 100 µm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The bottom panel shows a schematic cross-section with the distortion under pressure of the culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (b) Energy of the NV center zero-phonon line (ZPL) as a function of pres- sure and diamond volume, recorded for NV centers implanted in and out of the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Inset: typical PL spectra of the NV centers recorded at 0, 37 and 78 GPa (bottom to top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The arrows indicate the ZPL position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (c) Diamond Raman spectra recorded on a pressurized microstructured diamond anvil at 92 GPa, on and outside the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' In the spec- trum taken on the micropillar, the peak indicated by the star reveals hydrostatic compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (d) Raman frequency shift measured on the micropillar as a function of relative diamond volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Data from [16] is a reference of the Raman shift of diamond under hydrostatic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Four effects of stress on the ODMR signals are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' First, the zero-field center frequency D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='87 GHz increases almost linearly with a slope of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='6 MHz/GPa to a value D + δ, where δ is the pressure induced variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Second, a splitting ∆σ appears between the transition lines in the absence of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This splitting increases almost linearly with pressure with a slope of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='9 MHz/GPa and originates in deviatoric stress at the anvil culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Conse- quently, at a given pressure, the quasi-linear evolution of the Zeeman splitting due to the applied magnetic field can only be recovered above a compensating amplitude of the magnetic field that increases with pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This detrimental influence of stress hence weakens the NV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='8 51 GPa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='025 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='56 41 GPa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='000 MW frequency (GHz) α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='57 30 GPa PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=') α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='975 20 GPa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='4 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='950 10 GPa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='2 α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='900 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='875 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='850 0 5 10 5 10 5 10 5 10 5 10 α model fit B applied in [100] (mT)3 sensing magnetic sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Furthermore, the required larger applied bias magnetic field isn’t aligned with a given NV axis here, to overlap responses from all NV orientations, and thus mixes the sublevels of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This mixing perturbs the optically induced spin polarization and quenches the PL [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Third, the shape of the ODMR spectra differs from the conventional symmetrical pair of peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The contrast of the low frequency branch becomes gradually smaller than the high frequency branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' After vanishing at a pressure around 40 GPa, a slightly positive contrast reappears (increase of PL at resonance) above 50 GPa under high enough magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Finally, the overall observed ODMR contrast decreases severely under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' In the diamond lattice under mechanical stress (or equivalently strain), the Hamiltonian describing the NV center ground state is modified by a spin-mechanical in- teraction [18, 19] related to the stress tensor ↔σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The stress tensor must exhibit the cylindrical symmetry of the anvil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' At the anvil tip, the stress components parallel (σ∥) and perpendicular (σ⊥) to the surface differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Due to continuity of the normal stress component, σ⊥ is equal to the experimental pressure P in the DAC chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The tangential component, σ∥, is reduced by a factor α com- pared to σ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Using a simplified model of a semi-infinite anvil with a flat face and a circularly symmetric distribu- tion of pressure applied to this face, the α parameter was estimated about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='6 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Neglecting off-diagonal shear stress components, the stress tensor then reads as: ↔σ= � � αP 0 0 0 αP 0 0 0 P � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (1) Using this stress tensor, the diagonalization of the NV ground state Hamiltonian yields modified spin resonance frequencies which can be approximated to first order as: ν± = D + δ ± ∆/2 (2) where δ is the spectral shift due to compression, and ∆ = � ∆2σ + ∆2 B is the quadratic sum of the splittings respectively induced by the stress and by the magnetic field (see Supplementary Material for the full expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Since eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (2) is exact only for low off-axis magnetic field, a full numerical diagonalization was used to accurately fit the measured resonance frequencies, as shown by the green dashed lines in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Only two parameters, α and P, are hence needed to predict the magnetic field re- sponse under stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We obtained a value α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='56 that is essentially constant with pressure, quantifying deviatoric stress close to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='6 value given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Deviatoric stress thus introduces major modifications to the NV behavior as the anisotropic compression of the diamond host lattice distorts the C3v symmetry of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Here we quantified changes within the NV ground triplet states, but the stress dependence of the singlet states and the excited triplet states remains unexplored and is difficult to assess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' As a hypothesis, we attribute the observed modification and ultimate loss of ODMR contrast to the effect of deviatoric stress on these levels involved in the contrast mechanism [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This hypothesis is corroborated by recent results obtained on microdiamonds compressed quasi-hydrostatically inside the sample chamber of a DAC, for which the ODMR signal could be conserved up to 140 GPa [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' These results converge toward a possible circumventing strategy by ensuring hydrostatic compression of the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Restoring hydrostaticity with diamond microstructura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A strategy to try to mitigate deviatoric stress can be implemented by microstructuring the diamond anvil culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A successful geometry is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A pillar, 7 µm in diameter and with a 2 µm deep trench around it was FIB-machined on an NV-implanted dia- mond anvil culet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The pillar surface is thus disconnected from the anvil surface submitted to deviatoric stress in- duced by anvil cupping tension [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This also al- lows the pressure-transmitting medium (PTM) to fill the trench to immerse the pillar in a stress field close to hy- drostatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The pillar is then equivalent to a di- amond microdisk that would be integrated in the sample chamber of the DAC but ensures perfect reproducibility and removes any interface with the diamond culet to op- timize PL measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' As seen below, this design is also very robust and can withstand extreme pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The hydrostaticity of the stress exerted on diamond under pressure can be tested by measuring the Raman frequency of the diamond optical phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Under hy- drostatic conditions, the dependence of the frequency of the Raman scattering with diamond volume follows a Gruneisen relation of parameter γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='97(1) whereas the frequency shift is smaller under deviatoric stress [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2c, the Raman spectra measured at the dia- mond anvil culet on the micropillar and away from it dif- fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' In both cases, the broad asymmetric peak is associ- ated to the stress distribution within the thickness of the anvil that is optically probed and the high frequency edge is used to estimate the pressure [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' At the micropil- lar, a well separated peak appears with higher frequency shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The pressure evolution of its center wavenumber perfectly matches the value obtained for diamond under hydrostatic pressure [16] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This indi- cates that the tip of the micropillar hosting part of the NV center layer is then close to hydrostatic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Accordingly the PL spectrum of the NV layer in the micropillar shows a pressure induced blue shift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 2b) that can be quantified with the zero-phonon line (ZPL) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' While the NV ZPL dependence with pressure is not linear, its evolution becomes linear when plotted versus the compressed diamond volume estimated using 4 0 5 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 MW frequency (GHz) 20 GPa α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 α model fit 5 10 50 GPa α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 5 10 B applied in [100] (mT) 74 GPa α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 5 10 103 GPa α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 5 10 131 GPa α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='00 PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=') (a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5 Microwave frequency (GHz) PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=') |B| = 6 mT P = 73 GPa P = 103 GPa P = 131 GPa (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (a) ODMR spectra obtained from NV centers implanted in a micropillar at varying pressures, as a function of magnetic field applied along the diamond [100] axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Fitted values of the stress anisotropy parameter α ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 indicate quasi-hydrostatic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (b) ODMR spectra recorded for NV centers in the micropillar for a magnetic field of 6 mT amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The signals at 73 GPa, 103 GPa, and 131 GPa are normalized for clarity, with contrast values of 5%, 3% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' the diamond equation of state [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Linear fit gives a slope of −769±4 meV/(cm3·mol−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' A similar measure- ment performed on a non modified diamond anvil yields a weaker slope of −434 ± 2 meV/(cm3·mol−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This sig- nificant difference in the pressure dependence of the ZPL is another indication of the deviatoric stress reduction caused by the microstructuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' ODMR measurements were also performed for the NV centers hosted in the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3 corresponding to the pressure evolution up to 130 GPa, most of the detrimental effects previously observed and attributed to deviatoric stress are now suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The spectra consistently show a negative contrast remaining almost constant up to at least 100 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Increasing fur- ther the pressure up to 130 GPa (where the experiment was stopped by one of the anvils breaking), a slight decrease of the contrast was observed and is attributed to a degraded efficiency of the microwave excitation for frequencies higher than 4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The magnetic field response remains also unchanged across the whole tested pressure range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The ODMR spectra exhibit a very low zero-field splitting ∆σ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='03 MHz/GPa with increasing pressure, and a shift of the zero-field center frequency D + δ of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='14 MHz/GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 4 these values differ significantly from those measured for NV centers in standard anvils, and were consistent across four experimental runs performed on different anvils, with pillars machined either using a FIB or a femtosecond laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Applying the model described above for the spin-mechanical interaction, the evolution of the ODMR eigenfrequencies versus the applied magnetic field were well-fitted using an anisotropy parameter α ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='95 that stays constant within the pressure range tested (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Since α ≃ 1 (a) (b) 0 25 50 75 100 125 150 Pressure (GPa) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='5 D + δ (GHz) Standard anvil Micropillar run 1, 2, 3, 4 Doherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [11] Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [22] 0 25 50 75 100 125 150 Pressure (GPa) 0 50 100 150 200 250 ∆σ (MHz) Standard anvil Micropillar run 1, 2, 3, 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (a) Pressure dependence of ODMR center frequency D + δ, showing a quasi-linear shift of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='14 MHz/GPa on the micropillar compared to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='8 MHz/GPa on the standard anvil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' The extrapolation of the values measured up to 60 GPa in [11] and the fit up to 140 GPa from [22] are given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' (b) Pressure dependence of ODMR frequency splitting ∆σ at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' At the mi- cropillar, ∆σ increases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='03 MHz/GPa instead of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='06 MHz/GPa with the standard geometry of the anvil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' 5 would indicate perfect hydrostaticity, this result gives an independent confirmation of the almost hydrostatic pressure applied on the NV centers in the micropillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Consequently, the microstructuration strategy enables efficient magnetic field sensing at pressures higher than 100 GPa with a sensitivity improved by orders of magnitude compared to the use of a standard anvil with a flat tip (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Microstructuration of diamond anvils, implemented here by machining a micropillar on the culet, provides quasi-hydrostatic conditions for NV centers implanted in the anvil up to 100 GPa and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' With this design NV magnetic sensing can be implemented under such extreme pressures as if at ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This work opens the way to sensitive and spatially resolved magnetic measurements in the constrained environment of the DAC which should now be used for a convincing observation of the Meissner effect in super-hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' We are grateful to Olivier Marie and Gr´egoire Le Caruyer for machining of the diamond culets, to Flo- rent Occelli for assistance in DACs preparation and to Doroth´ee Colson and Anne Forget for annealing the dia- mond anvils after nitrogen implantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' This work has received funding from the EMPIR program co-financed by the Participating States and the European Union’s Horizon 2020 research and innovation program (20IND05 QADeT), from the Agence Nationale de la Recherche under the project SADAHPT and the ESR/EquipEx+ program (grant number ANR-21-ESRE-0031), and from the Paris ˆIle-de-France R´egion in the framework of DIM SIRTEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' JFR acknowledges support from Institut Uni- versitaire de France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' ∗ jean-francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='roch@ens-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='fr [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Loubeyre, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Occelli, and P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Ninet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Datchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Mezouar, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Loubeyre, Physical Review Letters 128, 165701 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Lesik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Plisson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Toraille, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Renaud, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Occelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Schmidt, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Salord, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Delobbe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Debuisschert, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Rondin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Loubeyre, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Roch, Science 366, 1359 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Hsieh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Bhattacharyya, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Zu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Mittiga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Smart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Machado, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Kobrin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' H¨ohn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Rui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Kamrani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Chatterjee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Choi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Zaletel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Struzhkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Moore, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Levitas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Jeanloz, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Yao, Science 366, 1349 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} 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Mashimo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} +page_content=' Wentzcovitch, and other members of the IPPS task group, High Pressure Research 40, 299 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE4T4oBgHgl3EQfdwyk/content/2301.05094v1.pdf'} diff --git a/ANFRT4oBgHgl3EQftjjG/content/tmp_files/2301.13628v1.pdf.txt b/ANFRT4oBgHgl3EQftjjG/content/tmp_files/2301.13628v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c7c654971735033a60a18c1662358c86ce4f421 --- /dev/null +++ b/ANFRT4oBgHgl3EQftjjG/content/tmp_files/2301.13628v1.pdf.txt @@ -0,0 +1,714 @@ +Coherent driving of direct and indirect excitons in a quantum dot molecule +Frederik Bopp,1, ∗ Johannes Schall,2 Nikolai Bart,3 Florian Vogl,1 Charlotte Cullip,1 Friedrich +Sbresny,4 Katarina Boos,4 Christopher Thalacker,1 Michelle Lienhart,1 Sven Rodt,2 Dirk Reuter,5 +Arne Ludwig,3 Andreas Wieck,3 Stephan Reitzenstein,2 Kai M¨uller,4 and Jonathan J. Finley1, † +1Walter Schottky Institut, School of Natural Sciences, and MCQST, +Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany +2Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany +3Faculty of Physics and Astronomy, Ruhr-Universit¨at Bochum, +Universit¨atsstraße 150, 44801 Bochum, Germany +4Walter Schottky Institut, School of Computation, Information and Technology, and MCQST, +Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany +5Paderborn University, Department of Physics, Warburger Straße 100, 33098 Paderborn, Germany +(Dated: February 1, 2023) +Quantum dot molecules (QDMs) are one of the few quantum light sources that promise deter- +ministic generation of one- and two-dimensional photonic graph states. +The proposed protocols +rely on coherent excitation of the tunnel-coupled and spatially indirect exciton states. Here, we +demonstrate power-dependent Rabi oscillations of direct excitons, spatially indirect excitons, and +excitons with a hybridized electron wave function. An off-resonant detection technique based on +phonon-mediated state transfer allows for spectrally filtered detection under resonant excitation. +Applying a gate voltage to the QDM-device enables a continuous transition between direct and +indirect excitons and, thereby, control of the overlap of the electron and hole wave function. This +does not only vary the Rabi frequency of the investigated transition by a factor of ≈ 3, but also +allows to optimize graph state generation in terms of optical pulse power and reduction of radiative +lifetimes. +I. +INTRODUCTION +The use of single photons as flying qubits facilitates +transmission of quantum information at the speed of +light. However, transfer over large distances unavoidably +comes with losses and decoherence. Encoding quantum +information on an ensemble of entangled photons, a so- +called graph state [1], instead of a single photon, provides +a possibility to mitigate the losses is transmission chan- +nels [2, 3]. +Furthermore, other specific forms of graph +states such as photonic cluster states promise realization +of measurement-based quantum computing [4] as well as +quantum error correction [5, 6]. +Following +the +Lindner-Rudolph +protocol [7], +one- +dimensional photonic cluster states can be deterministi- +cally generated by utilizing single spins in semiconductor +quantum dots (QDs). The polarization entanglement of +up to five photons has been achieved in a one-dimensional +cluster state has been achieved [8] and most recent ex- +periments demonstrate localizable entanglement over ten +photons [9]. While the nanophotonic environment of QDs +provides high photon emission rates, the cluster state cre- +ation fidelity is limited by spin dephasing and modified +selection rules in the presence of a transverse magnetic +field [9]. These challenges can be overcome by using a +pair of tunnel coupled and vertically stacked QDs, so +called quantum dot molecules (QDMs) [10]. Besides pro- +longing the spin coherence compared to single quantum +∗ frederik.bopp@wsi.tum.de +† finley@wsi.tum.de +dots [11], QDMs possess an unique level structure [12]. +This level structure enables, for example, spin rotations +and spin readout transitions without application of a +magnetic field. The ability to create spatially indirect +excitons, with one charge carrier occupying the upper +and one the lower QD [13], provides a cycling transi- +tion which can be used for generating time-bin entangled +photons [10]. Moreover, QDMs are proposed to generate +two-dimensional photonic cluster states by harnessing the +tunnel coupling between the two QDs and inter-dot con- +trol gates [14]. +The foundation for creating one- and two-dimensional +photonic cluster states is the occurrence of excitons in +spatially direct, spatially indirect, and hybridized config- +urations [15]. In these different configurations, the charge +carriers of an electron-hole pair are located in the same +QD, in different QDs, or one of the charge wave func- +tions is hybridized over both quantum dots, respectively. +In each configuration, the overlap of the electron and +hole wave functions and, therefore, the transition dipole +moment (TDM) of the corresponding optical transition +differs. This results in a change of both the lifetime of the +excited state and the pulse area needed for maximal pop- +ulation inversion [16]. While the lifetime influences the +cluster state creation efficiency and rate, the π-pulse area +sets the intensity of the required optical control pulses. +Hence, the TDM of the addressed transitions influences +the generation process of photonic cluster states. Fur- +thermore, the proposed protocols require coherent exci- +tation of electron-hole pairs in various exciton configura- +tions to control and readout the exciton spin state. +In this work, we demonstrate coherent Rabi oscillations +arXiv:2301.13628v1 [cond-mat.mes-hall] 31 Jan 2023 + +2 +of direct, spatially indirect, and hybridized excitons in a +single QDM. An off-resonant detection technique is in- +troduced and applied, relying on phonon-mediated state +transfers. We examine the dependence of the Rabi fre- +quency on the excitonic configuration, as the overlap of +the electron and hole wave functions changes. Tuning the +electric field via a gate voltage allows electrical control of +this wave function overlap and, therefore, of the pulse +area needed for population inversion. +In this way, we +demonstrate and quantify electric control of the TDM. +Finally, a simple one-dimensional model of a double-well +potential allows us to model the voltage-dependence of +the TDM. +II. +RESULTS +By vertically stacking two QDs with a separation in +the nm regime, charge wave functions can hybridize +across both QDss. +In addition, both direct and spa- +tially indirect excitons can form. Figure 1 (a) illustrates +a schematic band-diagram of a QDM. The two QDs +are depicted by a double-well potential, in which elec- +trons (filled circle) and holes (empty circle) are trapped. +The design of the investigated sample, described in Ap- +pendix A, energetically favours the location of a hole +in the top QD. Consequently, a direct/indirect exciton +(red/blue ellipse) forms, when an electron is trapped in +the top/bottom QD. The QDM is embedded in a p-i-n +diode structure; applying a gate voltage V facilitates tun- +ing of the energy levels of both QDs relative to each other. +In this way, the direct and indirect exciton energies can +be brought into resonance. At the resonance condition, +the electron wave function hybridizes across both dots, +molecular bonding and anti-bonding states form, and an +avoided crossing between the orbital states occurs. Since +we can control the tunnel coupling between the two QDs +by varying the gate voltage, we use this dependency to +investigate coherent driving of different exciton configu- +rations. +The most elemental charge state exhibiting the hy- +bridization of wave functions is the neutral exciton (X0). +Figure 1 (b) shows a voltage-dependent photolumines- +cence measurement of the X0. We make use of a two- +phase electrical and optical sequence to deterministi- +cally prepare the QDM in a zero-charge ground-state +and individually adjust the tunnel coupling [17]. Excit- +ing the energetically higher p-shell orbital of the upper +dot at 1353.6 meV enables the unimpeded detection of +the X0 s-shell emission for multiple coupling conditions. +At 0.16 V, the electron wave function hybridizes and an +avoided crossing forms. +The resulting electron eigen- +states are described by symmetric and antisymmetric +wave functions [13]. The corresponding lower and higher +energy transitions of the avoided crossing are denoted +LOW and UP in Figure 1 (b). The red and blue dashed +lines depict the energies of a direct and indirect exciton, +respectively. By increasing the gate voltage, the exciton +0 +50 +100 +150 +0 +5 +10 +0.05 +0.1 +0.15 +0.2 +1336 +1338 +1340 +1342 +Energy (meV) +Gate Voltage (V) +Power1/2 (nW1/2) +Emission (cts/3s) +kCounts (/s) +103 +102 +101 +V +z +E +(a) +(b) +AlGaAs +Excita�on +Emission +Emission +UP +LOW +cgs +UP +LOW +(c) +(d) +0.1 V +𝛾P +FIG. 1. Rabi oscillations of the neutral exciton in a QDM. +(a) Schematic band structure of a QDM represented by a +double-well potential. An AlGaAs barrier below the molecule +prolongs tunneling times for electrons while not affecting tun- +neling for holes. One hole (empty circle) is located in the up- +per QD, while electrons (filled circles) occur in both dots. As +a consequence, direct (red ellipse) and indirect (blue ellipse) +excitons arise. A gate voltage V applied to the sample facil- +itates tuning of the direct and indirect exciton energies rela- +tive to each other. (b) Voltage-dependent photoluminescence +of the neutral exciton. The red and blue dashed lines indicate +the energies of the direct and indirect excitons. tunnel cou- +pling between the two QDs leads to an avoided crossing with +a symmetric (pink) and an anti-symmetric (green) electron +eigenstate. The upper (lower) energy transition is called UP +(LOW). Triangles indicate the excitation energy and voltage +applied in Figure 2. (c) Neutral exciton state diagram illus- +trating the excitation and detection scheme for monitoring +Rabi oscillations. While a resonant light field (green) is driv- +ing UP, a phonon-mediated state transfer with rate γP (black +arrow) is enabling emission from both UP and the energet- +ically detuned LOW. (d) Power-dependent Rabi oscillations +when exciting UP and detecting UP (green) or LOW (pink) +at 0.1 V. + +ge1336 +0050.150.210e +1338 +uS +C1340n +01342SMSGaateVolta3 +character changes from direct to hybridized to indirect +for the upper energy branch, and vice versa for the lower +energy branch. As a result, the overlap of the electron +and hole wave functions changes. +The change of the wave function overlap is quanti- +fied by coherently driving Rabi oscillations on the ex- +citon transition. +The Rabi frequency of a resonantly +excited two-level system ΩR = +�� E0D +ℏ +�� is linearly depen- +dent on the TDM D, which in return is proportional to +the overlap of the electron and hole wave function [18]. +In addition, ΩR depends linearly on the electric driving +field amplitude E0. The E0-dependence allows the ob- +servation of power-dependent Rabi oscillations [19]. For +this purpose, a 5 ps laser pulse is applied to resonantly +drive the crystal ground state (cgs)-to-X0 transition in +the QDM. The occupation of the excited state is mon- +itored by detecting the photons emitted by the driven +two-level system. Commonly, emission from resonantly +excited states is detected in a cross-polarized setup con- +figuration to suppress the excitation laser [20]. At high +excitation power, however, laser light can leak into the +detection channel and reduce the signal-to-noise ratio. +We propose and demonstrate a readout technique utiliz- +ing a phonon-mediated state transfer [21], which detunes +the emitted photons energetically from the two-level sys- +tem. +Thereby, the limitation of an insufficiently sup- +pressed excitation laser is eliminated via spectral filter- +ing, and the visibility of the Rabi oscillations is increased. +Figure 1 (c) visualizes the state diagram of the X0. The +two excited states UP and LOW can both radiatively de- +cay into the cgs. A phonon emission process with rate +γP can transfer the electron from the UP to the LOW +configuration [21]. +Since the excitation pulse length is +short compared to the decay rates, the cgs-UP system +is well approximated by a two-level system. It is coher- +ently driven by a 5 ps laser pulse (green arrow). Fig- +ure 1 (d) shows the power-dependent resonance fluores- +cence emission of the UP transition as green data points. +The measurement is performed at 0.1 V, such that the +driven transition exhibits a direct exciton character, as +shown in Figure 1 (b). Rabi oscillations are observed up +to a pulse area of slightly above 2π and 602 nW. However, +a decreasing signal-to-noise ratio prevents the detection +of oscillations above 602 nW due to nsufficient suppres- +sion of the excitation laser. +To improve the signal-to-noise ratio, which decreases +with increasing power, +we make use of a phonon- +mediated state transfer. +The emission of a phonon +transfers the electron from the UP into the LOW +configuration. +This process can only occur as long as +the system is in the excited state. Thus, the ensemble +occupation of LOW is proportional to the ensemble +occupation of UP, and so is the number of emitted +photons of both transitions. +In addition, due to the +avoided crossing, the emission of LOW is at least +2.1 meV detuned from the driving energy for any gate +voltage, which allows the spectral filtering of the emis- +sion from the excitation laser pulse. Thus, the resonant +kCounts (/s) +Power1/2 (nW1/2) +UP +LOW +0.1 V +0.22 V +0 +50 +100 +0 +1 +2 +3 +0 +50 +100 +0 +5 +10 +0 +50 +100 +0 +2 +4 +0 +50 +100 +0 +0.1 +0.2 +FIG. 2. +Rabi oscillations of the UP and LOW branch at 0.1 V +(left) and 0.22 V (right) by phonon-mediated state transfers. +The red data points correspond to a direct, the blue to an +indirect driven transition. +excitation of the two-level system and the off-resonant +monitoring of its excited state occupation are achieved +simultaneously. +The power-dependent emission of the +LOW transition when exciting UP is shown by the +pink data points in Figure 1 (d). +Below 602 nW, both +readout techniques show the same Rabi frequency as +expected, confirming the proportionality of occupancy +between UP and LOW. However, in contrast to the +resonant detection (green), Rabi oscillations are well +resolvable up to a pulse area of 7π. +The reduction of +the oscillation amplitude arises from interactions with +phonons [22], while the increase of the mean is attributed +to a slightly chirped excitation laser pulse [23]. From the +relative intensities of both transitions, we can conclude +that the phonon induced relaxation rate is compara- +ble to the radiative decay rate of the direct UP transition. +Electric control of the tunnel coupling between the two +QDs allows coherent excitation of electron-hole pairs in +different occupation configurations. Figure 2 shows the +power-dependent emission of the QDM while resonantly +exciting UP and detecting LOW (green dashed box, UP). +The measurements are performed at 0.1 V (left) and +0.22 V (right), on either side of the avoided crossing. The +red and blue data points indicate a direct and indirect +character of the excited transition, respectively. We ob- +serve Rabi oscillations for both the direct and indirect +transitions, which confirms that coherent excitation of a +spatially indirect exciton is possible. However, the Rabi + +4 +0 +0.1 +0.2 +0.3 +Gate Voltage (V) +0 +0.05 +0.1 +0.15 +Rabi Frequency (1/nW1/2) +0 +0.2 +0.4 +0.6 +0.8 +1 +FIG. 3. +Measured voltage dependent Rabi frequency of +UP (green) and LOW (pink), plotted on the left axes. The +right axes visualizes the calculated overlap of the electron +and hole wave functions as a function of the voltage, where +the pink/green dashed line corresponds to the lowest/second- +lowest electron eigenenergy. The red/blue shaded background +indicates the direct/indirect character of the transition. +frequency of the indirect configuration is reduced com- +pared to the direct configuration. This is caused by the +reduced overlap of the electron and hole wave functions +and the accompanying decrease of the TDM for the in- +direct exciton. +A verification of these results is found by performing +the same experiments on the lower branch (Figure 2, +pink box, LOW). Similar to the previous case, a phonon +absorption process facilitates a state transfer from LOW +to UP and, therefore, the off-resonant detection of UP +while exciting LOW. This allows us to off-resonantly +monitor Rabi oscillations of the LOW branch. +The +investigated gate voltages of both excitation cases are +chosen to be ±0.06 V away from the avoided crossing. +Therefore, the electron configuration of the upper branch +at 0.1 V (0.22 V) resembles the electron configuration of +the lower branch at 0.22 V (0.1 V). This leads to a com- +parable Rabi frequency of the direct (red) and indirect +(blue) exciton of the upper and lower energy transition +at the two voltages. The difference in absolute counts +between the excitation of UP and LOW is attributed +to the underlying phonon process. +When exciting UP +(LOW), we rely on the emission (absorption) of a +phonon to detect the signal. +Since the measurements +are performed at 10 K, the probability of absorbing a +phonon is strongly reduced compared to the emission. +Since our QDM device allows continuous tuning the +gate voltage while maintaining the prepared charge state, +arbitrary exciton configurations can be set. +Thereby, +the overlap of the electron and hole wave functions is +analyzable for any coupling condition. Figure 3 shows +the power-dependent Rabi frequencies as a function of +the gate voltage for the upper (green) and lower branch +(pink). The Rabi frequencies are extracted by fitting a +sin2(laser power) function to the data, with an exponen- +tial decay to take phonon dephasing into account, and a +linear increase with intensity to compensate for a chirped +excitation pulse. We observe a continuous increase (de- +crease) of the frequency when transitioning from an indi- +rect (direct) to a direct (indirect) exciton. By raising the +gate voltage and following the UP transition, the elec- +tron occupation shifts from the top to the bottom dot. +The opposite holds for the LOW transition. This leads +to a continuous variation of the overlap of the electron +and hole wave functions and, consequently, to a change +in the Rabi frequency. Within the investigated range be- +tween 0.1 V and 0.22 V, we are able to electrically tune +the Rabi frequency by a factor of ≈ 3. +The wave function overlap is modeled by calculating +the eigenenergies and -values of a tilted, one-dimensional +double-squarewell potential representing the QDM. By +fitting the energy difference between the two lowest eigen- +states to the voltage-dependent separation of UP and +LOW, the depth of the squarewell potential and the effec- +tive electron mass are determined. A detailed description +of the model is provided in Appendix B. The right axes +of Figure 3 shows the overlap of the electron and hole +wave functions |⟨ψe|ψh⟩| for the lowest (pink) and sec- +ond lowest (green) electron eigenenergy by dashed lines. +The electron eigenenergies correspond to the LOW and +UP transition, respectively. The one-dimensional model +provides a remarkably good description of the measured +voltage-dependent Rabi frequencies. Thereby, the Rabi +frequency can be related to the TDM of direct, indirect, +and hybridized excitons, which allows determination the +π-pulse area as well as the difference in radiative lifetime +of the corresponding transition. +III. +DISCUSSION AND SUMMARY +Adressing direct, indirect, and hybridized excitons is +fundamental for using QDMs as spin-photon interfaces. +In addition, the electrical tuneability of the TDM of the +adressed transitions at and around the tunnel coupling +regime is one key parameter of a QDM. It not only de- +termines the π-pulse power of the addressed transition, +as shown in this work, but is also directly related to the +lifetime of the excited state. Therefore, it is one of the +parameters setting the creation rate for generating one- +and two- dimensional photonic cluster states as well as +for performing quantum-repeater protocols. +We have demonstrated the coherent excitation of di- +rect, indirect, and hybridized excitons – one of the el- +ementary building blocks for creating photonic cluster +states from QDMs. We use non-resonant readout, which +is facilitated by phonon-mediated charge relaxation and +excitation between the two lowest energy eigenstates of +the electron.. Voltage-dependent Rabi oscillations show +a continuous increase of the Rabi frequency when tran- +sitioning from an indirect to a direct exciton. +This is + +5 +attributed to an electrically controlled increase of the +TDM of a direct compared to an indirect transition. Fur- +thermore, we apply a one-dimensional model to calculate +the overlap of the X0 electron and hole wave functions. +Within the voltage range presented, we are able to tune +the Rabi frequency and consequently the TDM by a fac- +tor of ≈ 3. This corresponds to a variation of the radia- +tive lifetime between a direct and an indirect exciton by +a factor of ≈ 9, as it scales quadratically with the TDM. +The coherent excitation and the electrical tunability +between various exciton configurations in QDMs not only +paves the way towards the generation of entangled multi- +photon states. It might also enable protocols which uti- +lize fast electrical switching between the exciton config- +urations. This can reduce their lifetime and the π-pulse +power and highly improve the cluster state generation +rate. +ACKNOWLEDGMENTS +The authors gratefully acknowledge financial sup- +port from the German Federal Ministry of Educa- +tion and Research (BMBF) via Q.Link.X (16KIS0874, +16KIS086), QR.X (16KISQ027, 16KISQ014, 16KISQ012 +and 16KISQ009), the European Union’s Horizon 2020 re- +search and innovation program under grant agreement +862035 (QLUSTER) and the Deutsche Forschungsge- +meinschaft (DFG, German Research Foundation) via +SQAM (FI947-5-1), DIP (FI947-6-1), and the Excellence +Cluster MCQST (EXC-2111, 390814868). F.B. gratefully +acknowledges the Exploring Quantum Matter (ExQM) +programme funded by the State of Bavaria. F.S., K.B., +and K.M. gratefully acknowledge the BMBF for financial +support via project MOQUA (13N14846). +Appendix A: Sample +The investigated InAs QDM is enclosed in a GaAs +matrix and was grown by molecular beam epitaxy. +It +consists of two vertically stacked QDs. +The inter-dot +coupling strength is determined by a wetting layer- +to-wetting layer separation of 10 nm. +In addition, an +AlxGa(x−1)As barrier (x = 0.33) with a thickness of +2.5 nm is placed between the dots to reduce the coupling +strength. The height of the top (bottom) QD was fixed +to 2.9 nm (2.7 nm) via the indium-flush technique during +growth. +This height configuration facilitates electric +field-induced tunnel coupling of orbital states in the +conduction band. A 50 nm thick AlxGa(x−1)As tunnel +barrier (x = 0.33) was grown 5 nm below the QDM +to prolong electron tunneling times. +The molecule is +embedded in a p-i-n diode, with the doped regions used +as contacts to gate the sample. The diode contacts are +placed more than 150 nm away from the molecule to +prevent uncontrolled charge tunneling into the QDM. +Furthermore, a distributed Bragg reflector was grown +below the diode and a circular Bragg grating was +positioned deterministically via in-situ electron beam +lithography above an individual and pre-selected QDM +to improve photon in- and outcoupling efficiencies [24]. +All measurements are performed at 10 K. +Appendix B: Double-well potential model +To calculate the overlap of the electron and hole wave +functions, we set up a model consisting of a double- +squarewell potential representing the conduction band of +the QDM. We assume that the variation of the in-plane +wave functions is small compared to the wave functions +along the growth direction z, when changing the gate +voltage. This assumption is reasonable, since the confine- +ment of charges along the growth axes and the translation +introduced by the gate voltage along the growth axes ex- +ceeds the in-plane variation. We can, therefore, approach +the problem with a one-dimensional model and expect +an acceptable degree of accuracy. The potential V (z) is +designed to match the dimensions of the QDM with re- +spect to the tunnel barrier width and dot heights (see +Section A). z is here the growth direction of the sample. +In addition, we tilt the potential to imitate the presence +of an applied gate voltage. Solving the time-independent +Schr¨odinger equation for a given gate voltage allows us to +obtain the envelope functions Ψ and their eigenenergies +E of an electron with mass me trapped in the double-well +potential. The envelope functions are then used to rep- +resent the wave function of the electron since the Bloch +part of the wave functions are only weakly sensitive to +electric fields of the magnitude applied here. +To define the free parameters of the double-well model, +we determine the effective electron mass me and the po- +tential depth by fitting the difference between the cal- +culated two lowest eigenenergies to the measured energy +difference ∆E between UP and LOW. ∆E between the +two X0 branches is purely determined by the energy dif- +ference between the electron eigenstates. For calculating +the hole wave function, the potential is inverted to repre- +sent the valance band. Its depth is set to match half the +depth of the electron potential whereas the heavy hole +mass is set to match mh = 10 me [25]. Since we are inter- +ested in calculating the overlap of wave functions rather +than absolute transition energies, it is sufficient to work +with relative values in the model. +[1] H. J. Briegel and R. Raussendorf, Persistent entangle- +ment in arrays of interacting particles, Physical Review +Letters 5, 910 (2001). + +6 +[2] K. Azuma, K. Tamaki, and H. K. Lo, All-photonic quan- +tum repeaters, Nature Communications 6 (2015). +[3] K. Azuma and G. Kato, Aggregating quantum repeaters +for the quantum internet, Physical Review A 96, 032332 +(2017). +[4] R. Raussendorf and H. J. Briegel, A one-way quantum +computer, Physical Review Letters 86, 5188 (2001). +[5] D. Schlingemann and R. F. Werner, Quantum error- +correcting codes associated with graphs, Physical Review +A - Atomic, Molecular, and Optical Physics 65, 8 (2002). +[6] B. +A. +Bell, +D. +A. +Herrera-Mart´ı, +M. +S. +Tame, +D. Markham, W. J. Wadsworth, and J. G. 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Chauveau, A. Ludwig, A. D. Wieck, D. E. Reiter, +T. Kuhn, and R. J. Warburton, Demonstrating the de- +coupling regime of the electron-phonon interaction in a +quantum dot using chirped optical excitation, Physical +Review B 95, 241306 (2017). +[24] J. Schall, M. Deconinck, N. Bart, M. Florian, M. Hel- +versen, C. Dangel, R. Schmidt, L. Bremer, F. Bopp, +I. H¨ullen, C. Gies, D. Reuter, A. D. Wieck, S. Rodt, +J. J. Finley, F. Jahnke, A. Ludwig, and S. Reitzenstein, +Bright Electrically Controllable Quantum-Dot-Molecule +Devices Fabricated by In Situ Electron-Beam Lithogra- +phy, Advanced Quantum Technologies 4, 2100002 (2021). +[25] N. Bouarissa and H. Aourag, Effective masses of electrons +and heavy holes in InAs, InSb, GaSb, GaAs and some of +their ternary compounds, Infrared Physics & Technology +40, 343 (1999). + diff --git a/ANFRT4oBgHgl3EQftjjG/content/tmp_files/load_file.txt b/ANFRT4oBgHgl3EQftjjG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9473667b2b1718d782507101141129f8fd1c6037 --- /dev/null +++ b/ANFRT4oBgHgl3EQftjjG/content/tmp_files/load_file.txt @@ -0,0 +1,504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf,len=503 +page_content='Coherent driving of direct and indirect excitons in a quantum dot molecule Frederik Bopp,1, ∗ Johannes Schall,2 Nikolai Bart,3 Florian Vogl,1 Charlotte Cullip,1 Friedrich Sbresny,4 Katarina Boos,4 Christopher Thalacker,1 Michelle Lienhart,1 Sven Rodt,2 Dirk Reuter,5 Arne Ludwig,3 Andreas Wieck,3 Stephan Reitzenstein,2 Kai M¨uller,4 and Jonathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Finley1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' † 1Walter Schottky Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' and MCQST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Am Coulombwall 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Germany 2Technische Universit¨at Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Hardenbergstraße 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 10623 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Germany 3Faculty of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Ruhr-Universit¨at Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Universit¨atsstraße 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 44801 Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Germany 4Walter Schottky Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' School of Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Information and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' and MCQST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Am Coulombwall 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Germany 5Paderborn University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Warburger Straße 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 33098 Paderborn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Germany (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 2023) Quantum dot molecules (QDMs) are one of the few quantum light sources that promise deter- ministic generation of one- and two-dimensional photonic graph states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The proposed protocols rely on coherent excitation of the tunnel-coupled and spatially indirect exciton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Here, we demonstrate power-dependent Rabi oscillations of direct excitons, spatially indirect excitons, and excitons with a hybridized electron wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' An off-resonant detection technique based on phonon-mediated state transfer allows for spectrally filtered detection under resonant excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Applying a gate voltage to the QDM-device enables a continuous transition between direct and indirect excitons and, thereby, control of the overlap of the electron and hole wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This does not only vary the Rabi frequency of the investigated transition by a factor of ≈ 3, but also allows to optimize graph state generation in terms of optical pulse power and reduction of radiative lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' INTRODUCTION The use of single photons as flying qubits facilitates transmission of quantum information at the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' However, transfer over large distances unavoidably comes with losses and decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Encoding quantum information on an ensemble of entangled photons, a so- called graph state [1], instead of a single photon, provides a possibility to mitigate the losses is transmission chan- nels [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Furthermore, other specific forms of graph states such as photonic cluster states promise realization of measurement-based quantum computing [4] as well as quantum error correction [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Following the Lindner-Rudolph protocol [7], one- dimensional photonic cluster states can be deterministi- cally generated by utilizing single spins in semiconductor quantum dots (QDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The polarization entanglement of up to five photons has been achieved in a one-dimensional cluster state has been achieved [8] and most recent ex- periments demonstrate localizable entanglement over ten photons [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' While the nanophotonic environment of QDs provides high photon emission rates, the cluster state cre- ation fidelity is limited by spin dephasing and modified selection rules in the presence of a transverse magnetic field [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' These challenges can be overcome by using a pair of tunnel coupled and vertically stacked QDs, so called quantum dot molecules (QDMs) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Besides pro- longing the spin coherence compared to single quantum ∗ frederik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='bopp@wsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='de † finley@wsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='de dots [11], QDMs possess an unique level structure [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This level structure enables, for example, spin rotations and spin readout transitions without application of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The ability to create spatially indirect excitons, with one charge carrier occupying the upper and one the lower QD [13], provides a cycling transi- tion which can be used for generating time-bin entangled photons [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Moreover, QDMs are proposed to generate two-dimensional photonic cluster states by harnessing the tunnel coupling between the two QDs and inter-dot con- trol gates [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The foundation for creating one- and two-dimensional photonic cluster states is the occurrence of excitons in spatially direct, spatially indirect, and hybridized config- urations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In these different configurations, the charge carriers of an electron-hole pair are located in the same QD, in different QDs, or one of the charge wave func- tions is hybridized over both quantum dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In each configuration, the overlap of the electron and hole wave functions and, therefore, the transition dipole moment (TDM) of the corresponding optical transition differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This results in a change of both the lifetime of the excited state and the pulse area needed for maximal pop- ulation inversion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' While the lifetime influences the cluster state creation efficiency and rate, the π-pulse area sets the intensity of the required optical control pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Hence, the TDM of the addressed transitions influences the generation process of photonic cluster states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Fur- thermore, the proposed protocols require coherent exci- tation of electron-hole pairs in various exciton configura- tions to control and readout the exciton spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In this work, we demonstrate coherent Rabi oscillations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='13628v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='mes-hall] 31 Jan 2023 2 of direct, spatially indirect, and hybridized excitons in a single QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' An off-resonant detection technique is in- troduced and applied, relying on phonon-mediated state transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We examine the dependence of the Rabi fre- quency on the excitonic configuration, as the overlap of the electron and hole wave functions changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Tuning the electric field via a gate voltage allows electrical control of this wave function overlap and, therefore, of the pulse area needed for population inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In this way, we demonstrate and quantify electric control of the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Finally, a simple one-dimensional model of a double-well potential allows us to model the voltage-dependence of the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' RESULTS By vertically stacking two QDs with a separation in the nm regime, charge wave functions can hybridize across both QDss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, both direct and spa- tially indirect excitons can form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Figure 1 (a) illustrates a schematic band-diagram of a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The two QDs are depicted by a double-well potential, in which elec- trons (filled circle) and holes (empty circle) are trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The design of the investigated sample, described in Ap- pendix A, energetically favours the location of a hole in the top QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Consequently, a direct/indirect exciton (red/blue ellipse) forms, when an electron is trapped in the top/bottom QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The QDM is embedded in a p-i-n diode structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' applying a gate voltage V facilitates tun- ing of the energy levels of both QDs relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In this way, the direct and indirect exciton energies can be brought into resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' At the resonance condition, the electron wave function hybridizes across both dots, molecular bonding and anti-bonding states form, and an avoided crossing between the orbital states occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Since we can control the tunnel coupling between the two QDs by varying the gate voltage, we use this dependency to investigate coherent driving of different exciton configu- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The most elemental charge state exhibiting the hy- bridization of wave functions is the neutral exciton (X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Figure 1 (b) shows a voltage-dependent photolumines- cence measurement of the X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We make use of a two- phase electrical and optical sequence to deterministi- cally prepare the QDM in a zero-charge ground-state and individually adjust the tunnel coupling [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Excit- ing the energetically higher p-shell orbital of the upper dot at 1353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='6 meV enables the unimpeded detection of the X0 s-shell emission for multiple coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='16 V, the electron wave function hybridizes and an avoided crossing forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The resulting electron eigen- states are described by symmetric and antisymmetric wave functions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The corresponding lower and higher energy transitions of the avoided crossing are denoted LOW and UP in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The red and blue dashed lines depict the energies of a direct and indirect exciton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' By increasing the gate voltage, the exciton 0 50 100 150 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='2 1336 1338 1340 1342 Energy (meV) Gate Voltage (V) Power1/2 (nW1/2) Emission (cts/3s) kCounts (/s) 103 102 101 V z E (a) (b) AlGaAs Excita�on Emission Emission UP LOW cgs UP LOW (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V 𝛾P FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Rabi oscillations of the neutral exciton in a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' (a) Schematic band structure of a QDM represented by a double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' An AlGaAs barrier below the molecule prolongs tunneling times for electrons while not affecting tun- neling for holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' One hole (empty circle) is located in the up- per QD, while electrons (filled circles) occur in both dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' As a consequence, direct (red ellipse) and indirect (blue ellipse) excitons arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' A gate voltage V applied to the sample facil- itates tuning of the direct and indirect exciton energies rela- tive to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' (b) Voltage-dependent photoluminescence of the neutral exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The red and blue dashed lines indicate the energies of the direct and indirect excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' tunnel cou- pling between the two QDs leads to an avoided crossing with a symmetric (pink) and an anti-symmetric (green) electron eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The upper (lower) energy transition is called UP (LOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Triangles indicate the excitation energy and voltage applied in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' (c) Neutral exciton state diagram illus- trating the excitation and detection scheme for monitoring Rabi oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' While a resonant light field (green) is driv- ing UP, a phonon-mediated state transfer with rate γP (black arrow) is enabling emission from both UP and the energet- ically detuned LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' (d) Power-dependent Rabi oscillations when exciting UP and detecting UP (green) or LOW (pink) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' ge1336 0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='210e 1338 uS C1340n 01342SMSGaateVolta3 character changes from direct to hybridized to indirect for the upper energy branch, and vice versa for the lower energy branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' As a result, the overlap of the electron and hole wave functions changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The change of the wave function overlap is quanti- fied by coherently driving Rabi oscillations on the ex- citon transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The Rabi frequency of a resonantly excited two-level system ΩR = �� E0D ℏ �� is linearly depen- dent on the TDM D, which in return is proportional to the overlap of the electron and hole wave function [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, ΩR depends linearly on the electric driving field amplitude E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The E0-dependence allows the ob- servation of power-dependent Rabi oscillations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' For this purpose, a 5 ps laser pulse is applied to resonantly drive the crystal ground state (cgs)-to-X0 transition in the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The occupation of the excited state is mon- itored by detecting the photons emitted by the driven two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Commonly, emission from resonantly excited states is detected in a cross-polarized setup con- figuration to suppress the excitation laser [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' At high excitation power, however, laser light can leak into the detection channel and reduce the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We propose and demonstrate a readout technique utiliz- ing a phonon-mediated state transfer [21], which detunes the emitted photons energetically from the two-level sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Thereby, the limitation of an insufficiently sup- pressed excitation laser is eliminated via spectral filter- ing, and the visibility of the Rabi oscillations is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Figure 1 (c) visualizes the state diagram of the X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The two excited states UP and LOW can both radiatively de- cay into the cgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' A phonon emission process with rate γP can transfer the electron from the UP to the LOW configuration [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Since the excitation pulse length is short compared to the decay rates, the cgs-UP system is well approximated by a two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' It is coher- ently driven by a 5 ps laser pulse (green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Fig- ure 1 (d) shows the power-dependent resonance fluores- cence emission of the UP transition as green data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The measurement is performed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V, such that the driven transition exhibits a direct exciton character, as shown in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Rabi oscillations are observed up to a pulse area of slightly above 2π and 602 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' However, a decreasing signal-to-noise ratio prevents the detection of oscillations above 602 nW due to nsufficient suppres- sion of the excitation laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' To improve the signal-to-noise ratio, which decreases with increasing power, we make use of a phonon- mediated state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The emission of a phonon transfers the electron from the UP into the LOW configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This process can only occur as long as the system is in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Thus, the ensemble occupation of LOW is proportional to the ensemble occupation of UP, and so is the number of emitted photons of both transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, due to the avoided crossing, the emission of LOW is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 meV detuned from the driving energy for any gate voltage, which allows the spectral filtering of the emis- sion from the excitation laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Thus, the resonant kCounts (/s) Power1/2 (nW1/2) UP LOW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V 0 50 100 0 1 2 3 0 50 100 0 5 10 0 50 100 0 2 4 0 50 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Rabi oscillations of the UP and LOW branch at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V (left) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V (right) by phonon-mediated state transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The red data points correspond to a direct, the blue to an indirect driven transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' excitation of the two-level system and the off-resonant monitoring of its excited state occupation are achieved simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The power-dependent emission of the LOW transition when exciting UP is shown by the pink data points in Figure 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Below 602 nW, both readout techniques show the same Rabi frequency as expected, confirming the proportionality of occupancy between UP and LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' However, in contrast to the resonant detection (green), Rabi oscillations are well resolvable up to a pulse area of 7π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The reduction of the oscillation amplitude arises from interactions with phonons [22], while the increase of the mean is attributed to a slightly chirped excitation laser pulse [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' From the relative intensities of both transitions, we can conclude that the phonon induced relaxation rate is compara- ble to the radiative decay rate of the direct UP transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Electric control of the tunnel coupling between the two QDs allows coherent excitation of electron-hole pairs in different occupation configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Figure 2 shows the power-dependent emission of the QDM while resonantly exciting UP and detecting LOW (green dashed box, UP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The measurements are performed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V (left) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V (right), on either side of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The red and blue data points indicate a direct and indirect character of the excited transition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We ob- serve Rabi oscillations for both the direct and indirect transitions, which confirms that coherent excitation of a spatially indirect exciton is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' However, the Rabi 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='3 Gate Voltage (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='15 Rabi Frequency (1/nW1/2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='8 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Measured voltage dependent Rabi frequency of UP (green) and LOW (pink), plotted on the left axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The right axes visualizes the calculated overlap of the electron and hole wave functions as a function of the voltage, where the pink/green dashed line corresponds to the lowest/second- lowest electron eigenenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The red/blue shaded background indicates the direct/indirect character of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' frequency of the indirect configuration is reduced com- pared to the direct configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This is caused by the reduced overlap of the electron and hole wave functions and the accompanying decrease of the TDM for the in- direct exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' A verification of these results is found by performing the same experiments on the lower branch (Figure 2, pink box, LOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Similar to the previous case, a phonon absorption process facilitates a state transfer from LOW to UP and, therefore, the off-resonant detection of UP while exciting LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This allows us to off-resonantly monitor Rabi oscillations of the LOW branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The investigated gate voltages of both excitation cases are chosen to be ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='06 V away from the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Therefore, the electron configuration of the upper branch at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V) resembles the electron configuration of the lower branch at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This leads to a com- parable Rabi frequency of the direct (red) and indirect (blue) exciton of the upper and lower energy transition at the two voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The difference in absolute counts between the excitation of UP and LOW is attributed to the underlying phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' When exciting UP (LOW), we rely on the emission (absorption) of a phonon to detect the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Since the measurements are performed at 10 K, the probability of absorbing a phonon is strongly reduced compared to the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Since our QDM device allows continuous tuning the gate voltage while maintaining the prepared charge state, arbitrary exciton configurations can be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Thereby, the overlap of the electron and hole wave functions is analyzable for any coupling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Figure 3 shows the power-dependent Rabi frequencies as a function of the gate voltage for the upper (green) and lower branch (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The Rabi frequencies are extracted by fitting a sin2(laser power) function to the data, with an exponen- tial decay to take phonon dephasing into account, and a linear increase with intensity to compensate for a chirped excitation pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We observe a continuous increase (de- crease) of the frequency when transitioning from an indi- rect (direct) to a direct (indirect) exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' By raising the gate voltage and following the UP transition, the elec- tron occupation shifts from the top to the bottom dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The opposite holds for the LOW transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This leads to a continuous variation of the overlap of the electron and hole wave functions and, consequently, to a change in the Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Within the investigated range be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='1 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='22 V, we are able to electrically tune the Rabi frequency by a factor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The wave function overlap is modeled by calculating the eigenenergies and -values of a tilted, one-dimensional double-squarewell potential representing the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' By fitting the energy difference between the two lowest eigen- states to the voltage-dependent separation of UP and LOW, the depth of the squarewell potential and the effec- tive electron mass are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' A detailed description of the model is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The right axes of Figure 3 shows the overlap of the electron and hole wave functions |⟨ψe|ψh⟩| for the lowest (pink) and sec- ond lowest (green) electron eigenenergy by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The electron eigenenergies correspond to the LOW and UP transition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The one-dimensional model provides a remarkably good description of the measured voltage-dependent Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Thereby, the Rabi frequency can be related to the TDM of direct, indirect, and hybridized excitons, which allows determination the π-pulse area as well as the difference in radiative lifetime of the corresponding transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' DISCUSSION AND SUMMARY Adressing direct, indirect, and hybridized excitons is fundamental for using QDMs as spin-photon interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, the electrical tuneability of the TDM of the adressed transitions at and around the tunnel coupling regime is one key parameter of a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' It not only de- termines the π-pulse power of the addressed transition, as shown in this work, but is also directly related to the lifetime of the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Therefore, it is one of the parameters setting the creation rate for generating one- and two- dimensional photonic cluster states as well as for performing quantum-repeater protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We have demonstrated the coherent excitation of di- rect, indirect, and hybridized excitons – one of the el- ementary building blocks for creating photonic cluster states from QDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We use non-resonant readout, which is facilitated by phonon-mediated charge relaxation and excitation between the two lowest energy eigenstates of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='. Voltage-dependent Rabi oscillations show a continuous increase of the Rabi frequency when tran- sitioning from an indirect to a direct exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This is 5 attributed to an electrically controlled increase of the TDM of a direct compared to an indirect transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Fur- thermore, we apply a one-dimensional model to calculate the overlap of the X0 electron and hole wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Within the voltage range presented, we are able to tune the Rabi frequency and consequently the TDM by a fac- tor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This corresponds to a variation of the radia- tive lifetime between a direct and an indirect exciton by a factor of ≈ 9, as it scales quadratically with the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The coherent excitation and the electrical tunability between various exciton configurations in QDMs not only paves the way towards the generation of entangled multi- photon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' It might also enable protocols which uti- lize fast electrical switching between the exciton config- urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This can reduce their lifetime and the π-pulse power and highly improve the cluster state generation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors gratefully acknowledge financial sup- port from the German Federal Ministry of Educa- tion and Research (BMBF) via Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='Link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='X (16KIS0874, 16KIS086), QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='X (16KISQ027, 16KISQ014, 16KISQ012 and 16KISQ009), the European Union’s Horizon 2020 re- search and innovation program under grant agreement 862035 (QLUSTER) and the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) via SQAM (FI947-5-1), DIP (FI947-6-1), and the Excellence Cluster MCQST (EXC-2111, 390814868).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' gratefully acknowledges the Exploring Quantum Matter (ExQM) programme funded by the State of Bavaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' gratefully acknowledge the BMBF for financial support via project MOQUA (13N14846).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Appendix A: Sample The investigated InAs QDM is enclosed in a GaAs matrix and was grown by molecular beam epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' It consists of two vertically stacked QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The inter-dot coupling strength is determined by a wetting layer- to-wetting layer separation of 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, an AlxGa(x−1)As barrier (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='33) with a thickness of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='5 nm is placed between the dots to reduce the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The height of the top (bottom) QD was fixed to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='9 nm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='7 nm) via the indium-flush technique during growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This height configuration facilitates electric field-induced tunnel coupling of orbital states in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' A 50 nm thick AlxGa(x−1)As tunnel barrier (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content='33) was grown 5 nm below the QDM to prolong electron tunneling times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The molecule is embedded in a p-i-n diode, with the doped regions used as contacts to gate the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The diode contacts are placed more than 150 nm away from the molecule to prevent uncontrolled charge tunneling into the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Furthermore, a distributed Bragg reflector was grown below the diode and a circular Bragg grating was positioned deterministically via in-situ electron beam lithography above an individual and pre-selected QDM to improve photon in- and outcoupling efficiencies [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' All measurements are performed at 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Appendix B: Double-well potential model To calculate the overlap of the electron and hole wave functions, we set up a model consisting of a double- squarewell potential representing the conduction band of the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We assume that the variation of the in-plane wave functions is small compared to the wave functions along the growth direction z, when changing the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' This assumption is reasonable, since the confine- ment of charges along the growth axes and the translation introduced by the gate voltage along the growth axes ex- ceeds the in-plane variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' We can, therefore, approach the problem with a one-dimensional model and expect an acceptable degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The potential V (z) is designed to match the dimensions of the QDM with re- spect to the tunnel barrier width and dot heights (see Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' z is here the growth direction of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' In addition, we tilt the potential to imitate the presence of an applied gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Solving the time-independent Schr¨odinger equation for a given gate voltage allows us to obtain the envelope functions Ψ and their eigenenergies E of an electron with mass me trapped in the double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' The envelope functions are then used to rep- resent the wave function of the electron since the Bloch part of the wave functions are only weakly sensitive to electric fields of the magnitude applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' To define the free parameters of the double-well model, we determine the effective electron mass me and the po- tential depth by fitting the difference between the cal- culated two lowest eigenenergies to the measured energy difference ∆E between UP and LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' ∆E between the two X0 branches is purely determined by the energy dif- ference between the electron eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' For calculating the hole wave function, the potential is inverted to repre- sent the valance band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'} +page_content=' Its depth is 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Jia-Xin Yin1, Nana Shumiya1, Tyler A. Cochran1, Xian P. Yang1, +Maksim Litskevich1, Nan Yao9, Kenji Watanabe10, Takashi Taniguchi11, Hua Zhang3,12,13†, Luis Balicas2, M. +Zahid Hasan1,14† +1Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton +University, Princeton, New Jersey, USA. +2National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA. +3Department of Chemistry, City University of Hong Kong, Hong Kong, China. +4Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China. +5Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan +6Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological +University, Singapore 637371, Singapore. +7Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551, Singapore +8Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore +9Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, NJ, USA. +10Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan. +11International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. +12Hong Kong Branch of National Precious Metals Material Engineering Research Center (NPMM), City University of +Hong Kong, Hong Kong, China. +13Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China. +14Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. + +* These authors contributed equally to this work +†Corresponding +to: +qz9@princeton.edu; +mdsh@princeton.edu; +hua.zhang@cityu.edu.hk; +mzhasan@princeton.edu +Ongoing advances in superconductors continue to revolutionize technology thanks to the increasingly +versatile and robust availability of lossless supercurrent. In particular high supercurrent density can lead to +more efficient and compact power transmission lines, high-field magnets, as well as high-performance +nanoscale radiation detectors and superconducting spintronics. Here, we report the discovery of an +unprecedentedly high superconducting critical current density (17 MA/cm2 at 0 T and 7 MA/cm2 at 8 T) in +1T′-WS2, exceeding those of all reported two-dimensional superconductors to date. 1T′-WS2 features a +strongly anisotropic (both in- and out-of-plane) superconducting state that violates the Pauli paramagnetic +limit signaling the presence of unconventional superconductivity. Spectroscopic imaging of the vortices +further substantiates the anisotropic nature of the superconducting state. More intriguingly, the normal state +of 1T′-WS2 carries topological properties. The band structure obtained via angle-resolved photoemission +spectroscopy and first-principles calculations points to a Z2 topological invariant. The concomitance of +topology and superconductivity in 1T′-WS2 establishes it as a topological superconductor candidate, which +is promising for the development of quantum computing technology. + +Since the discovery of superconductivity by Heike +Kamerlingh Onnes back in 1911 [1], superconductors have +revolutionized science and technology through numerous +applications ranging from superconducting qubits to high- +field magnets [2-4]. High-field magnets fabricated from +superconductors with high critical current density, have +enabled scientific discoveries across physical, chemical, +and biological sciences [5-7]. On the other hand, +superconducting materials exhibiting topological properties +offer possibilities beyond this classical application +paradigm, opening a new frontier to implement fault- +tolerant quantum information technologies. Recently, two- +dimensional +(2D) +transition +metal +dichalcogenides +(TMDCs) attracted considerable interests thanks to their +abundant crystal structures and novel physical properties +[8-11]. Specifically, hole-doped TMDCs have been +considered as candidates for topological superconductivity +based on momentum-space-split spinless fermions [8]. For + +example, the coexistence of superconductivity with a +topologically non-trivial electronic state makes 2M-WS2 a +good candidate for topological superconductivity [12]. +Here, we access both avenues and demonstrate an +unprecedentedly high superconducting critical current +density and topological features in the 2D superconductor +1T′-WS2. +1T′-WS2 is composed of a distorted [WS6] octahedral and +crystallizes in a monoclinic layered structure [13], as shown +in Fig. 1a. High purity 1T′-WS2 crystals were synthesized +via a previously reported method [13]. The single-phase +nature can be observed in the cross-sectional scanning +transmission electron microscope (STEM) image (Fig. 1b). +STEM image unveils the atomic stacking pertaining to a +monoclinic and distorted structure. This atomic-resolution +characterization confirms the high crystallinity and phase +purity of the as-synthesized 1T′-WS2 crystals, consistent +with the previous report [13]. After characterizing the bulk +material, we fabricated devices based on few-layer 1T′-WS2 +for transport measurements. Thin flakes of 1T′-WS2 +obtained via mechanical exfoliation were transferred onto a +SiO2 (285 nm)/Si substrate (inset of Fig. S1a). The Raman +spectrum of the as-prepared 1T′-WS2 flake shows a series +of peaks at ~112, ~178, ~270, ~316, and ~407 cm-1 (Fig. +S1a), consistent with single-phase 1T′-WS2 [13]. The +thickness of the flakes used in our measurements is ~6.1 +nm as measured by the atomic force microscope (Fig. S1b). +The +device +was +fabricated +following +a +Hall-bar +configuration and measured from T = 300 K to 2.0 K in a +Physical Property Measurement System. Figure 1c depicts +the four-probe resistance as a function of temperature and +captures the electrical transport behavior of the sample. At +high temperatures, it exhibits metallic behavior (dR/dT > 0), +indicating phonon-scattering-dominated transport [14]. The +superconducting transition occurs at 7.7 K, which is slightly +lower than the bulk critical temperature (Tc) of 1T′-WS2 +(8.6 K) [13]. We also measured the Hall effect of 1T′-WS2 +above the critical temperature. Strikingly, the carrier +concentration in 1T′-WS2 approaches 1015~1016 cm-2 at T = +10 K (Fig. S2a and S2b). This value is much higher than the +typical +carrier +concentration +(~1014 +cm-2) +of +2D +superconductors with electrostatic gating [15]. +To investigate the superconducting state of 1T′-WS2, we +performed magneto-transport measurements (Fig. 1d). We +start with the angular dependence of the upper critical +magnetic field (Hc2), defined as the magnetic field at which +the resistance drops to 50% of its normal state value. The +details of the angular dependent measurement are described +in Fig. S3. For a clear visualization, we normalized the +resistance by Rn, i.e., the normal state resistance for all the +samples. Figure 1e summarizes the magnetic field +dependence of the resistance at different angles (θ) at T = +0.33 K, where θ is the angle between the z-axis and the +magnetic field direction (Fig. 1d). As the sample is rotated +from the perpendicular (θ = 0º) to a parallel field (θ = 90º) +configuration, the transition towards superconductivity +progressively shifts to higher fields, manifesting a clear +superconducting anisotropy (Fig. 1e). In Fig. 1f, we present +a plot of Hc2 as a function of θ, showing that the highest Hc2 +occurs when the magnetic field is applied parallel to the +sample plane. To understand the anisotropic nature of Hc2, +we fitted our data to the Tinkham formula, which describes +the angular dependence of Hc2 for a 2D superconductor +[16]: + +where Hc2,⊥ and Hc2,// are the upper critical field for fields +perpendicular and parallel to the plane of the sample, +respectively. As shown in Fig. 1f, the blue fitting curve +matches the data quite well and thus confirms the 2D nature +of the superconductivity in 1T′-WS2. The fitting of the +angle dependent critical field for smaller angular regimes to +the 2D Tinkham formula is shown in Fig. S4. + After exploring the anisotropy of Hc2 along the out-of- +plane directions, we then examined how Hc2 evolved along +the in-plane directions. As the device is rotated from the x- +axis (φ = 0°; φ is the angle between the magnetic field and +the x-axis as shown in Fig. 1d) to the y-axis (φ = 90°), the +superconducting transition progressively shifts from higher +fields to lower fields (Fig. 1g). Careful measurements were +performed to rule out the possibility of an accidental out-of- +plane component (Fig. S5). Such a planar anisotropy is +likely to result from the reduced crystal symmetry due to +the distorted structure of 1T′-WS2, as clearly seen in Fig. +1a. Figure 1h, which shows Hc2 as a function of φ, reveals +an emergent two-fold symmetry. Furthermore, we observed +that the largest value of the in-plane Hc2 (28 T). To obtain a +quantitative understanding of such a large value, we +compared it to the expected Pauli paramagnetic limiting +field. In conventional superconductors, a sufficiently high +external magnetic field can suppress superconductivity +through the orbital [17] and spin Zeeman effect [18,19]. For +a few-layer sample, the suppression from the orbital effect +is nearly absent when the magnetic field is parallel to the +sample plane. Consequently, the Zeeman effect imposes an +upper bound on Hc2, known as the Pauli limit (Hp = 1.84×Tc +T/K) [20]. We find that the in-plane Hc2 (28 T) in 1T′-WS2 +clearly violates the Pauli limit (14 T for Tc = 7.7 K). Such a +violation combined with the emergence of two-fold +symmetry for the in-plane Hc2 suggests unconventional +superconductivity in 1T′-WS2. +We further explored the superconducting transition via +systematic temperature dependent measurements. Figures +1i and 1j show such data taken when the magnetic field was +perpendicular and parallel to the sample plane, respectively. +In both cases, the superconducting transition shifts +gradually to lower magnetic fields as the temperature +increases. The temperature dependence of the out-of-plane +upper critical field (Hc2,⊥) and in-plane upper critical field + +COS 0 +FIG.1 Crystal structure and Superconductivity of 1T′-WS2. a, Schematic illustration of the structure of 1T′-WS2. Top +panel: side view of the crystallographic structure; bottom panel: top view of a typical monolayer. b, Cross-sectional STEM +image of 1T′-WS2. Inset: high-magnification STEM image of layered structure with atomic resolution. c, Temperature- +dependent electrical resistance of the mechanically exfoliated 1T′-WS2 without magnetic field. Insets: optical image of the +1T′-WS2 device covered by h-BN with Hall-bar configuration (top) and small range Rxx-T plot of 1T′-WS2 around Tc shown in +the area within the red rectangle (bottom). d, Schematic illustration of a 1T′-WS2 device and the rotation experiment setup, +where the x-axis is parallel to c-axis of the crystal and z-axis is perpendicular to crystalline plane. θ is the angle between the +out-of-plane magnetic field and the z-axis; φ is the angle between the in-plane magnetic field and the x-axis. e, Magnetic field +dependence of the normalized resistance of the 1T′-WS2 device at T = 0.33 K with different out-of-plane rotation angles θ. f, +The θ-dependence of the upper critical field. The blue curve denotes a fit to the data following the Tinkham formula for a 2D +superconductor. g, Magnetic field dependence of the normalized resistance of the 1T′-WS2 device at T = 0.33 K with +different in-plane rotation angles φ. h, The φ-dependence of the upper critical field. The green dashed line indicates the Pauli +limit. i, j, Superconducting transition of the 1T′-WS2 device under a perpendicular magnetic field (i) and under a parallel +magnetic field (j) at different temperature. k, Temperature dependence of the upper critical field with magnetic field +directions parallel and perpendicular to the crystal plane. The red curve represents the linear relationship between Hc2,⊥ and T +according to the 2D GL theory. + +c +0.3 +10 um +BN +0.2 +IT'WS +0.05 +C +0.1 +0.0 +12 +T (K) +0 +100 +200 +300 +T (K) +2D-Tinkham (Hc2,//) are summarized in Fig. 1k. Hc2,⊥ displays a linear +dependence on temperature, that is well fitted by the +standard +Ginzburg-Landau +(GL) +theory +for +2D +superconductors [16]: + +where +(0) is the zero-temperature GL in-plane +coherence length, Φ0 is the magnetic flux quantum, and Tc +is the critical temperature at which the resistance drops to +50% of its value in the normal state. From the fit we can +estimate the coherence length +(0) ≈ 9.6 nm. The +temperature dependence of Hc2,//, on the other hand, follows +the GL formula expected for 2D superconductors [16]: + +where dSC is the superconducting thickness. From the +fitting of Hc2,//, the superconducting thickness is around 3.2 +nm, which is smaller than +(0) and consistent with 2D +superconductivity. + + +FIG.2 Scanning tunneling microscopy measurements on +1T′-WS2. a, Topographic image of the bc plane of 1T′- +WS2. Top inset: a zoom-in view of the topographic image +showing the atomic arrangements. Bottom inset: Fast +Fourier transform of the topographic image. b, A zero-bias +conductance map of vortices at 1 T. Inset: Fourier +transform of the dI/dV map. c, d, The conductance map of a +single vortex at 1 T with zero bias (c) and 1 mV bias (d). e, +Tunneling spectroscopy spectrum taken at 4.2 K, revealing +a superconducting gap. Light blue curves are the +differential spectra taken at different positions on the +surface; the dark blue curve denotes the average spectra. f, +Field dependence of tunneling spectroscopy taken at 0 T, 1 +T and 5 T. + To further characterize the anisotropic superconductivity +in 1T′-WS2, we performed scanning tunneling microscopy +(STM) measurements and directly imaged the vortices +under magnetic field. A single crystal was cleaved in-situ at +T = 77 K and measured at T = 4.2 K. Figure 2a shows the +topography of 1T′-WS2 over a large area. The atomically +resolved STM topographic image reveals a clean surface +featuring zigzag chains along the b-axis of the crystal (top +inset of Fig. 2a). In addition, the corresponding fast Fourier +transform pattern also exhibits the distorted octahedral +coordination feature (bottom inset of Fig. 2a). A zero- +energy conductance map under 1 T applied perpendicularly +to the bc plane is shown in Fig. 2b. The Fourier transform +of the dI/dV map is two-fold symmetric (inset of Fig. 2b). +The conductance maps of a single vortex at 0.1 T taken at V += 0 mV (Fig. 2c) and 1 mV (Fig. 2d) further highlight the +anisotropic nature of the superconductivity. Consistent with +the anisotropy observed in our transport data, the vortices +are anisotropic and elongated along the b direction, +reflecting the anisotropy of the Ginzburg-Landau coherence +length between both directions. Tunneling differential +conductance collected from an atomically resolved lattice +illustrates a superconducting gap with sharp coherence +peaks (Fig. 2e). This superconducting gap disappears +gradually as the magnetic field is increased (Fig. 2f). + Subsequently, we performed critical current density (Jc) +measurements. As alluded in the introduction, an important +aspect of a superconductor is its Jc, which dictates several +practical +applications. +The +higher +the +Jc +of +a +superconductor, the smaller and more efficient the +superconducting devices that can be fabricated from it or +the larger the magnetic fields that can be generated. We +measured differential resistance of the 1T′-WS2 device with +thickness of 6 nm as a function of direct current (DC) bias +current at different temperatures (Fig. 3a). Note that, Jc is +defined as the current density at which the differential +resistance (dV/dI) reaches its maximum, as reported in +previous works [21,22]. Remarkably, as seen in Fig. 3b, +1T′-WS2 exhibits ultrahigh critical current densities +reaching 17 MA/cm2 at T = 0.33 K. Figure 3b highlights the +temperature dependence of the critical current density, +featuring an enormous Jc = 13 MA/cm2 at liquid He +temperature (4.2 K). In addition, we systematically +measured the critical currents of samples with different +layer thicknesses, as shown in Fig. S6. The thickness +dependence of the critical current density is summarized in +Fig. 3c. There is no obvious difference among the samples +with thicknesses exceeding 20 nm. The critical current +densities increase as the devices become thinner, which is +also observed in atomically thin TaS2 [23]. Furthermore, we +evaluated the field dependence of Jc (Fig. S7). The critical +current density falls rapidly as the perpendicular magnetic +field increases (Fig. 3d). In contrast, the critical current +density is rarely influenced by a parallel magnetic field +since 1T′-WS2 shows extremely high in-plane upper critical + +GJH +100nm +0 +0.2nS +100nm +0mV +e +T=4K +T=4K +0.2 +0.2 +(su) +(su) Λp/Ip +my +ΛP/Ip +0.1 +0 +5T +1T +-OT +0 +-10 +-5 +0 +5 +10 +-10 +-5 +0 +5 +10 +20nm +Bias (mV) +Bias (mV)fields. Even under an 8 T in-plane magnetic field, Jc is +substantially large (7 MA/cm2). +Experimentally, numerous 2D superconducting transition +metal dichalcogenides have been studied [20-33]. In-plane +anisotropic upper critical fields were observed in 2H-NbSe2 +[24] and Td-MoTe2 [25]. 2H-NbSe2 [20] and ionic-gated +2H-MoS2 [26] also exhibited high in-plane upper critical +fields. However, we emphasize that 1T′-WS2 is the only 2D +material to our knowledge that shows the suitable critical +temperature and high critical current under high in-plane +magnetic field, which are crucial for building high-field +magnets. Even for a thick sample, the in-plane critical field +surpasses 8 T at 4 K (Fig. S8). We summarize the +parameters of 2D superconductors in Fig. 3e. As for 1T- +MoS2 [27], 2H-TaS2 [23], 3R-TaSe2 [28], Td-MoTe2 [25], +2H-NbS2 [29], their critical temperatures are below the +temperature of liquid helium (4.2 K), rendering the +construction of high-field magnets impractical. Gated MoS2 +displays a relatively high critical temperature and also very +high critical fields, but superconductors under ionic gating +are not suitable for applications [30]. Lastly, 2H-NbSe2 is +comparable to 1T′-WS2 in critical fields and critical +temperatures. However, its critical current density is two +orders of magnitude lower than that of 1T′-WS2 [31]. The +significance of our work is that we +report +an +unprecedentedly high superconducting critical current +density (17 MA/cm2 at 0 T) in 1T′-WS2, which exceeds +those of all the known 2D superconductors to date [21-33]. +Notably, it even exceeds the Jc of MgB2 films [34], a well- +known superconductor for high-critical-current applications +(Fig. 3e). Even under an 8 T in-plane magnetic field, the Jc +of 1T′-WS2 is substantially large (7 MA/cm2). As a +reference, the critical currents of commercial magnet +building materials are listed here, such as Nb-Ti alloy (0.1 +MA/cm2 at 10 T) and Nb3Sn (0.5 MA/cm2 at 10 T) [35]. +The large Jc at zero and finite magnetic fields makes 1T′- +WS2 a potential candidate for future study on building next- +generation superconducting magnets. +Having explored the superconductivity of 1T′-WS2, we +turn to the topological features pertinent to its electronic +band structure using a series of theoretical calculations and +angle-resolved +photoemission +spectroscopy +(ARPES) +experiments. The calculated bulk band structure is shown in +Fig. S9. Besides the continuous energy gap between +conduction band and valence band around the Fermi level, +we observe a band inversion at the -point between W d +and S p orbitals, which leads to a strong topological +insulating phase. Furthermore, the surface-projected +calculation shows the topological Dirac surface state +emerging from the valence band and merging into +conduction bands (Fig. 4a). The corresponding ARPES data +(Fig. 4b), taken at T = 10 K (above Tc) matches the first +principles calculations below EF. In particular we identify +the linear-dispersed hole pocket at +to be the lower cone +of the topological surface state, as it shows no photon- + +FIG.3 Ultrahigh critical current density of 1T′-WS2. a, +Differential resistance of a 6-nm-thick 1T′-WS2 sample as a +function of the direct current (DC) bias at different +temperatures. b, Critical current density for a 1T′-WS2 +device as a function of the temperature. c, Critical current +density of the 1T′-WS2 device plotted as a function of +sample thickness. d, Critical current densities of the 1T′- +WS2 device plotted as a function of perpendicular and +parallel magnetic fields. e, Comparison of critical current +densities among 1T′-WS2 and other representative 2D +superconductors, such as twisted bilayer graphene (TBG) +and transition metal dichalcogenides. Commercial magnet +building materials are also included for reference. Here, the +superconducting critical temperatures Tc of the different +materials were determined under zero magnetic field. +energy dependence and agrees well with the calculated +dispersion of the Dirac state (More details of ARPES data +analysis are shown in Figs. S10-12). ARPES Fermi surface +map also visualizes the highly anisotropic Fermi surface +(Fig. 4c), which possibly contributes to the extremely +anisotropic +Hc2 +in +1T′-WS2. +The +calculated +superconducting gap of 1T′-WS2 on the Fermi surface is +presented in Fig. 4d. These results lend crucial credence to +the in-plane anisotropy of superconductivity. It is noted that +so far there is no clear relationship between the topological +nature and high critical current. + +0.4K - + 1.8K—2.5K—3.5K +4.5K—5.5K— +6.5K—7.5K +8.4nm 1T-MoS2 +2nm 1T-WS2 +7nm 3R-TaSe2 +6nm 1T'-WS2★ +6.5nm T.-MoTe2 +1T" WS, @8T +1.6nm TBG +6nm 2H-NbS2 +7.6nm a-Mo2C +MgB- filn +4.2nm 2H-TaS2 +Nb,Sn @10T +3nm 2H-NbSe2 +1.6nm gated +Nb-Ti @10T +MoS2 +FIG.4 Topological features of 1T′-WS2. a, Calculated surface band structure of 1T′-WS2 at ky = 0, featuring a topological +Dirac surface state near the Fermi level. b, Energy-momentum cut acquired through ARPES. c, Fermi surface of 1T′-WS2. d, +Calculated superconducting gap which all kz are projected in the surface Brillouin zone at 2.5 K on the Fermi surface. The +unit of the color bar is meV. +In summary, combining a series of experimental and +numerical techniques, we comprehensively studied 1T′- +WS2 and find a unique blend of ultrahigh critical +supercurrent density, large superconducting anisotropy (in- +plane versus out-of-plane) along with topological features. +Our findings not only provide a promising material +platform for high magnetic field technologies but also +unveil a promising platform for future exploration of +topological superconductivity, which may be used to +fabricate topologically protected qubits for future quantum +computing schemes. + +ACKNOWLEDGEMENTS. Experimental and theoretical +work at Princeton University was supported by the Gordon +and 286 Betty Moore Foundation (GBMF4547; M.Z.H.). +The material characterization is supported by the United +States 287 Department of Energy (US DOE) under the +Basic Energy Sciences program (grant number DOE/BES +DE-FG-288 02-05ER46200). L.B. is supported by DOE- +BES through award DE-SC0002613. The National High + + +Magnetic Field Laboratory acknowledges support from the +US-NSF Cooperative agreement Grant number DMR- +1644779 and the state of Florida. The authors acknowledge +the sample characterization of Imaging and Analysis Center +(IAC) at Princeton University, partially supported by the +Princeton Center for Complex Materials (PCCM) and the +NSF-MRSEC program (MRSEC; DMR-2011750). G.C. +acknowledges the support of the National Research +Foundation, Singapore under its Fellowship Award (NRF- +NRFF13-2021-0010) +and +the +Nanyang +Assistant +Professorship +grant +from +Nanyang +Technological +University. J.Y.Y. and Y.P.F. is supported by the Ministry +of Education, Singapore, under its MOE AcRF Tier 3 +Award MOE2018-T3-1-002. 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Cryogenics 48, 283-292 (2008). + + + diff --git a/DtFJT4oBgHgl3EQfBiw4/content/tmp_files/load_file.txt b/DtFJT4oBgHgl3EQfBiw4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..76823f7536f1f64b6e43079701bdfbb54dbf2ec8 --- /dev/null +++ b/DtFJT4oBgHgl3EQfBiw4/content/tmp_files/load_file.txt @@ -0,0 +1,543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf,len=542 +page_content='Anomalously high supercurrent density in a two-dimensional topological material Qi Zhang1*†, Md Shafayat Hossain1*†, Brian Casas2, Wenkai Zheng2, Zi-Jia Cheng1, Zhuangchai Lai3,4, Yi-Hsin Tu5, Guoqing Chang6, Yao Yao3, Siyuan Li3, Yu-Xiao Jiang1, Sougata Mardanya5, Tay-Rong Chang5, Jing-Yang You7, Yuan-Ping Feng7,8, Guangming Cheng9, Jia-Xin Yin1, Nana Shumiya1, Tyler A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Cochran1, Xian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Yang1, Maksim Litskevich1, Nan Yao9, Kenji Watanabe10, Takashi Taniguchi11, Hua Zhang3,12,13†, Luis Balicas2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Zahid Hasan1,14† 1Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New Jersey, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3Department of Chemistry, City University of Hong Kong, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 4Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 5Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan 6Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 7Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551, Singapore 8Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore 9Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, NJ, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 10Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 11International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 12Hong Kong Branch of National Precious Metals Material Engineering Research Center (NPMM), City University of Hong Kong, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 13Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 14Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' These authors contributed equally to this work †Corresponding to: qz9@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' mdsh@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' hua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='zhang@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' mzhasan@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='edu Ongoing advances in superconductors continue to revolutionize technology thanks to the increasingly versatile and robust availability of lossless supercurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In particular high supercurrent density can lead to more efficient and compact power transmission lines, high-field magnets, as well as high-performance nanoscale radiation detectors and superconducting spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Here, we report the discovery of an unprecedentedly high superconducting critical current density (17 MA/cm2 at 0 T and 7 MA/cm2 at 8 T) in 1T′-WS2, exceeding those of all reported two-dimensional superconductors to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1T′-WS2 features a strongly anisotropic (both in- and out-of-plane) superconducting state that violates the Pauli paramagnetic limit signaling the presence of unconventional superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Spectroscopic imaging of the vortices further substantiates the anisotropic nature of the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' More intriguingly, the normal state of 1T′-WS2 carries topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The band structure obtained via angle-resolved photoemission spectroscopy and first-principles calculations points to a Z2 topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The concomitance of topology and superconductivity in 1T′-WS2 establishes it as a topological superconductor candidate, which is promising for the development of quantum computing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Since the discovery of superconductivity by Heike Kamerlingh Onnes back in 1911 [1], superconductors have revolutionized science and technology through numerous applications ranging from superconducting qubits to high- field magnets [2-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' High-field magnets fabricated from superconductors with high critical current density, have enabled scientific discoveries across physical, chemical, and biological sciences [5-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' On the other hand, superconducting materials exhibiting topological properties offer possibilities beyond this classical application paradigm, opening a new frontier to implement fault- tolerant quantum information technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Recently, two- dimensional (2D) transition metal dichalcogenides (TMDCs) attracted considerable interests thanks to their abundant crystal structures and novel physical properties [8-11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Specifically, hole-doped TMDCs have been considered as candidates for topological superconductivity based on momentum-space-split spinless fermions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' For example, the coexistence of superconductivity with a topologically non-trivial electronic state makes 2M-WS2 a good candidate for topological superconductivity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Here, we access both avenues and demonstrate an unprecedentedly high superconducting critical current density and topological features in the 2D superconductor 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1T′-WS2 is composed of a distorted [WS6] octahedral and crystallizes in a monoclinic layered structure [13], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' High purity 1T′-WS2 crystals were synthesized via a previously reported method [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The single-phase nature can be observed in the cross-sectional scanning transmission electron microscope (STEM) image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' STEM image unveils the atomic stacking pertaining to a monoclinic and distorted structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' This atomic-resolution characterization confirms the high crystallinity and phase purity of the as-synthesized 1T′-WS2 crystals, consistent with the previous report [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' After characterizing the bulk material, we fabricated devices based on few-layer 1T′-WS2 for transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Thin flakes of 1T′-WS2 obtained via mechanical exfoliation were transferred onto a SiO2 (285 nm)/Si substrate (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The Raman spectrum of the as-prepared 1T′-WS2 flake shows a series of peaks at ~112, ~178, ~270, ~316, and ~407 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S1a), consistent with single-phase 1T′-WS2 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The thickness of the flakes used in our measurements is ~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 nm as measured by the atomic force microscope (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The device was fabricated following a Hall-bar configuration and measured from T = 300 K to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='0 K in a Physical Property Measurement System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figure 1c depicts the four-probe resistance as a function of temperature and captures the electrical transport behavior of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' At high temperatures, it exhibits metallic behavior (dR/dT > 0), indicating phonon-scattering-dominated transport [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The superconducting transition occurs at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='7 K, which is slightly lower than the bulk critical temperature (Tc) of 1T′-WS2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='6 K) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We also measured the Hall effect of 1T′-WS2 above the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Strikingly, the carrier concentration in 1T′-WS2 approaches 1015~1016 cm-2 at T = 10 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S2a and S2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' This value is much higher than the typical carrier concentration (~1014 cm-2) of 2D superconductors with electrostatic gating [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' To investigate the superconducting state of 1T′-WS2, we performed magneto-transport measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We start with the angular dependence of the upper critical magnetic field (Hc2), defined as the magnetic field at which the resistance drops to 50% of its normal state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The details of the angular dependent measurement are described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' For a clear visualization, we normalized the resistance by Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=', the normal state resistance for all the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figure 1e summarizes the magnetic field dependence of the resistance at different angles (θ) at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='33 K, where θ is the angle between the z-axis and the magnetic field direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As the sample is rotated from the perpendicular (θ = 0º) to a parallel field (θ = 90º) configuration, the transition towards superconductivity progressively shifts to higher fields, manifesting a clear superconducting anisotropy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1f, we present a plot of Hc2 as a function of θ, showing that the highest Hc2 occurs when the magnetic field is applied parallel to the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' To understand the anisotropic nature of Hc2, we fitted our data to the Tinkham formula, which describes the angular dependence of Hc2 for a 2D superconductor [16]: where Hc2,⊥ and Hc2,// are the upper critical field for fields perpendicular and parallel to the plane of the sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1f, the blue fitting curve matches the data quite well and thus confirms the 2D nature of the superconductivity in 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The fitting of the angle dependent critical field for smaller angular regimes to the 2D Tinkham formula is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' After exploring the anisotropy of Hc2 along the out-of- plane directions, we then examined how Hc2 evolved along the in-plane directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As the device is rotated from the x- axis (φ = 0°;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' φ is the angle between the magnetic field and the x-axis as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1d) to the y-axis (φ = 90°), the superconducting transition progressively shifts from higher fields to lower fields (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Careful measurements were performed to rule out the possibility of an accidental out-of- plane component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Such a planar anisotropy is likely to result from the reduced crystal symmetry due to the distorted structure of 1T′-WS2, as clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figure 1h, which shows Hc2 as a function of φ, reveals an emergent two-fold symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Furthermore, we observed that the largest value of the in-plane Hc2 (28 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' To obtain a quantitative understanding of such a large value, we compared it to the expected Pauli paramagnetic limiting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In conventional superconductors, a sufficiently high external magnetic field can suppress superconductivity through the orbital [17] and spin Zeeman effect [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' For a few-layer sample, the suppression from the orbital effect is nearly absent when the magnetic field is parallel to the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Consequently, the Zeeman effect imposes an upper bound on Hc2, known as the Pauli limit (Hp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='84×Tc T/K) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We find that the in-plane Hc2 (28 T) in 1T′-WS2 clearly violates the Pauli limit (14 T for Tc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='7 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Such a violation combined with the emergence of two-fold symmetry for the in-plane Hc2 suggests unconventional superconductivity in 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We further explored the superconducting transition via systematic temperature dependent measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figures 1i and 1j show such data taken when the magnetic field was perpendicular and parallel to the sample plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In both cases, the superconducting transition shifts gradually to lower magnetic fields as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The temperature dependence of the out-of-plane upper critical field (Hc2,⊥) and in-plane upper critical field COS 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 Crystal structure and Superconductivity of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' a, Schematic illustration of the structure of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Top panel: side view of the crystallographic structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' bottom panel: top view of a typical monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' b, Cross-sectional STEM image of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Inset: high-magnification STEM image of layered structure with atomic resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' c, Temperature- dependent electrical resistance of the mechanically exfoliated 1T′-WS2 without magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Insets: optical image of the 1T′-WS2 device covered by h-BN with Hall-bar configuration (top) and small range Rxx-T plot of 1T′-WS2 around Tc shown in the area within the red rectangle (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' d, Schematic illustration of a 1T′-WS2 device and the rotation experiment setup, where the x-axis is parallel to c-axis of the crystal and z-axis is perpendicular to crystalline plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' θ is the angle between the out-of-plane magnetic field and the z-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' φ is the angle between the in-plane magnetic field and the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' e, Magnetic field dependence of the normalized resistance of the 1T′-WS2 device at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='33 K with different out-of-plane rotation angles θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' f, The θ-dependence of the upper critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The blue curve denotes a fit to the data following the Tinkham formula for a 2D superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' g, Magnetic field dependence of the normalized resistance of the 1T′-WS2 device at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='33 K with different in-plane rotation angles φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' h, The φ-dependence of the upper critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The green dashed line indicates the Pauli limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' i, j, Superconducting transition of the 1T′-WS2 device under a perpendicular magnetic field (i) and under a parallel magnetic field (j) at different temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' k, Temperature dependence of the upper critical field with magnetic field directions parallel and perpendicular to the crystal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The red curve represents the linear relationship between Hc2,⊥ and T according to the 2D GL theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='3 10 um BN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content="2 IT'WS 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='05 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='0 12 T (K) 0 100 200 300 T (K) 2D-Tinkham (Hc2,//) are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Hc2,⊥ displays a linear dependence on temperature, that is well fitted by the standard Ginzburg-Landau (GL) theory for 2D superconductors [16]: where (0) is the zero-temperature GL in-plane coherence length, Φ0 is the magnetic flux quantum, and Tc is the critical temperature at which the resistance drops to 50% of its value in the normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' From the fit we can estimate the coherence length (0) ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='6 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The temperature dependence of Hc2,//, on the other hand, follows the GL formula expected for 2D superconductors [16]: where dSC is the superconducting thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' From the fitting of Hc2,//, the superconducting thickness is around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 nm, which is smaller than (0) and consistent with 2D superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 Scanning tunneling microscopy measurements on 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' a, Topographic image of the bc plane of 1T′- WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Top inset: a zoom-in view of the topographic image showing the atomic arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Bottom inset: Fast Fourier transform of the topographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' b, A zero-bias conductance map of vortices at 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Inset: Fourier transform of the dI/dV map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' c, d, The conductance map of a single vortex at 1 T with zero bias (c) and 1 mV bias (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' e, Tunneling spectroscopy spectrum taken at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 K, revealing a superconducting gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Light blue curves are the differential spectra taken at different positions on the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' the dark blue curve denotes the average spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' f, Field dependence of tunneling spectroscopy taken at 0 T, 1 T and 5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' To further characterize the anisotropic superconductivity in 1T′-WS2, we performed scanning tunneling microscopy (STM) measurements and directly imaged the vortices under magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' A single crystal was cleaved in-situ at T = 77 K and measured at T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figure 2a shows the topography of 1T′-WS2 over a large area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The atomically resolved STM topographic image reveals a clean surface featuring zigzag chains along the b-axis of the crystal (top inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In addition, the corresponding fast Fourier transform pattern also exhibits the distorted octahedral coordination feature (bottom inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' A zero- energy conductance map under 1 T applied perpendicularly to the bc plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The Fourier transform of the dI/dV map is two-fold symmetric (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The conductance maps of a single vortex at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 T taken at V = 0 mV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2c) and 1 mV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2d) further highlight the anisotropic nature of the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Consistent with the anisotropy observed in our transport data, the vortices are anisotropic and elongated along the b direction, reflecting the anisotropy of the Ginzburg-Landau coherence length between both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Tunneling differential conductance collected from an atomically resolved lattice illustrates a superconducting gap with sharp coherence peaks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' This superconducting gap disappears gradually as the magnetic field is increased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Subsequently, we performed critical current density (Jc) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As alluded in the introduction, an important aspect of a superconductor is its Jc, which dictates several practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The higher the Jc of a superconductor, the smaller and more efficient the superconducting devices that can be fabricated from it or the larger the magnetic fields that can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We measured differential resistance of the 1T′-WS2 device with thickness of 6 nm as a function of direct current (DC) bias current at different temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Note that, Jc is defined as the current density at which the differential resistance (dV/dI) reaches its maximum, as reported in previous works [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Remarkably, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3b, 1T′-WS2 exhibits ultrahigh critical current densities reaching 17 MA/cm2 at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='33 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Figure 3b highlights the temperature dependence of the critical current density, featuring an enormous Jc = 13 MA/cm2 at liquid He temperature (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In addition, we systematically measured the critical currents of samples with different layer thicknesses, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The thickness dependence of the critical current density is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' There is no obvious difference among the samples with thicknesses exceeding 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The critical current densities increase as the devices become thinner, which is also observed in atomically thin TaS2 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Furthermore, we evaluated the field dependence of Jc (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The critical current density falls rapidly as the perpendicular magnetic field increases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In contrast, the critical current density is rarely influenced by a parallel magnetic field since 1T′-WS2 shows extremely high in-plane upper critical GJH 100nm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2nS 100nm 0mV e T=4K T=4K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 (su) (su) Λp/Ip my ΛP/Ip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 0 5T 1T OT 0 10 5 0 5 10 10 5 0 5 10 20nm Bias (mV) Bias (mV)fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Even under an 8 T in-plane magnetic field, Jc is substantially large (7 MA/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Experimentally, numerous 2D superconducting transition metal dichalcogenides have been studied [20-33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In-plane anisotropic upper critical fields were observed in 2H-NbSe2 [24] and Td-MoTe2 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 2H-NbSe2 [20] and ionic-gated 2H-MoS2 [26] also exhibited high in-plane upper critical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' However, we emphasize that 1T′-WS2 is the only 2D material to our knowledge that shows the suitable critical temperature and high critical current under high in-plane magnetic field, which are crucial for building high-field magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Even for a thick sample, the in-plane critical field surpasses 8 T at 4 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' We summarize the parameters of 2D superconductors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As for 1T- MoS2 [27], 2H-TaS2 [23], 3R-TaSe2 [28], Td-MoTe2 [25], 2H-NbS2 [29], their critical temperatures are below the temperature of liquid helium (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2 K), rendering the construction of high-field magnets impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Gated MoS2 displays a relatively high critical temperature and also very high critical fields, but superconductors under ionic gating are not suitable for applications [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Lastly, 2H-NbSe2 is comparable to 1T′-WS2 in critical fields and critical temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' However, its critical current density is two orders of magnitude lower than that of 1T′-WS2 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The significance of our work is that we report an unprecedentedly high superconducting critical current density (17 MA/cm2 at 0 T) in 1T′-WS2, which exceeds those of all the known 2D superconductors to date [21-33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Notably, it even exceeds the Jc of MgB2 films [34], a well- known superconductor for high-critical-current applications (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Even under an 8 T in-plane magnetic field, the Jc of 1T′-WS2 is substantially large (7 MA/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' As a reference, the critical currents of commercial magnet building materials are listed here, such as Nb-Ti alloy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='1 MA/cm2 at 10 T) and Nb3Sn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5 MA/cm2 at 10 T) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The large Jc at zero and finite magnetic fields makes 1T′- WS2 a potential candidate for future study on building next- generation superconducting magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Having explored the superconductivity of 1T′-WS2, we turn to the topological features pertinent to its electronic band structure using a series of theoretical calculations and angle-resolved photoemission spectroscopy (ARPES) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The calculated bulk band structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Besides the continuous energy gap between conduction band and valence band around the Fermi level, we observe a band inversion at the -point between W d and S p orbitals, which leads to a strong topological insulating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Furthermore, the surface-projected calculation shows the topological Dirac surface state emerging from the valence band and merging into conduction bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The corresponding ARPES data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 4b), taken at T = 10 K (above Tc) matches the first principles calculations below EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In particular we identify the linear-dispersed hole pocket at to be the lower cone of the topological surface state, as it shows no photon- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='3 Ultrahigh critical current density of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' a, Differential resistance of a 6-nm-thick 1T′-WS2 sample as a function of the direct current (DC) bias at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' b, Critical current density for a 1T′-WS2 device as a function of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' c, Critical current density of the 1T′-WS2 device plotted as a function of sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' d, Critical current densities of the 1T′- WS2 device plotted as a function of perpendicular and parallel magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' e, Comparison of critical current densities among 1T′-WS2 and other representative 2D superconductors, such as twisted bilayer graphene (TBG) and transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Commercial magnet building materials are also included for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Here, the superconducting critical temperatures Tc of the different materials were determined under zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' energy dependence and agrees well with the calculated dispersion of the Dirac state (More details of ARPES data analysis are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' S10-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' ARPES Fermi surface map also visualizes the highly anisotropic Fermi surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 4c), which possibly contributes to the extremely anisotropic Hc2 in 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The calculated superconducting gap of 1T′-WS2 on the Fermi surface is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' These results lend crucial credence to the in-plane anisotropy of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' It is noted that so far there is no clear relationship between the topological nature and high critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='4K - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='8K—2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K—3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K—5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K— 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K—7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5K 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content="4nm 1T-MoS2 2nm 1T-WS2 7nm 3R-TaSe2 6nm 1T'-WS2★ 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5nm T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='-MoTe2 1T" WS, @8T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='6nm TBG 6nm 2H-NbS2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='6nm a-Mo2C MgB- filn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='2nm 2H-TaS2 Nb,Sn @10T 3nm 2H-NbSe2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='6nm gated Nb-Ti @10T MoS2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='4 Topological features of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' a, Calculated surface band structure of 1T′-WS2 at ky = 0, featuring a topological Dirac surface state near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' b, Energy-momentum cut acquired through ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' c, Fermi surface of 1T′-WS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' d, Calculated superconducting gap which all kz are projected in the surface Brillouin zone at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='5 K on the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The unit of the color bar is meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' In summary, combining a series of experimental and numerical techniques, we comprehensively studied 1T′- WS2 and find a unique blend of ultrahigh critical supercurrent density, large superconducting anisotropy (in- plane versus out-of-plane) along with topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Our findings not only provide a promising material platform for high magnetic field technologies but also unveil a promising platform for future exploration of topological superconductivity, which may be used to fabricate topologically protected qubits for future quantum computing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' ACKNOWLEDGEMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' Experimental and theoretical work at Princeton University was supported by the Gordon and 286 Betty Moore Foundation (GBMF4547;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The material characterization is supported by the United States 287 Department of Energy (US DOE) under the Basic Energy Sciences program (grant number DOE/BES DE-FG-288 02-05ER46200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' is supported by DOE- BES through award DE-SC0002613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The National High Magnetic Field Laboratory acknowledges support from the US-NSF Cooperative agreement Grant number DMR- 1644779 and the state of Florida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' The authors acknowledge the sample characterization of Imaging and Analysis Center (IAC) at Princeton University, partially supported by the Princeton Center for Complex Materials (PCCM) and the NSF-MRSEC program (MRSEC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' DMR-2011750).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' acknowledges the support of the National Research Foundation, Singapore under its Fellowship Award (NRF- NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang Technological University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' is supported by the Ministry of Education, Singapore, under its MOE AcRF Tier 3 Award MOE2018-T3-1-002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' thanks the support from ITC via the Hong Kong Branch of National Precious Metals Material Engineering Research Center (NPMM), the Research Grants Council of Hong Kong (AoE/P-701/20), the Start-Up Grant (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 9380100) and grant (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' 1886921) from the City University of Hong Kong, and the Science Technology and Innovation Committee of Shenzhen Municipality (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFJT4oBgHgl3EQfBiw4/content/2301.11425v1.pdf'} +page_content=' JCYJ20200109143412311).' metadata={'source': 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Constructor Institute +BERTRAND MEYER, Constructor Institute +YINLING LIU, Institut de Recherche en Informatique de Toulouse, UT2J +ALIYU ALEGE, Constructor Institute and National University of Singapore +ABSTRACT Te technology of formal sofware verifcation has made spectacular advances, but how much +does it actually beneft the development of practical sofware? Considerable disagreement remains about the +practicality of building systems with mechanically-checked proofs of correctness. Is this prospect confned to +a few expensive, life-critical projects, or can the idea be applied to a wide segment of the sofware industry? +To help answer this question, the present survey examines a range of projects, in various application areas, +that have produced formally verifed systems and deployed them for actual use. It considers the technologies +used, the form of verifcation applied, the results obtained, and the lessons that can be drawn for the sofware +industry at large and its ability to beneft from formal verifcation techniques and tools. +ACM Reference format: +Li Huang, Sophie Ebersold, Alexander Kogtenkov, Alexandr Naumchev, Bertrand Meyer, Yinling Liu, and Aliyu +Alege. 2021. Lessons from Formally Verifed Deployed Sofware Systems. ACM Comput. Surv. 1, 1, Article 1 +(January 2021), 37 pages. +DOI: 10.1145/3448975 +1 +INTRODUCTION +Te ever more central role that sofware plays in all processes of the modern world brings to the +forefront the critical question of program correctness. How do we know that sofware systems +perform as expected? Formal verifcation is the task of proving that a program fulflls its specifcation. +It is a long-established research area of sofware engineering, but disagreement persists on its +relevance to mainstream system development. Many practitioners have not even heard of formal +methods, and those who have ofen dismiss them as too hard to apply to mainstream sofware +projects. Against this view, proponents of formal verifcation argue that the technology has now +reached a high level of maturity and applicability. Which of these two views is correct? In other +words, how realistic is the prospect of applying formal methods to production projects? To help +answer this question, it is important to have an objective basis: an assessment of existing atempts +to apply formal methods in practice. Te present survey provides such an assessment, by reviewing +sofware systems fulflling two properties: they have actually been deployed; and they were the +subject, during their development, of formal verifcation. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for proft or commercial advantage and that copies bear this notice and +the full citation on the frst page. Copyrights for components of this work owned by others than ACM must be honored. +Abstracting with credit is permited. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specifc permission and/or a fee. Request permissions from permissions@acm.org. +© 2021 ACM. 0360-0300/2021/1-ART1 $15.00 +DOI: 10.1145/3448975 +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. +arXiv:2301.02206v1 [cs.SE] 5 Jan 2023 + +1:2 +Bruel et al. +Two or three decades ago, one could legitimately phrase the underlying question simply as: +“Can formal verifcation be applied in industry?”. In that form, it is no longer open: a number of +well-publicized industrial projects have used formal verifcation, even if initially for relatively small +programs in mission-critical and particularly life-critical areas such as transportation and defense, +where the consequences of incorrectness in programs are so great as to justify any difculties and +extra costs that formal verifcation might imply. Te contemporary version of the question does +not ask any more about feasibility (which has been established) but about practical aspects, such +as: How large a system can formal verifcation handle? What special qualifcations or training does +it require for the development team? What extra costs, if any, does it imply? +While it is beyond the scope of this survey to provide defnitive answers to these and other +questions on the practicality of formal verifcation, it introduces a factual basis for discussing them. +It analyzes a number of deployed, formally verifed systems according to a set of criteria listed in +section 2.2. Te lessons learned from this analysis appear in section 6. +Previous surveys have covered part of the scope of this article, not necessarily with a focus on +actual deployment of the verifed systems. Tey include the following: +• Surveys of systems verifed with a specifc approach: Event-B in [2], SPARK in [14]. +• Surveys of verifed systems in a specifc application area or of a specifc kind: separation +kernels [96]; distributed systems [28]. +• Surveys of approaches based on the concept of proof assistant [72]. +Te following references are more general: +• [88], from to 2009, presents a questionnaire-based summary of projects that had applied +formal methods to some degree, not necessarily at the level of formally verifed code. +• [92] and [91] are more recent and present verifed systems that the respective authors +found most important. Tey capture the essence of the reviewed approaches, without going +into technical details of each. +Te present article, which does include a fair amount of technical detail, resulted from studying +a signifcant set of formally verifed and deployed sofware systems of widely diferent kinds, +extending across a variety of implementation languages and verifcation techniques. Te projects +were selected using a combination of literature review and responses to a questionnaire widely +circulated by the authors. Tis article helps answer the following questions : +• In what areas of the industry have formally verifed IT systems been deployed? +• What are the properties of the formally verifed systems’ projects in terms of required +initial developer expertise, learning efort and efect on the sofware process? +• What approaches (programming languages, mathematical basis, verifcation techniques, +verifcation tools, verifcation schemes) have been applied to verify deployed systems? +• What are the potential of and obstacles to generalizing the results to the sofware industry +as a whole? Are there specifc kinds of systems that do not lend themselves to formal +verifcation? +Te rest of the article is structured as follows: +• Section 2 presents the selection criteria for the analyzed systems, and the analysis criteria +for their study. +• Sections 3 and 4 provide a short tutorial on formal verifcation, focusing on the methods +actually used in the projects under study. +• Section 5 is the core of the article; it describes and assesses the individual projects and their +use of formal verifcation, according to the criteria of section 2.2. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:3 +• Section 6 draws the general lessons of the analysis and examines its limitations as well as +its signifcance for the generalization of formal verifcation in industry. +2 +SELECTED SYSTEMS +Te sofware systems under review must be both “formally verifed” and “deployed”. +A system is formally verifed if it has been mathematically proven to possess properties specifed +as part of its requirements. +A system is deployed if it is either: +• Publicly available on the Web (in which case the authors of this article were able to use it). +• Not publicly available, but with strong evidence that it is used in production by at least +several users. +Omiting the second category would have excluded commercial, proprietary systems, which account +for some of the most signifcant applications of formal verifcation. Te downside is that analysis +of these systems has to rely on available documents ofen writen by the authors of the respective +systems, or interviews with these authors, rather than direct examination of the sofware. +2.1 +Selection process +Te selection of relevant systems used a combination of literature search and a questionnaire. +Te questionnaire, which remains available [70], was distributed to various relevant communities +and Internet channels starting in December 2020. It yielded responses on 20 systems, of which 10 +were deemed relevant. +Examination of the documentation on these systems, suggestions from various sources, a list +of companies that use formal verifcation methods in sofware engineering [18], and the general +literature on formal verifcation led, by transitive closure on the references, to the identifcation of +65 potentially relevant systems from December 2020 to October 2021. +Afer application of the selection criteria described below, thirty two systems remained (Ta- +ble 1). For reasons of space, the present article focuses on eleven of them, selected for their +representativeness of the various kinds of application areas and verifcation approaches. +2.2 +Criteria for the analysis +Te description of the selected systems uses the following criteria: +• Scope: What system was verifed, in what application domain and for what purpose? +• Components: What are the verifed components and their roles? +• Verifed properties: What properties of the system were verifed? +• Project context: What is the motivation behind the project? +• Decisions: What key decisions were made to help the verifcation process? +• Tool stack: What tools were used for developing and verifying the sofware? +• Style: What is the underlying approach to specifcation? +• Sofware characteristics: What results did the verifcation efort produce? +• Project characteristics: What amount of resources was spent to verify the sofware? +• Lessons: What are the conclusions about the verifcation experience? +Blank entries express that the corresponding information was not available. +3 +VERIFICATION: BASIC CONCEPTS +Te term “verifcation” covers techniques that ascertain the correctness of programs, subject to the +following observations. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:4 +Bruel et al. +Table 1. Deployed and Verified Systems Surveyed +Category +Name +Verifed properties +Input PL +Output PL +Proof +engine +Spec/ +Imp +KLOC +Efort +(py) +References +Compiler +CompCert +semantics preservation +Coq +OCaml, C +Coq +1 +135 +6 +[53] [54] +[43] +CakeML +semantics preservation +CakeML +CakeML +HOL4 +– +100 +– +[48] [78] +[32] +Vellvm +semantics preservation +Coq +OCaml, C +Coq +1 +32 +– +[95] [93] +[94] +Vericert +semantics preservation +Coq +OCaml +Coq +3.63 +2.5 +1.5 +[35] +Vermillion +functional correctness +Coq +OCaml +Coq +– +8 +0.75 +[49] +Q*Cert +functional correctness +Coq +OCaml +Coq +– +– +– +[4] +Library +EifelBase2 +functional correctness +Eifel +Eifel +AutoProof, +Boogie, Z3 +– +9.37 +6 +[66] +HACL* +security, functional cor- +rectness, memory safety +F* +C +Z3 +3.3 +31 +< 1 +[97] [33] +DICE* +security, functional cor- +rectness, memory safety +F* +C +Z3, Meta-F* +4.7 +29 +- +[79] +Signal* +security, functional cor- +rectness, memory safety +F* +WebAssembly Z3, ProVerif +- +4 +- +[68] +Amazon +s2n +functional correctness +C +C +Coq, SAW +0.86 +0.7 +> 3 +[16] [73] +OS +seL4 +functional +correctness, +security +C +C +Isabelle, Z3, +Sonolar +100 +> 10 +31.2 +[29], [47] +ProvenCore +security +C +C +ProvenTools +– +– +3 +[55] +Verve +safety +Beat +C#, TAL +Boogie, Z3 +3 +7.55 +0.75 +[89] +mCertiKOS +security, functional cor- +rectness +Coq +ClightX, +LAsm +Coq +6 +3 +1 +[30] [31] +[21] +mC2 +security, functional cor- +rectness +Coq +ClightX, +LAsm +Coq +17 +6.5 +2 +[31] +Hyper-V +functional +correctness, +safety +C +C, +Assem- +bly +Boogie, Z3 +5 +105 +1.5 +[50] +PikeOS +functional +correctness, +security +C +C, +Assem- +bly +Boogie, Z3 +5 +– +1.5 +[8] +ExpressOS +security +C#, Dafny +C# +Z3 +0.028 +– +– +[59] +Aeronautics +SHOLIS +functional correctness, +security, +timing/memory constraints +SPARK +SPARK +Simplifer, +Proof +Checker +3.07 +27 +19 +[45] +Lockheed +C130J +functional correctness +SPARK +SPARK +Simplifer, +Proof +Checker +– +350 +– +[22] +NATS +iFACTS +absence of run-time ex- +ceptions, functional cor- +rectness, memory safety +SPARK +SPARK +Simplifer +0.3 +250 +> 50 +[14] [15] +EuroFighter +Typhoon +functional correctness +Ada +Ada +Supertac, +ProofPower +– +35 +1.5 +[77] [65] +Transport +Roissy +Shuttle +security +B, Ada +Ada +EDiT +B, +Bertille +1.16 +158 +– +[6] +Dutch +Tunnel CS +safety and liveness +mCRL2 +Java +mCRL2, Ver- +Cors +0.15 +37 +8 +[64] +Nuclear +Power +Sizewell B +functional correctness +B +PL/M-86, +Assembly +MALPAS +– +150 +250 +[82] +Network +Ironclad +Apps +functional +correctness, +security +Dafny +Dafny +Boogie, Z3 +0.5 +85 +3 +[27] +Qark +security +Coq +OCaml +Coq, Ynot +0.2 +976 +0.83 +[41] [40] +[69] +CoCon +security +Isabelle/HOL +Scala +Isabelle/HOL +BD-Security +0.67 +15.2 +0.25 +[67] [42] +CoSMed +security +Isabelle/HOL +Scala +Isabelle/HOL +BD-Security +1 (ker- +nel) +0.33 +(app) +7.4 +0.33 +[7] +DataBase +Ynot +functional correctness +Coq +OCaml +Coq +1 +3.03 +8 +[1] +Verifcation +Tool +Flover +soundness with respect +to standard semantics +Coq +HOL4 +Coq , HOL4 +10 +3 +2 +[9] [26] +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:5 +3.1 +Definitions +Te meanings of “verifcation” has both broad and narrow meanings: +• In practice. the sofware industry mostly uses “verifcation” to mean testing. In the +programming research literature, the term is used instead to denote techniques for proving +correctness mathematically, the meaning also retained in the present article. +• Te sofware engineering literature sometimes distinguishes verifcation from validation, +using “V & V” to cover their combination. In this terminology, verifcation is internal +(checking consistency, as in “the system is doing things right” ) and validation external +(checking the program against its specifcation, as in “checking that the program does the +right things”). In keeping with the usual sense of “formal verifcation” in the programming +research community, this article uses “verifcation” to cover both parts of “ V & V”. +• Verifcation is “dynamic” if it needs to execute the program, as in the case of testing. It +is “static” if it works only from the program text Tis article focuses on static techniques +(which, since they do not perform any execution, require neither a compiler nor input +data). +• Program proving is the most ambitious form of static verifcation, meant to ascertain that +the program satisfes its specifcation. If the specifcation covers all relevant aspects of the +program behavior, the proof will demonstrate “full functional correctness”. +• Other forms of static analysis only analyze the program for the presence or absence of +specifc faults, such as deadlock or memory overfow. +Te following characteristics apply to both static and dynamic forms of verifcation: +• Verifcation is not debugging. It may fnd faults (“bugs”), but correcting them is not its job. +• Te words “succeed” and “fail” mean something else for verifcation tools than for programs. +A verifcation tool succeeds if it either fnds faults or ascertains (mathematically in the case +of formal techniques) that the program contains no faults of the specifed kinds. Te tool +fails if it is unable to decide either way. (In contrast, a program succeeds if it completes its +job according to its specifcation, and fails otherwise. Verifcation can succeed on a failing +program — by identifying the fault — and conversely.) +3.2 +Inherent theoretical and practical limitations +It is well known that a passing test, or any number of them, do not demonstrate that the program +is correct. (A non-passing test does prove that the program is incorrect.) Tis observation is the +basis for asserting the superiority of mathematical proofs (and static techniques in general) over +tests. Limitations remain, however. +First, a failed proof (as defned above) does not indicate that the program is incorrect. It simply +indicates that the proof tool is unable to decide either way – correct or incorrect program. Tis +frequently arising case is due to both theoretical and practical reasons: +• On the theoretical side, program correctness for any useful programming language is an +undecidable problem in the sense that it is not possible to produce a tool that will always +prove or disprove the correctness of any program in fnite time. (Tis property does not +preclude the production of tools that will yield proofs for some programs.) +• On the practical side, a failed proof atempt is not the end of the story but just a step. +(“Losing a batle, not the war”.) Such failure is in fact the most common experience in the +day-to-day practice of verifcation: try to verify the program; fnd that the tool either reveals +a fault or fails to produce a result; in either case, tweak the program, the specifcation or +both to try to correct the problem; repeat. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:6 +Bruel et al. +Such an iterative scheme is reminiscent of the “run a test, fx a bug, repeat” of traditional non-formal +development using tests. Te diference is that in static approaches the basic iterative step involves +analyzing the text of the program (rather than executing it. But in both cases the process is iterative +(as opposed to an ideal two-step process of writing the program then running a tool to prove +its correctness). As a result, even though modern verifcation tools are described as permiting +“automatic” or “mechanical” verifcation, their actual use still involves human efort. How much +efort is an important practical factor to consider when assessing verifcation tools and techniques. +Tis survey atempts, whenever information is available, to provide assessments of the efort that +was involved in the verifcation of the reviewed systems. +3.3 +The annotation issue +It is only meaningful to verify a program against specifed properties. As noted above, these +properties can specify the entire relevant behavior of the program (“full correctness”) or only +some of its generic characteristics, for example that a square root computation will not produce an +arithmetic overfow. +Te frst approach obviously yields more benefts, but it requires an extra annotation efort, since +it needs, in addition to the program, a specifcation of the intended properties. Te programmer, or +whoever is performing the verifcation, must equip the program of these properties and ofen, in +practice, intermediate “proof obligations”. Te choice between the verifcation approaches reviewed +next is in part a tradeof between the amount of annotation efort and the extent of properties +proved. +4 +VERIFICATION APPROACHES +Te verifcation of the systems reviewed here relies on a variety of approaches and frameworks. +4.1 +Axiomatic semantics +Axiomatic semantic (also called Hoare logic or Floyd-Hoare-Dijkstra semantics) is one of the most +widespread formal frameworks. It uses the most annotations and addresses full correctness. +Axiomatic semantics assumes that the program is equipped with “verifcation conditions”, also +called “assertions”, which may include routine preconditions and postconditions, loop invariants, +class invariants (in object-oriented programming), as well as conditions included at arbitrary +program places. A verifcation condition is a boolean-valued function on program states (or, in the +case of postconditions, on two program states, initial and fnal). Proofs of termination of loops and +recursive routines additionally require integer “variants”. An example of a routine annotated with +a precondition and a postcondition is, in the notation of the Eifel programming language +1 +sqrt (x: REAL, epsilon: REAL): REAL +2 +-- Non-negative square root of 'x' with precision 'epsilon' +3 +require +4 +x ≥ 0 +5 +do +6 +.. . Algorithm to compute into Result the square root approximation .. . +7 +ensure +8 +Result ≥ 0 +9 +abs (Result ˆ2 − x) ≤ epsilon +10 +end +Te precondition (require) expresses a property of x in the original state, at the time of a call; +the postcondition (ensure) expresses a property of the Result, in relation to the value of x. Te +combination of the precondition and postcondition of a routine are its specifcation, or “contract”. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:7 +Verifcation consists of proving that the implementation matches this specifcation. It relies on a +set of rules (axioms and inference rules) associated with the programming language. An example +inference rule (with {P} A {Q} stating that if P, a verifcation condition, is satisfed prior to the +execution of A, then Q will hold aferwards) characterizes the sequencing of instructions as usually +represented by the “;” symbol in programming languages: +{𝑃}𝐴{𝑄}{𝑄}𝐵{𝑅} +{𝑃}𝐴; 𝐵{𝑅} +(1) +Te part above the line is a hypothesis; if that hypothesis is satisfed, the inference rule makes it +possible to infer the conclusion below the line. Te rule states that the sequence A ; B, assuming +P on start, will produce R on end if there is an intermediate condition Q such that A ensures Q +from P and B ensures R from Q. Tis sequencing rule is typical of how the rules of axiomatic +semantics enable reasoning formally about programs. Tey can be automated and fed into a proof +tool. Tey make it possible to specify the efect of a program, typically through preconditions and +postconditions, and to use the proof tool to prove that the program actually produces that efect. To +achieve this result, it is necessary to provide enough verifcation conditions to guide the proof tool. +4.2 +Abstract interpretation +Abstract interpretation is a mathematical framework for performing static analyses of programs, +based on mapping concrete domains of execution values onto more abstract domains, so that +analyses of important properties (such as arithmetic overfow, pointer nullness or safety constraints), +which would be prohibitive on a concrete domain, can be performed on the abstract domain with +its results still valid at the concrete level. Abstractions satisfying such properties are so-called +Galois connections, enjoying conservation properties both ways (concrete to abstract and back). +Determining specifc properties of relevant program properties involves writing a set of equations +applying on the variables of the program, based on its control fowgraph, data fow graph or both. +Tese equations are mutually recursive, implying that the way to obtain a solution is to compute a +fxpoint of the equations through an iterative method. Such fxpoint computation, and prior to it +the construction of the graph and its conversion into a set of equations, are usually impractical or +even impossible for the program being verifed, if only because the “concrete domains” in which +program variables take their values are very large or infnite. Abstract interpretation will instead +perform the computation on an abstracted version of the program, in an abstract domain. To ensure +that the fxpoint on the abstract domain is reached afer a fnite set of iterations, it may be necessary +to simplify the abstracted computation further through narrowing and widening operations. +A simple example of abstraction could serve to determine whether a variable x can ever be zero +at a certain program point where the instruction involves a division by x. Assuming for simplicity +that the original values (concrete domain) are integers, the abstract domain will only include fve +values, representing zero (Z), positive (P), Negative (N), Botom (representing impossible values) +and Top (representing all possible values), with a partial order relation “less defned than” defning +a latice structure. Te operations on numbers transpose in the abstract domain: for example Z + Z += Z - Z = Z, P + P = P - N = P, N + N = N - P = N, but P + N = N + P = N - N = P - P = Top (since +the last operations may yield a result of any of the categories). Te static analysis in the abstract +domain can determine whether the abstract value of x can ever be Z, much more easily than if we +atempt to perform a similar analysis on the concrete program and domain. Under the appropriate +conditions, results obtained in the abstract domain can be mapped back to the original. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:8 +Bruel et al. +4.3 +Model Checking +Model checking is the result of a “why not?” reaction to the accepted wisdom about the impossibility +of certain tasks. Exhaustive testing is well known to be impossible in practice, since the number of +cases would be infnite. On further analysis, however: +• Computers are fnite automata; integers as represented by computers, for example, do not +form an infnite set but one limited to (typically) 232 or 264 values. Similarly, the number +of times the body of an ordinary “while” loop can be executed is unbounded in principle; +but in practice the number of iterations is, in any program execution, not only fnite but +bounded (since a loop taking 100 years to execute would be of no interest). +• Tese sizes are extraordinarily large at the human scale, making the number of possible +states for any realistic program appear, at frst sight, intractable. But modern computers +are very powerful, executing billions of operations a second, which may make it possible +to achieve the seemingly absurd goal of exhaustive state-space search, perhaps not for the +program itself (except in elementary cases) but for a simplifed version known as a model. +An elementary example of a model of a program is a “boolean model” which replaces every integer +variable by a boolean variable, with False standing for 0 and True for any other integer value, +dramatically reducing the number of possible states (since an integer variable now yields two +states instead of, for example, 264). Another example of a model-checking technique for fghting +the phenomenon of “state explosion” is loop unfolding, which replaces every loop by a scheme +executing the body 1 to N times, ofen for a small N (typically 2 or 3). +Te model-checking algorithm will then verify a specifed property, such as liveness (non- +deadlock) or non-starvation for a concurrent program, by constructing the set of all possible states +of the model program and trying to fnd a “counter-example”: an execution path that leads to a +state violating the desired property. Tis step may or may not be the end of the story: +• If there is a violation of the property (a fault) in the original program, it will (for the +appropriate kinds of property) persist in the model. Ten if the state space exploration does +not produce a counter-example, it defnitely proves that the original program is free from +the fault. +• Finding the fault in a state of the model program is, however, not conclusive: the fault +might be in the original, or it might be an artifact of the reduction to a model. For example, +the property m ≠ n between integer variables is true if m = 1 and n = 2, but a model-checker +using a boolean program will fnd a counter-example since both values map to True. In +this case it is necessary to refne the model to fnd out more. +4.4 +B +Most approaches to verifcation analyze a program to determine whether it is correct, regardless of +how it was produced in the frst place. “Correctness by construction” denotes a diferent process: +build sofware so that it is correct, intertwining the construction and verifcation eforts. Te B +method [36] applies this idea, combined with the notion of refnement. Most systems using B rely +on the “Event B” variant, which adds to the basic framework the notion of event. +A refnement process starts with a very high-level view of the program, involving variables +defning a state, abstract events afecting that state, and invariants. As an example, in a system +controlling access to a road segment undergoing repair work, an invariant could state that all +cars in the system are either stopped or traveling one-way (all East-West or all West-East, if these +are the two directions). Te events might be “let a car enter East” , “Let a car exit East” and the +same for West, “Car arrives East” and “Car arrives West” . Variables include: numbers 𝑤𝑒 and +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:9 +𝑤𝑤 of cars waiting on the East and West sides, current direction 𝑑 of travel (boolean), number +𝑛 of cars currently traveling on the road segment, maximum number 𝑁 on that segment. Each +event is defned by its efect on the variables (and hence the state); for example each “Enter” event +increases 𝑛 by one, decreases the respective waiting variable (𝑤𝑒 or 𝑤𝑤) by one, and leaves the +other variables unchanged. Invariants in this example include 𝑛 ≥ 0 and 𝑛 ≤ 𝑁. Events can have +guards, meaning conditions that must be satisfed for an event to occur; for example the guard +for “enter east” is that the direction of travel is East to West, 𝑤𝑒 > 0 and 𝑛 < 𝑁. Te fundamental +correctness rule is that every event, when executed with its guard satisfed as well as the invariants, +must ensure that the invariant is satisfed again. +Te B method works by step-by-step refnement. Each step, refning an existing (“abstract”) +model into a new (“concrete”) one, may introduce new events, new invariant properties, and more +specifc versions of the abstract events. Te refnement is correct if the new events and the event +refnement preserve both the concrete and abstract invariants. For example, we might refne the +road construction model by introducing trafc lights on both ends, with events such as going +from green to orange, orange to red etc. on either of them. New invariant properties appear (for +example, if the East light is green the West light must be red, and so on). We may refne the “let a car +enter East” by including the change of light to green. All new versions must satisfy the preceding +properties as well as the new ones. +Refnement proceeds until it has reached a level of detail where the result is explicit enough +to be directly implemented in a programming language. With the possible exception of this last +translation step, the result is correct by construction, since every step has been proved correct in +the sense of invariant preservation. +B-specifc proof tools support the method, to prove at each step that new events preserve +invariants and that the new invariants imply those of the preceding refnement level. +4.5 +Proof techniques: SMT solvers +One way to prove a “universal” property, stating that that all elements of a set 𝐸 satisfy a property +𝑃, is to prove the absence of a counter-example; in other words, to prove that it is impossible to +fnd an element of E that satisfes ¬𝑃, the negation of 𝑃. +Te properties 𝑃 of interest, and their negations, are boolean formulae. Disproving 𝑃 means +fnding a variable assignment that satisfes ¬𝑃. Te boolean satisfability problem is NP-complete, +potentially requiring unrealistic computation time, but for theories meeting specifc criteria, known +as satisfability modulo theories (SMT), efective algorithms are possible. SMT solvers apply this +discovery and lie at the basis of many modern proof tools. +4.6 +Z +Te Z specifcation language, a predecessor of B, is a formal specifcation language based on set +theory. Z makes it possible to specify systems in terms of sets and operations on them represented +by mathematical functions and relations. Associated tools support both consistency proofs of the +specifcations themselves and proofs that program in specifc languages satisfy the specifcation. +4.7 +HOL +HOL (Higher-Order Logic) is a mathematical-logic framework underlying by two of the frameworks +used by systems in this survey, HOL4 [37] and Isabelle/HOL [39, 63]. As refected by the name, +it can include logic of several increasing orders: propositional (zero-order), predicate calculus, +second-order (with quantifcation over relations), third-order (with quantifcation over sets of sets). +It also supports typed 𝜆-calculus with object-level polymorphism [34]. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:10 +Bruel et al. +4.8 +Coq +Te Coq framework [20] relies on a diferent logical basis: constructive intuitionistic type theory. A +formal proof in Coq, according to the Curry–Howard correspondence, has an associated program +with a suitable type. to extract a program directly from the proof. +4.9 +Labeled Transition Systems +A Labeled Transition System (LTS) [54] is a mathematical relation 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑠𝑡𝑎𝑡𝑒 +𝑡𝑟𝑎𝑐𝑒 +−→ 𝑛𝑒𝑥𝑡 𝑠𝑡𝑎𝑡𝑒 +that describes one step of execution of a program and its efect on the program state. Te sequence +of transitions from an initial state defnes the observable behavior. A fnite sequence describes a +terminating program execution, where the program terminates either normally or with a run-time +error. An infnite sequence of transitions describes a program execution that runs forever. +5 +THE SYSTEMS: DESCRIPTIONS +Te present section describes the selected systems and reviews them through the ten criteria of the +columns in table 1 (2.2). Te criteria are mostly self-explanatory, but note the following: +• Verifed Properties: properties being verifed, e.g. some properties only (termination, +liveness…), or full functional correctness. +• Programming language(s): languages used for the implementation (not the specifcation). +• Specifcation/implementation (Spec/Imp) ratio : estimate of ratio between lines of specifca- +tion/annotation and lines of implementation code. +• Efort (py): estimate of development and verifcation efort, in person-years. +5.1 +CompCert +5.1.1 +Scope. CompCert is a compiler for the C programming language, intended.for compiling +life- and mission-critical sofware that must meet high levels of assurance. +5.1.2 +Components. Te CompCert project focuses on compilation, excluding preprocessor, +assembler, and linker. Te later components are unverifed and come from a legacy compilation +tool chain. Te compiler supports almost all of the C language (ISO C99) [54], generating code for +the PowerPC, ARM, RISC-V and x86 processors. +5.1.3 +Verified properties. Te compiler includes multiple passes. It formally verifes semantic +preservation between the input and output of every pass. To that end it provides formal semantics +for every source, intermediate and target language, from C to assembly; such semantics defnes the +set of all possible program behaviors [54], including termination (normal or abnormal). CompCert +guarantees that every compilation step preserves all behaviors. +Tese formally verifed properties only apply to the correctness of the compiler itself, with no +guarantee as to the correctness of the compiled sofware and the absence of harmful events such as +null-pointer dereferencing. +CompCert’s formal semantics for C and assembly is hand-crafed; while there is a high degree of +confdence that it faithfully describes these languages, no formal guarantee is possible since the +languages, like most, are not formally defned. +5.1.4 +Project context. Te goal of CompCert is to avoid incorrect compilation, which would +generate incorrect machine code from a correct source program. Many production compilers have +bugs due to the complexity of code generation and optimization. Ten even if the source code of a +program has been proved correct, the version actually executed may produce the very condition +violations that the program’s formal proof was supposed to have rooted out. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:11 +5.1.5 +Decisions. Te compiler is split into 20 passes using 11 intermediate languages [43]. Te un- +verifed parts deal with initial preprocessing, type refnement, and language-specifc simplifcations. +Te formally verifed passes follow the conventional multi-pass compiler structure and include +parsing, front-end compiler, back-end compiler and assembling. Te fnal assembling and linking +steps involve standard non-verifed tools. Te part with most phases is the back-end compiler, with +12 passes implementing various optimizations and covering specifc target architectures. +Te staged implementation enables modular reasoning about every pass and postponing ver- +ifcation of some passes (like parsing, assembling, and linking) to later development. As long +as an intermediate language between two successive passes is the same, the proof of semantics +preservation for both passes automatically guarantees semantics preservation for their combination. +Te verifed compiler code is extracted automatically from the proofs in Coq. Te code of +the extractor is not mechanically checked, though a proof on paper is available [56]. Te Coq +documentation explicitly states that there are cases when the translation from Coq to OCaml (used +for code extraction) may be unsound (e.g., due to the diference between the type systems), and +CompCert developers are responsible for making sure no bad events (such as integer overfow or +exceptions) occur when the compiler runs. +A validation tool called Valex compensates for current absence of formal proofs for assembling +and linking. It reads and disassembles the generated executable, then compares it with the output +produced by CompCert to ensure that there are no injections or other changes. +5.1.6 +Tool stack. Te verifed components of the compiler are writen in Coq to guarantee their +correctness. Given the formal specifcation and the associated constructive proof (Coq works within +the theory of the calculus of inductive constructions), Coq extracts a certifed program from the +proof in OCaml. Te extracted code becomes part of CompCert. +5.1.7 +Style. Te semantics of every source, intermediate and target language (from C to assembly) +is specifed in small-step operational style as a labeled transition system (section 4.9). Because +the behavior of a program can be nondeterministic (due to multiple possible execution orders or +undefned behavior of the source language), the specifcation relies on a refnement of the allowed +behaviors, replacing every non-deterministic behavior with a more deterministic one, by proving +15 simulation diagrams for each intermediate translator independently and then composing them +to establish semantic preservation for the whole compiler. +Te specifcations and proofs are writen in Coq. Te soundness theorem [53] states that if the +source code 𝑆 satisfes the specifcation 𝑆𝑝𝑒𝑐 (writen 𝑆 |= 𝑆𝑝𝑒𝑐), i.e. all observable behaviors 𝐵 of 𝑆 +satisfy 𝑆𝑝𝑒𝑐 (∀𝐵. 𝑆 ⇓ 𝐵 =⇒ 𝑆𝑝𝑒𝑐(𝐵)), so does the compiled program 𝐶: +𝑆 |= 𝑆𝑝𝑒𝑐 =⇒ 𝐶 |= 𝑆𝑝𝑒𝑐 +Tis result comes from two other theorems. One states that every intermediate translator +guarantees that the target program 𝑃𝑡𝑔𝑡 simulates the source program 𝑃𝑠𝑟𝑐: 𝑃𝑠𝑟𝑐 ≿ 𝑃𝑡𝑔𝑡. Another +one [76] shows that in this case, the behavior of the target program (the set of its execution traces) +reproduces the one of the source program: Beh(𝑃𝑠𝑟𝑐) ⊇ Beh(𝑃𝑡𝑔𝑡). +5.1.8 +Sofware characteristics. In 2016, authors reported [54] that the source code has 100 000 +lines of Coq. CompCert is empirically more reliable than GCC and LLVM: the test generation tool +Csmith [90] found 79 bugs in GCC and 202 in LLVM, but none in the verifed parts of CompCert. +5.1.9 +Project characteristics. Te project was started around 2005 [19] and resulted in more +than 30 publications (conferences, journals and books), plus 3 PhD and 1 habilitation theses. Te +frst public version of CompCert (1.2) was released in 2008. Te frst commercial version has been +available since 2015 [54]. As of 2016, the project had taken [54] 6 person-years (code + verifcation). +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:12 +Bruel et al. +5.1.10 +Lessons. Te project demonstrates the feasibility of writing a formally verifed compiler +for a popular programming language. Te correctness comes at a price: the generated programs +run 10–20% slower (depending on the CPU type and optimization level) compared to GCC 4 with +optimizations turned on. Verifed compilation alone, however, turns out to be insufcient for +industrial applications. Established tool chains require further extension of the compiler (e.g., to +produce debug information [54]), as well as refactoring of existing source code (to move compiler- +dependent pragmas and hand-coded inline assembly code to the run-time environment [43]). Te +pipe-lined nature of compilation enables almost independent development and verifcation of more +than 10 compilation passes. Unverifed parts of the compiler rely on additional validation tools to +guarantee correctness of the output. +5.2 +HACL* +5.2.1 +Scope. HACL* [97] is a verifed portable C library of cryptographic primitives for a new +mandatory ciphersuite in TLS 1.3 [71], also intended as the main cryptographic provider for the +miTLS [11] verifed implementation and integrated in Mozilla’s NSS cryptographic library. +5.2.2 +Components. All cryptographic algorithms in HACL* are correct by construction. +5.2.3 +Verified properties. Te developers of HACL* have verifed the following properties: +• Memory safety. Te code never reads or writes memory at invalid locations, such as null +or freed pointers, unallocated memory, or out-of-bounds of allocated memory. Also, any +locally allocated memory is eventually freed (exactly once). +• Functional correctness. Te code for each primitive conforms to its published standard +specifcation on all inputs. +• Mitigations against Side-Channel Atacks. Te code does not reveal any secret inputs to an +adversary, even if the adversary can observe low-level runtime behavior such as branching, +memory access paterns, cache hits and misses, power consumption, etc. +• Cryptographic security. Te code for each cryptographic construction implemented by the +library is indistinguishable (with high probability) from some standard security defnition, +under well-understood cryptographic assumptions on its underlying building blocks. +5.2.4 +Project context. Te project’s primary goal was to build a reference implementation in C +and prove that it conformed to computations described in the corresponding RFC standards. +5.2.5 +Decisions. Te main implementation vehicles for HACL* are the F* functional language +and Low*, an embedding in F* of a safe subset of C. HACL* code never allocates memory on the +heap; all temporary states are stored on the stack to simplify proofs of memory safety and avoid the +need for explicit memory management. Te Low* source code is broken into many small functions, +in order to improve readability, modularity and code sharing, and to reduce the complexity of +each proof. Consequently, the default translation of this code to C would result in a set of small C +functions, which can be overly verbose and hurts runtime performance with some compilers like +CompCert (Section 5.1). To allow beter control over the generated code, the KreMLin compiler is +sometimes directed (via program annotations) to inline certain functions and unroll certain loops. +A large chunk of the bignum verifed code is shared across Poly1305, Curve25519 and Ed25519, +meaning that this code is verifed once but used in three diferent ways. Te sharing has no impact +on the quality of the generated code because KreMLin inlines the generic code and specializes it for +one particular set of bignum parameters. Te net result is that Poly1305 and Curve25519 contain +separate, specialized versions of the original Low* bignum library. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:13 +5.2.6 +Tool stack. Te developers frst write a high-level specifcation (Spec) for the primitive in a +higher-order purely functional subset of F* (Pure F*). Tey then write an optimized implementation +(Code) in Low*, a low-level subset of F* that can be efciently compiled to C. Te code is then +verifed, using the F* Z3-based typechecker, for conformance to the Spec and to ensure that it +respects the logical preconditions and type abstractions required by the F* standard library. If +type checking fails, there potentially may be a bug in the code, or it may be that the type checker +requires more annotations to prove the code correct. Finally, the Low* code for the primitive is +translated via KreMLin to C code. More precisely, it is translated to Clight, a subset of C that can be +compiled with CompCert (Section 5.1). In practice, however, the resulting Clight code is compiled +with GCC due to perfromance considerations. +5.2.7 +Style. Fig. 1a contains an example of a pure F* executable specifcation. Te meaning of +this specifcation is as follows. To implement prime feld arithmetic for the Poly1305 algorithm +on 64-bit platforms, one strategy is to represent each 130-bit feld element as an array of three +64-bit limbs, where each limb uses 42 or 44 bits and so has room to grow. When adding two such +feld elements, one can simply add the arrays point-wise, and ignore carries, and the fsum function +above does exactly that. Fig. 1b contains a contract of the future implementation (fadd) of the +abstraction (fsum) expressed in pure F* (Fig. 1a). +1 +type limbs = b:buffer uint64_s{length b = 3} +2 +let fsum (a:limbs) (b:limbs) = +3 +a.(0ul) ← a.(0ul) + b.(0ul); +4 +a.(1ul) ← a.(1ul) + b.(1ul); +5 +a.(2ul) ← a.(2ul) + b.(2ul) +(a) The pure F* abstraction. +1 +val fsum: a:limbs → b:limbs → Stack unit +2 +(requires (𝜆 h0 → live h0 a ∧ live h0 b +3 +∧ disjoint a b +4 +∧ index h0.[a] 0 + index h0.[b] 0 reading ≤ danger levelLessons from Formally Verified Deployed Sofware Systems +1:21 +5.6.8 +Project characteristics. Te research and development team of the project comprises over +140 engineers from NATS and Altran and the project spanned over 10 years. A one-week formal +training was carried out, including the training on Z notation and SPARK. +5.6.9 +Lessons. Verifcation of iFACTS project scaled much beter than previous SPARK projects +such as SHOLIS and C130J, owing to the considerable improvement in computing resources and +theorem-provers. However, the further reduction of time consumption of the overall project is +limited due to the rudimentary tool support for Z. Functional specifcation in Z was writen in word +documents, which makes merging diferent versions of documents difcult and time-consuming +when they documents are large (over 1000 pages documents). +In general, iFACTS is a ground-breaking predictive tool for air trafc controllers and has delivered +signifcant increase of efciency of controllers and capacity of airspace. Te scale of implementation +made iFACTS the most ambitious SPARK project to date. Formal methods are applicable to all +phases of the lifecycle in the project. Training engineers is not a barrier. It was shown that engineers +of diverse programming backgrounds can easily pick up verifcation languages/tools [83]. +5.7 +Ynot +5.7.1 +Scope. Relational database management systems (RDBMSs) enable application developers +to have a high-level specifcation for the behavior of the data manager (one of the modules in the +RDBMS) and be suitable for formal reasoning about application-level security and correctness +properties. It is one of the systems implemented by the team of Te Ynot Project. +5.7.2 +Components. Te Ynot RDBMS is a lightweight, fully verifed, in-memory database man- +agement system. Users can use a command line interface to create tables, load tuples into a table, +save/restore a table to/from disk, and query the tables using a subset of SQL (Structured Qery +Language). +5.7.3 +Verified properties. Te main verifcation task focuses on the correctness of executing +queries in the RDBMS regarding denotational semantics of SQL and relations. +5.7.4 +Project context. RDBMSs are omnipresent in modern application sofware. Tey are used +to store data whose integrity and confdentiality must be maintained. Te implementation of the +data manager should ensure that a bug cannot bring about accidental corruption or disclosure. +5.7.5 +Decisions. Te design decisions of RDBMSs aim to avoid complicated proofs and reduce +computations. Te introduction of ptrees and the association of them with B+ tree [1] as ghost +states help avoid this complication. Te avoidance of disjunctions of any favor when something +can be readily computed help reduce computations. A ptree is a defned functional model of the +tree, simplifying the task of defning predicates that describe when a heap contains a tree with a +valid shape. +5.7.6 +Tool stack. Te verifcation of the system involved two tools: the formal proof management +system Coq and the library Ynot extension to Coq. Te proof that implementation meets the +specifcation is writen and verifed in Coq. Te Ynot extension to Coq is designed to support +writing and proving correct pointer-based code, using a variant of separation logic. +5.7.7 +Style. Te specifcation language is Coq. It allowed the team to develop mathematical +theories and prove specifcations of programs. Another language OCaml was used to create Coq +plugins for adding novel tactics or functionality. +An example of Coq specifcations is illustrated as follows: +1 +Definition accurate (m: DbInfo) (G: Ctx) (E: Env G) : Prop := +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:22 +Bruel et al. +2 +forall s I, getRelation s I E = empty < − > isEmpty (m s I) +Tis specifcation is expressed by the Defnition statement, where 𝐷𝑏𝐼𝑛𝑓 𝑜 means database +information; 𝐶𝑡𝑥 means context; 𝐸𝑛𝑣 implies environment; 𝑠 denotes string; 𝐼 means schema. Tis +specifcation ensures that semantic optimizations are used correctly. +5.7.8 +Sofware characteristics. Te RDBMS is a proven lightweight memory system, in which +the functional specifcation of system behavior, system implementation, and proof of achieving +compliance with the specifcation were writen and verifed in Coq. +5.7.9 +Project characteristics. Four people spent two years fnishing this project. +5.7.10 +Lessons. To be practically useful for real systems, there are a number of tasks to be +completed. Some of these tasks necessitate relatively modest extensions to the current implementa- +tion. For instance, key information can be integrated to enable efcient point and range queries. +Besides, reifying the low-level query plan will help fuse operations to avoid materializing transient +data. Te low-level queries are those executed by the RDBMS using a sequence of operations +over imperative fnite maps. Other tasks demand a lot of efort. Realizing the ACID (Atomicity, +Consistency, Isolation, Durability) guarantee of concurrency, transactional atomicity and isolation, +and fault-tolerant storage is a challenging task. Implementing and verifying the correctness of +high-performance, concurrent B+ tree is another challenging task. +5.8 +Roissy Shutle +5.8.1 +Scope. Te Roissy VAL shutle is a system for running driverless shutles in Charles de +Gaulle airport in Paris. In this project, the B Method was applied to develop the safety-critical +sofware that control the speed of the shutles. +5.8.2 +Components. A shutle is an automatic light train that runs on the line that connects +Roissy terminal 1 to Roissy terminal 2. Te line is made up of 5 sections. For each section, a wayside +control unit (WCU) is equipped to control the speed the driverless shutle in the section. Each +section is decomposed into a set of blocks. WCUs should localize the shutle by checking whether +a block is occupied. WCUs receive the commands from the trafc control center and drive the +shutles based on the commands. Te control logic of WCU is implemented as a safety critical +sofware, called WCU-SCS. +5.8.3 +Verified properties. Tere are three phases in the development, i.e. abstract model con- +struction, concrete model construction and code generation. In the abstract model phase, the +informal sofware specifcations, given in natural language, were formalized into an abstract model. +In the concrete model phase, a concrete model is built from the not implementable parts of the +abstract model. In the code generation phase, the Ada code is generated based on both the abstract +model and concrete model. Te verifed properties include safety requirements for the abstract +model, compliance between the concrete model and the abstract model, compliance between the +code and the models. +5.8.4 +Project context. Te whole VAL system is developed by Siemens Transportation Systems +and the development of WCU-SCS has been subcontracted to ClearSy. Te objective of the project +is to ensure that WCU-SCS satisfy a set of safety properties and that its development is within +budget and schedule. +5.8.5 +Decisions. ClearSy applied the Siemens B Method, a process designed for using B language +to build correct sofware by construction. B is a high-level programming language that supports +proving properties. Te functionalities of WCU-SCS were frstly modeled in B. Te B models are +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:23 +then translated into Ada code afer their correctness is established. Unit testing is not performed +since more of the efort is devoted to the early specifcation phase, to build correct sofware directly. +5.8.6 +Tool stack. Refnement from abstract model to concrete model was performed using two +semi-automatic refnement tools, EDiT B and Bertille. Te proof of the B models is performed by +the automatic prover of Atelier B. Generation of ADA code uses the Digisafe-ADA technology. +5.8.7 +Style. Te high-level properties of models are specifed using abstract data types such as +sets of scalar types, relations, partial and total functions. An example of a property for the abstract +model is illustrated below. +∀𝑏𝑙𝑜𝑐𝑘(𝑏𝑙𝑜𝑐𝑘 ∈ 𝑡 𝑏𝑙𝑜𝑐𝑘 ∧ ((𝑐𝑡𝑥 𝑏𝑙𝑜𝑐𝑘 𝑏𝑠 𝑢𝑝[{𝑏𝑙𝑜𝑐𝑘}] ∪ 𝑐𝑡𝑥 𝑏𝑙𝑜𝑐𝑘 𝑏𝑠 𝑑𝑜𝑤𝑛[{𝑏𝑙𝑜𝑐𝑘}]∩ +𝑐𝑢𝑡 𝑏𝑒𝑎𝑚 𝑠𝑒𝑛𝑠𝑜𝑟𝑠 ≠ ∅ ∨ 𝑐𝑡𝑥 𝑏𝑙𝑜𝑐𝑘 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟 [{𝑏𝑙𝑜𝑐𝑘}] ⊆ 𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑏𝑙𝑜𝑐𝑘 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟𝑠) ⇒ +𝑏𝑙𝑜𝑐𝑘 ∈ 𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑏𝑙𝑜𝑐𝑘𝑠) +𝑡 𝑏𝑙𝑜𝑐𝑘 represents the block type and 𝑐𝑡𝑥 𝑏𝑙𝑜𝑐𝑘 𝑥𝑥𝑥[{𝑏𝑙𝑜𝑐𝑘}] denotes abstract variables for a given +𝑏𝑙𝑜𝑐𝑘; Te property specifes that a block is regarded as occupied when a beam sensor located at +one of the block borders is cut (denoted by 𝑐𝑢𝑡 𝑏𝑒𝑎𝑚 𝑠𝑒𝑛𝑠𝑜𝑟𝑠 ≠ ∅) or when the block detector is +occupied (denoted by 𝑐𝑡𝑥 𝑏𝑙𝑜𝑐𝑘 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟 [{𝑏𝑙𝑜𝑐𝑘}] ⊆ 𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑏𝑙𝑜𝑐𝑘 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟𝑠). +5.8.8 +Sofware characteristics. Automatic refnement can lower the costs of a sofware devel- +opment, with an acceptable price that the produced code is usually 10% slower than handwriten +code. Moreover, although the approach can ensure that the code correctly implements the B +model, it is difcult for developers to link the auto-generated code with the corresponding sofware +specifcation. +Proving the concrete model was easier than the abstract model. Te ratio between the efort of +abstract model phase and concrete model phase is 2 : 1. Te proof of abstract model consists of +complex and long interactive demonstrations, making the proof difcult to reuse. Te proof for the +concrete model is broken down into small steps with the same paterns, which is easy to repeat. +With respect to maintenance, it is ensured that the modifed release will remain consistent as +along as the proof is fully redone and that the cost will be also limited since all the development +environment was set up and the rules for refnement and proof are likely to be reused. +5.8.9 +Project characteristics. Te interactive verifcation took about 50 person-days. +5.8.10 +Lessons. Te method used in this project is also suitable for other industrial domains. It +suits sofware mainly based on a discrete logic description (specifed using fnite sets, booleans, +integers etc.) but does not suit sofware based on continuous calculus or foating point numbers +that cannot be regarded as decimal numbers. +Although the compliance between the code and the model was established by mechanical proof, +the compliance between the model and the natural language specifcations is checked manually, +which still leaves room for potential errors. +5.9 +Dutch Tunnel Control System +5.9.1 +Scope. Te considered sofware component is a safety-critical component of a control +system for a trafc tunnel that is currently in use in the Netherlands. It is responsible for handling +emergencies (fres for example). Te Dutch government imposes very high reliability demands on +the trafc tunnel control sofware, and in particular on this emergency component. +5.9.2 +Components. Tis project [64] investigates how formal methods can help developers to +fnd potential problems in their specifcation and (Java) implementation, with realistic efort, and +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:24 +Bruel et al. +preferably at an early stage of development. Te tunnel was already deployed and the verifcation +was done aferwards. +5.9.3 +Verified properties. Te main properties of interest concern reliability and recoverability: +Does the system always go into the Calamity state in real emergency situations? And is it always +possible to recover from calamities, and thereby go back to the Normal operational state? +5.9.4 +Project context. Te tunnel control sofware is developed by Technolution, a Dutch sof- +ware and hardware development company which has a big experience in developing safety-critical, +industrial sofware. Te development process of the trafc tunnel control system come together +with an elaborate process of quality assurance/control, to satisfy the high demands on reliability. +5.9.5 +Decisions. Technolution invested signifcantly in an extensive design phase, to ensure +the quality of the control system and to cope with the high reliability demands. During design +(involving domain experts), the intended behavior of the control sofware was writen out in +pseudocode. Te pseudocode specifcations were further structured into a fnite state machine +(FSM). Te states of this FSM are the operational states of the tunnel system (operating normally, +under repair, evacuating, etc.), while the transitions are the pseudocode descriptions of the system +behavior. Te FSM thus illustrates how the diferent behaviors/events of the tunnel system should +change its operational state. +5.9.6 +Tool stack. A formal process-algebraic model of the informal pseudocode description +and the FSM of the tunnel control sofware is defned using mCRL2. An analysis of this mCRL2 +model is then done via state-space exploration and by checking desired 𝜇-calculus properties on +the model, like deadlock-freedom or strong connectivity. Ten, to deductively verify whether the +control system is correctly implemented with respect to the pseudocode specifcation, a machine- +checked proof that the (Java) implementation adheres to the pseudocode specifcation is provided, +by proving that the program refnes the mCRL2 model, using the automated verifer VerCors [38]. +5.9.7 +Style. Tis work aims at establishing whether +(1) the specifcation is itself consistent, by not being able to reach problematic states, for +example deadlocks in the FSM; +(2) the Java code implementation is writen correctly with respect to the pseudocode specifca- +tion of the intended behavior. +Once the mCRL2 model obtained (which formalizes the specifcation), desired properties are +formulated as 𝜇-calculus formulae, and checked on the reduced model. Here is an example of such +a formula, expressing that the StandBy state can only ever be reached via the Normal state. +5.9.8 +Sofware characteristics. Tanks to this project, an undesired behavior was detected: an +internal deadlock due to an intricate combination of timing and events. +5.9.9 +Project characteristics. Two weeks of a PhD student specializing in formal methods where +necessary to formally model and analyse the trafc tunnel system. Re-running the verifcation +would be no efort at all. Te step that probably will cost the most is to generate the model with +MCRL2, and this takes around 4 minutes. Ten verifying each property should be instantaneous. +Te Verifcation with VerCors should take less than a minute also. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Vr.([(-enter (Normal))* . enter (StandBy)]false ^ +[true*·enter(StandBy)])Lessons from Formally Verified Deployed Sofware Systems +1:25 +5.9.10 +Lessons. During the development phase, signifcant time and efort were invested in +ensuring that the code was correctly implemented with respect to this specifcation. Tis was done +primarily via unit testing and code reviewing. But unit testing and code review revealed to not +prevent from a serious bug in the code, that the current verifcation work pointed out. Tis work +demonstrated that formal methods can really help to detect undesired behaviors within reasonable +time, that would otherwise be hard to fnd. Despite of that, the actual mCRL2 verifcation was +done completely separately from the development and deployment and there has been no relation +between the two projects (no feedback of verifcation results to the actual sofware implementation). +5.10 +Sizewell B +5.10.1 +Scope. Sizewell ‘B’ is a Westinghouse designed Nuclear Pressurised Water Reactor (PWR) +built in Sizewell, Sufolk in the UK. It possesses two diverse protection systems whose role is +to provide an automatic reactor trip when plant conditions reach safety limits and to actuate +emergency safeguard features to limit consequences of a failure condition. +5.10.2 +Components. Te Sizewell B PWR has been constructed and operated by Nuclear Electric +plc. Te overall protection is provided by two systems of diverse technology: the Primary Protection +System (PPS) and the secondary protection system (SPS). Te PPS uses microprocessor-based logic, +while the SPS uses magnetic core logic (laddic) technology. Te design requirement for the PPS +was to provide reactor trip and engineered safety feature (ESF) actuation for all design faults. Te +design requirement for the SPS was to provide reactor trip and ESF actuation, in parallel with the +PPS, for faults with a frequency in excess of about once in 1000 years. Te reliability targets of the +two systems were therefore set at 1 in 10 000 for the PPS and 1 in 100 000 for the SPS. Tis could +have been achieved by hardware, but the highest standards of sofware production available at that +time and of demonstration of integrity had to be applied to provide assurance of this. A full code +analysis using the analyser MALPAS was performed, with compliance analysis, which was able to +show the full match of the requirements to the source code. +5.10.3 +Verified properties. Te conducted verifcation ensures that complex protective functions +are implemented accurately in code and that there is a high degree of confdence in this. +5.10.4 +Project context. Te confrmatory assessment of the sofware using MALPAS (consisting +of static and semantic analysis tools for all safety sofware) was conducted under the TACIS +programme of the European Union. It took place in a large acceptance process. An extensive +programme of sofware verifcation was carried out on the PPS sofware to detect and remove +any errors generated in the design and coding phases. All of the assessments were carried out by +independent companies or independent entities of the Nuclear Electric plc. +5.10.5 +Decisions. Te analysis of the PPS sofware was conducted on a procedure-by-procedure +basis in a botom-up manner. Te analysis commenced with procedures that call no others and then +progresses up the call hierarchy until the top level application code is reached. All the MALPAS +analysers were run on each procedure and the results verifed against two levels of specifcation: a +higher level specifcation (the Sofware Design Requirements, SDR) and a lower level (the Sofware +Design Specifcation, SDS). Te SDR was the primary document against which the code was verifed, +with supporting information provided by the SDS. All the code that can be accessed during on-line +operation was subjected to MALPAS analysis. +5.10.6 +Tool stack. Te MALPAS toolset comprises fve specifc analysis tools that address various +properties of a program. Te input to the analysers needs to be writen in MALPAS Intermediate +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:26 +Bruel et al. +Language (IL) which is produced by an automated translation tool from the original source code, +assisted by an analyst which can make suitable manual changes. +Te MALPAS toolset consists of fve analysers: +(1) Control Flow Analyser examines the program structure, and provides a summary report +drawing atention to undesirable constructs and an indication of the complexity of the +program structure. +(2) Data Use Analyser separates the variables and parameters used by the program into distinct +classes and identifes errors. +(3) Information Flow Analyser identifes the data and branch dependencies for each output +variable or parameter. +(4) Semantic Analyser reveals the exact functional relationship between all inputs and outputs +over all semantically feasible paths through the code. +(5) Compliance Analyser compares the mathematical behaviour of the code with its formal IL +specifcation to ensure that the code of each procedure: +• conforms to the functional aspects of its specifcation (SDR, supported by SDS), +• respects the state invariants, +• performs its specifed functions without corrupting the computing environment nor +data owned by any other module or procedure, +• conforms to the language in which it is writen. +5.10.7 +Style. We have not found an example of Sizewell B specifcation in the literature, so +this section will exceptionally not include an illustration. Te IL text is fed into MALPAS, which +constructs a directed graph and associated semantics for the program under analysis. Te IL +specifcation is writen as pre- and postconditions, as well as optional code assertions. Te MALPAS +Compliance Analyser requires the specifcation to be represented as pre- and postconditions for +each procedure, along with any necessary assert() statements within the body of the code, and the +analyser will then show whether the code meets the specifcation. +Te frst part of the Compliance Analysis process is the construction of the mathematical +specifcation from the natural language SDR and SDS. Tis work ensures both that the interpretation +of the existing specifcations is correct and that the important functionality and properties are +modelled in the mathematical specifcations. +5.10.8 +Sofware characteristics. Sizewell B has required the major efort by Westinghouse, NE, +the NNC and others to ensure that complex protective functions are implemented accurately in code +and that there is a high degree of confdence in this. Te quantity of sofware involved in the PPS +is large but not too large to have been verifed meticulously and in its entirety. Te PPS sofware +is verifed with very demanding reliability requirements (imposed by UK Nuclear Installations +Inspectorate (NII)), that atest to its suitability for reactor protection. About 2000 comments have +been raised during the verifcation process (One for 30 lines of code). 40% of them induced minor +specifcation changes or checks but no major issue was detected. +We didn’t fnd any representation of this semantics in the literature +5.10.9 +Project characteristics. Te team grew from 15 person at the beginning of year 1992, to +95 at the end of the project. It took four and a half years to complete the project; Te Compliance +Analysis was the most important part of the MALPAS analysis of the PPS sofware and also involved +the majority of the efort. +5.10.10 +Lessons. Te analysis does not prove the safety of the sofware, but does prove formally +that it meets its specifcations. It also increased the integrity of the sofware (through code +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:27 +and documentation modifcations that have resulted from detected anomalies) and confdence +in its correctness. Five main measures were adopted to ensure its accuracy, consistency and +reproducibility: (1) all work was conducted under a defned quality system and in accordance with +the requirements of ISO90011 ; (2) a detailed “standards and procedures” document defned very +precisely how the analysis was to be conducted; (3) full recording of all aspects of the analysis +was ensured; (4) peer reviews of all analytical work were carried out; (5) strict confguration +management of all analytical documents and results was applied, both in hard copy and magnetic +media. +Te Malpas analysis work conducted in the scope if this project has shown the feasibility of +conducting a rigorous retrospective analysis of a large sofware system. Even if the costs are high +in absolute terms, they are low in relation with any potential sofware malfunction for example. +Conducting rigorous static analysis, including Compliance Analysis, has been then considered +to be essential for sofware controlling systems with the level of criticality and potential failure +consequences similar to that of the Sizewell ‘B’ PPS. +5.11 +CoCon +5.11.1 +Scope. CoCon is a conference management system, providing a web interface. +5.11.2 +Components. Cocon involves 5 types of stakeholders. Each of them has a specifc role. +CoCon supports 45 operations, some of them being applicable by certain kind of stakeholders +only. Tese 45 op erations are dispatched in 5 categories (creation, update, nondestructive update, +reading, listing). Besides, a conference go through 7 phases (No-Phase, Setup, Submission, Bidding, +Reviewing, Discussion, Notifcation and Closing). +5.11.3 +Verified properties. Confdentiality properties of CoCon are verifed through a framework, +called verifcation infrastructure for Bounded-Deducibility (BD) security. It is a general framework +for the verifcation of rich information fow properties of input/output automata. BD security is +parameterized by declassifcation bounds and triggers (in a context of a fxed topology of bounds +and triggers). Informally, BD Security can be summarized as follows: If trigger T never holds then +atacker Obs can learn nothing about secrets Sec beyond B. CoCon’s verifcation also involves a +form of traceback properties and several safety properties proofs. All of them are described by +Isabelle scripts (A zip archive with the Isabelle formalization is available here2). +5.11.4 +Project context. A Conference Management System (CMS) presents confdentiality, in- +tegrity and security problems that have to be under control. In the same time, a CMS must guarantee +the selective availability of information. Te object of CoCon’s team is to address information fow +security problems of realistic web-based systems by interactive theorem proving—using a proof +assistant, Isabelle/HOL. +5.11.5 +Decisions. Te architecture of CoCon follows the paradigm of security by design [24] +(the guarantees do not apply to the application layer), as depicted on Figure 7. +5.11.6 +Tool stack. Cocon relies on a three layers stack as follows: +1) Formalization and verifcation of the kernel of the system in the Isabelle proof assistant +2) Automatic translation of the formalization obtained in 1) in the functional programming +language Scala +3) Wrapping of the translated program coming from 2) in a web application (using trusted +components) +1htps://www.iso.org/standard/16533.html +2htps://www.andreipopescu.uk/papers/Formal Scripts CoCon.zip (accessed in September 2021) +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:28 +Bruel et al. +Fig. 7. CoCon’s architecture +5.11.7 +Style. To ensure that all these properties are handled, CoCon’s kernel is formalized in +Isabelle as an executable input/output automaton (extracted from the Isabelle specifcation to a +Scala program using Isabelle’s code generator). A state of the CoCon’s I/O Automaton stores the +lists of registered conference IDs, user IDs and paper IDs; and, for each ID, the state stores actual +conference, user or paper information. For user IDs, the state also stores (hashed) passwords. In the +context of a conference, each user is assigned one or more of the roles described by the following +Isabelle datatype: +𝑑𝑎𝑡𝑎𝑡𝑦𝑝𝑒𝑅𝑜𝑙𝑒 = 𝐶ℎ𝑎𝑖𝑟|𝑃𝐶|𝐴𝑢𝑡𝑃𝑎𝑝𝑒𝑟𝐼𝐷|𝑅𝑒𝑣𝑃𝑎𝑝𝑒𝑟𝐼𝐷𝑁𝑎𝑡 +Te initial state of the system, 𝑖𝑠𝑡𝑎𝑡𝑒 ∈ 𝑆𝑡𝑎𝑡𝑒, is the one with a single user, the voronkov as +SuperChair (the superChair is the frst user of the system, and his role is to approve new-conference +requests), and no conference (Fig. 8). +Fig. 8. Specification of the initial state +5.11.8 +Sofware characteristics. CoCon has so far been used to manage the submission and +reviewing process of two international conferences, TABLEAUX 2015 and ITP 2016, hosting +approximately 70 users and 110 users, respectively—consisting of PC members and authors. +5.11.9 +Project characteristics. Te verifcation took 3 person-months, which also counts the +development of reusable proof infrastructure and automation. Te development of the web appli- +cation wrapped around the kernel took longer, and much of it was done incrementally while the +system was already operational; there were two MSc students working on it; the second changed +the implementation completely. Tis took them a total of 6 person-months. +5.11.10 +Lessons. Te second application of CoCon proved difcult. Indeed, it revealed a bug that +was difcult to catch. Tis bug allowed confdentiality violations, which is precisely what CoCon +aims at avoiding. In reality, this violation was caused by the web interface and had nothing to do +with CoCon’s verifcation. But this highlighted that the API layer, and all the trusted components, +need to be verifed too. Te restrictions on CoCon’s information fow doesn’t consider any dynamic +variations of roles. (For example, we can imagine that a PC reviewer U1 goes through some papers, +producing recommendations and reviews, then cancels the role and applies as an author). So +it is possible to get access to some secrets by changing the roles without having diferent roles +simultaneously (unless the model takes history of roles into account). +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +code generation +Isabelle +Functional +Specification +Program +Web +Applicationistate= +confIDs = l +conf=(>cid.emptyConf) +userlDs =["voronkov"] +pass=(αuid.emptyPass) +user=(αuid.emptyUser) +roles=(ciduid.) +paperlDs = (α cid. ) +paper=(>^pid.emptyPaper) +pref=(auid pid.NoPref) +voronkov="voronkov" +news = (α cid. [l) +phase=(acid.noPh)Lessons from Formally Verified Deployed Sofware Systems +1:29 +Sizewell B +Lockheed C130J +SHOLIS +EuroFighter +seL4 +CompCert +Roissy Shutle +NATS iFACTS +Hyper-V +PikeOS +Ynot +ProvenCore +Verve +EifelBase2 +mCertiKOS +CakeML +QUARK +Vellvm +CoCon +ExpressOS +Ironclad Apps +mC2 +Amazon s2n +Dutch Tunnel +CoSMed +FloVer +Q*Cert +HACL* +Signal* +Vermiliion +Vericert +DICE* +1989 +1993 +1997 +2001 +2005 +2009 +2013 +2017 +2021 +Starting year +Proof-based +Auto-active +Refnement +Fig. 9. Project starting year +Te formal guarantees provided in Isabelle have to be combined with a few trusted steps to apply +to the whole system. Te verifcation targets only the system’s implementation logic — atacks +such as browser-level forging are out of its reach, but are orthogonal issues that should be handled. +6 +DISCUSSION AND LESSONS LEARNED +Te analysis of deployed and formally verifed sofware systems (from which eleven are presented +here) led us to identify some generalities and provided insights in formal verifcation of sofware +systems. +6.1 +General observations +Figure 9 presents the starting years of the projects: the development periods of the surveyed systems +range from 1989 to 2021; afer 2004 new formally verifed systems, which were then deployed, +appear at an almost constant rate (the red line). Earlier atempts have stabilised, and there is now a +steady growth. Proof-based and auto-active approaches to verify such systems completely replaced +refnement-based ones by 2007. Both of them continue to materialize, but, starting from 2011, the +proof-based techniques take over the auto-active ones that were more popular at the beginning. +Model checking alone is never used for verifying code, limited by rather rigid use cases with a +well-known state space. It is used as an underlying technique (for example, Dutch Tunnel Control +System). Refnement-based systems are also very rare (Roissy Shutle is one of them). +General observations on technologies applied in the projects can be derived as follows: the +mostly frequently used programming languages are Coq and C — 8 projects use Coq as their +programming language, among which 6 of them also use Coq as the specifcation languages; the +most frequently used specifcation languages are Coq (used in 11 projects) and Z notation (used +in 4 projects); 7 projects apply annotations in the programs to specify the verifed properties. We +note also a binding between programming language and specifcation language. Examples giving, +a verifcation using HOL seems to be suitable for systems writen in C / SML / OCaml, SPARK code +is specifed in Z and SPARK. Isabelle is used for Scala and C. +Te most widely used proof engines are Isabelle, Coq, Z3, and HOL4. So the most widely used +theoretical framework for verifcation is Hoare logic. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:30 +Bruel et al. +6.2 +Formal verification penetration level +Most verifed systems belong to the category of system sofware, i.e. programs providing a platform +for running other sofware on the computer (compilers, operating systems and hypervisors, crypto- +graphic libraries, databases, …). Verifed application sofware systems (developed to accomplish +specifc business tasks) cover such safety-critical areas as aeronautics, rail transportation and +nuclear power plants. Non-safety-critical verifed applications include a web browser and few +social applications like a conferencing system and a social media platform. +For some reason, we did not see any formally verifed system in another safety critical area — +healthcare. Te reason is that the method on the verifcation of the systems ([13, 57, 80]) we found is +model checking. However, this method verifes the system from the model level, contradicting our +criteria that systems should be verifed from the code level. Other sofware domains where verifed +systems were not reported yet include desktop applications for text processing, spreadsheets and +alike, sofware for ML and AI. +6.3 +Human factors +Most of the verifed systems studied here involved researchers as the key force to do the verifcation. +Tis may not scale well if the demand for bug-free sofware grows. In that case, the industry +would need professionals specialized in proof engineering [46]. Compared to sofware engineering, +they should have strong background in formal reasoning and theory of programming. However, +regarding the projects studied here, the training of engineers does not seem to be an obstacle. +Even if involved people were used to developing mission-critical and can be suspected to pay +more atention to verifcation, the iFACTS project has shown that trained engineers from diferent +programming backgrounds can easily become familiar with verifcation languages and tools. In the +same way, in the SHOLIS project (not presented here), only one 5-day Z-spec training course and +one 10-day SPARK course were conducted, for a verifcation efort of 19 person-years. +6.4 +Verification results +Tanks to the market needs, some projects turned out very successful (according to their number +of installations and/or use in other projects): CompCert, Hyper-V, HACL* to name a few of them. +Many formally verifed sofware systems run in critical areas of the society, as nuclear or +transports. Teir verifcation is so strongly directed towards critical properties like security or +safety. Te Dutch Tunnel Control system and Sizewell B are some of them. +Verifying a system could also establish semantics preservation, memory safety, functional cor- +rectness, absence of run-time exceptions as is the case of CompCert, CakeML, VeriCert. +6.5 +Threats to validity +Te process of writing this survey and conducting the analysis of the numerous selected systems +presents some main threats to validity: +• We did not conduct an SLR for several reasons, including the fact that many of the studies +that can be found by conducting an SLR are academic, some of the approaches studied +here and to which we had access are not referenced. Nevertheless, the review protocol we +followed is the one recommended for conducting an SLR ([44]) +• It is possible that we overlooked systems which might be interesting and relevant to this +investigation because we did not know about them or had no way of fnding the information. +We may so have ignored atempts to apply formal verifcation in sofware companies that +continually deploy sofware and do not think of publishing their experience. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:31 +• Te fnal selected papers were related to European and US systems only and we have +analyzed systems described in English. +• A limitation of the surveyed papers themselves is that they tend to focus on verifcation suc- +cesses and not to report possible failures. Nevertheless, one team explains that the method +used, if it did not succeed in verifying security, allowed them to fnd a complementary +method to verify it (Ironclad app, not presented here). +• It has been difcult to fnd information on updates (and whether updates have been checked) +for many projects. +• Contacting people for geting missing information (any information about the systems we +didn’t get in publications) may sometimes be difcult, as they may have lef the organisation +and are no longer contactable. +Nevertheless, the analysis of thirty two deployed and formally verifed sofware systems allowed +us to draw some lessons, which are reported below. +6.6 +Lessons +Lessons learned by the project teams: +Tis section discusses some fndings in the lessons learned by the teams who worked on verifying +the described systems. +People working on fourteen of the systems reported performance issues in diferent spaces: +• verifed systems, compared to competing non-verifed systems, perform slower; +• derived systems (like binaries compiled using a verifed compiler) perform slower than +non-verifed systems; +• client systems (like a browser working on top of a verifed OS kernel) perform slower than +those depending on non-verifed components; +• verifcation tools may slow down the development process — failed verifcation atempts +may take longer times to terminate, or may not terminate at all. +Te developers of the HACL* library (section 5.2), however, report that the verifcation process +helped them optimize an existing implementation of the Curve25519 algorithm and then prove +the optimization correct. Te optimized implementation is about 2.2% faster than the formerly +fastest one. Optimizations ofen make the code less readable and thus more likely to contain defects. +Tis property encourages developers to be more conservative in optimizing good enough code. +Formal verifcation tools, as they prove correctness of optimizations, can make the developers +more determined and confdent about making the code more efcient. Tis efect is observed in the +HACL* project. +People who worked on twenty two systems confess they had to depend on an unverifed trusted +computing base (TCB). In the grep and HACL* projects, for example, the resulting C code is +compiled using GCC rather than the verifed CompCert compiler, because GCC produces more +efcient binaries. Possible sources of unverifed TCB include: +• reuse of unverifed existing components in the implementation (like parsing, assembling, +and linking in CompCert). +• unverifed features of the verifcation tool (like the code extraction feature of Coq, used in +many projects). Example giving, the Ironclad Apps project resulted in fnding an actual +bug in the verifcation tools. +• produced code which requires further processing by trusted tools (like the code produced +by the Verilog compiler). +• program design that makes it difcult to specify and verify properties of interest (HACL*); +• insufcient computing resources to conduct exhaustive verifcation (SHOLIS). +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:32 +Bruel et al. +• use of unverifed trusted components (an API layer in CoCon). +• assumption that future users of the system will not atempt certain malicious actions +(Ironclad Apps). +• bugs that may be contained in the formal specifcations (3 bugs were detected afer manually +inspecting Ironclad Apps’ specs); +• natural language requirements (Roissy Shutle). +Sometimes the projects teams would sacrifce some part of the functionality that was difcult to +verify, rather than rely on it as if it was correct. Te decision to restrict functionality was made in +the projects CakeML, HACL*, Verve, EuroFighter Typhoon Aircraf FCS, Ynot. +Seven projects (CompCert, CakeML, EifelBase2, PikeOS, Ynot, Roissy Shutle, Ironclad Apps) +reports indicate that the verifcation technology they were using would be difcult to practice by +an average developer. +For some projects, the teams reported measurable reduction of costs afer introducing formal +verifcation into the development process: Lockheed-Martin C130J, NATS iFACTS, EuroFighter +Typhoon Aircraf FCS, Dutch Tunnel Control. One of them, NATS iFACTS, recommend applying +formal methods on all phases of the sofware development life cycle. +Te Roissy Shutle project reported signifcant efort invested into validating formalization of +input requirements expressed in natural language. Even though the validation was performed +rigorously, it was still a manual human-driven process. Te researchers were not 100% sure that +the formal specifcation used for verifying these systems correctly formalized the actual needs. +Lessons we learned: +Te success of several formally verifed systems demonstrates that the idea of bug-free sofware is +no longer a miracle, but a real-life opportunity. Unlike well-tested systems, the freedom from bugs +is guaranteed up to the quality of the verifcation tools. It does not depend on the number of use +cases or on the possibility of having untested scenarios. Te advances in the verifcation tools are +huge, enabling sofware with thousands of lines of code to be verifed [10]. However, verifcation is +still not mainstream. +Obstacles to a generalization of verifcation Te key obstacle is the high entry level: each +project needs a verifcation expert. It looks much easier to fnd many testers than a developer with +the background sufcient to carry out all verifcation tasks. +Another issue is the rate of changes in modern sofware systems. A new version of almost any +program (apart from safety-critical and highly regulated ones from space programs, transportation +and energy plants) is now released several times a day. Doing that together with re-verifcation +instantly increases the maintenance cost not only in terms of money, but also — that might be +more important — in terms of time, thus slowing down the release rate. For long-running projects, +developers would prefer to use proof-based verifcation scheme, mostly because both code and +proofs remain in synchronization most of the time. Other verifcation schemes do not ofer such a +beneft. Tis raises a question of tool-supported continuous verifcation similar to the technique of +continuous integration, successfully applied in day-to-day sofware development process. +Applications with potentially complicated internals and simple input-output streams seem to be +much easier to verify: there is no interaction with the (unverifed) external world. Te systems +with such interactions also sufer from the regular changes, as mentioned earlier. We have not seen +an application with formally verifed graphical user interface, for example, apart from research +prototypes [3]. +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +Lessons from Formally Verified Deployed Sofware Systems +1:33 +Lack of reuse It is quite surprising to see that Microsof retired the VCC project, initially developed +for the verifcation of Hyper-V and PikeOS. Te tool proved to be successful, and the reasons for +this decision remain unclear. +Tis is a case of a fairly general phenomenon. Many projects, once verifed, do not apparently +continue the verifcation efort and do not have their results transposed to other projects of the same +company or (if it is a tool — and compiler-production company like Microsof) make any atempt to +integrate the results into their tools. In other words, there is litle follow-up and scaling-up of even +successful results. A large percentage of the projects initially selected were abandoned immediately +or shortly afer completion, as these projects, although deployed, were purely for research purposes. +Only a few of them, such as Roissy Shutle were deployed in public systems for constant usage. +We noticed also that some of the verifed projects were not accepted by the users even though +they have been proven (hence beter than their predecessors) due to lack of compatibility of the +new verifed project with legacy code, or other reasons. Example of this is ExpressOS. +Positive efects of formal verifcation attempts While elimination of sofware bugs is the main +driver for the collected projects, many verifcation projects were initiated to achieve the certifcation +of certain industrial standards. For example, the verifcation projects for microkernels, ProvenCore, +aim to achieve the Common Criteria EAL7 3, which requires that the system design should be +formally verifed and tested. For the Lockheed-Martin C130J and PikeOS project, the DO-178B 4 +was applied to determine if the sofware will perform reliably in an airborne environment. Besides, +formal verifcation project could also be driven by the needs of product users. In the s2n project, by +using formal analysis, Amazon aims to provide the customers with concrete information about +how security is established. +Some projects (Hyper-V, Amazon s2n, CoCon) report that formal verifcation could reveal +sofware design defects or subtle bugs that had not been detected using the regular quality assurance +process. In the Lockheed-Martin C130J project, some errors that can be immediately uncovered by +formal analysis (e.g., conditional initialization errors) may only emerge afer very extensive testing. +Furthermore, HACL* project suggests that formal verifcation is benefcial to identify and verify +potential optimizations. +Te entry cost for formal methods seems considerable; however, the cost of repeated verifcation +can decrease drastically. Re-verifcation, done in lots of the projects as seL4, Cocon, Amazon s2n, +iFACTS or Dutch tunnel, takes much less efort than the initial verifcation atempt. In particular, +in the seL4 project it was reported that a new and large feature (new data structure for API calls) +would cost about 1.5–2 person-years to re-verify, which is less than 10% of the initial verifcation +efort. Te re-verifcation of small changes in the iFACTS project can be completed in seconds +while its initial verifcation run took 3 hours. +Verifed sofware can be reusable and employed as a reliable component in another verifcation +project. For instance, part of the verifed CompCert compiler has been reused in VeriCert, which +compiles C programs into hardware designs writen in Verilog. In VeriCert, the compilation from C +programs to three-address code was adapted from CompCert and thus did not need to be re-verifed. +7 +CONCLUSION +Te survey of thirty two sofware systems, from which eleven are depicted in the present article, +shows that there is a wide range of methods, tools, specifcation styles and specifcation languages +used in formally verifed sofware systems. Although there is no generalized or formally accepted +3htps://www.commoncriteriaportal.org +4htps://www.academia.edu/24446830/SOFTWARE CONSIDERATIONS IN AIRBORNE SYSTEMS AND EQUIPMENT +CERTIFICATION +ACM Computing Surveys, Vol. 1, No. 1, Article 1. Publication date: January 2021. + +1:34 +Bruel et al. +method or tool(s) and even less a universal solution that can unify development and verifcation, +some approaches emerge. In particular in safety critical areas such as nuclear or aeronautics, where +deployed sofware systems are successfully formally verifed, formal languages such as Z or B seem +to be the most widely used. +Te verifcation comes up against obstacles to its generalization, such as the need for a high level +of qualifcation of the engineers, risks of inadequacy with the reality of the feld of use and an entry +cost which seems considerable. Verifcation is obviously not without cost and this appears to be its +main drawback. 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Publication date: January 2021. + diff --git a/G9E0T4oBgHgl3EQfRQDg/content/tmp_files/load_file.txt b/G9E0T4oBgHgl3EQfRQDg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..540a6c7b0de76bd5506e29f571f7db291cd09ff4 --- /dev/null +++ b/G9E0T4oBgHgl3EQfRQDg/content/tmp_files/load_file.txt @@ -0,0 +1,2181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf,len=2180 +page_content='1 Lessons from Formally Verified Deployed Sofware Systems LI HUANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Constructor Institute SOPHIE EBERSOLD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Institut de Recherche en Informatique de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' UT2J ALEXANDER KOGTENKOV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Constructor Institute ALEXANDR NAUMCHEV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Constructor Institute BERTRAND MEYER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Constructor Institute YINLING LIU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Institut de Recherche en Informatique de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' UT2J ALIYU ALEGE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Constructor Institute and National University of Singapore ABSTRACT Te technology of formal sofware verifcation has made spectacular advances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' but how much does it actually beneft the development of practical sofware?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Considerable disagreement remains about the practicality of building systems with mechanically-checked proofs of correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Is this prospect confned to a few expensive, life-critical projects, or can the idea be applied to a wide segment of the sofware industry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To help answer this question, the present survey examines a range of projects, in various application areas, that have produced formally verifed systems and deployed them for actual use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It considers the technologies used, the form of verifcation applied, the results obtained, and the lessons that can be drawn for the sofware industry at large and its ability to beneft from formal verifcation techniques and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Reference format: Li Huang, Sophie Ebersold, Alexander Kogtenkov, Alexandr Naumchev, Bertrand Meyer, Yinling Liu, and Aliyu Alege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verifed Deployed Sofware Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, 1, Article 1 (January 2021), 37 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1145/3448975 1 INTRODUCTION Te ever more central role that sofware plays in all processes of the modern world brings to the forefront the critical question of program correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' How do we know that sofware systems perform as expected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Formal verifcation is the task of proving that a program fulflls its specifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It is a long-established research area of sofware engineering, but disagreement persists on its relevance to mainstream system development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Many practitioners have not even heard of formal methods, and those who have ofen dismiss them as too hard to apply to mainstream sofware projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Against this view, proponents of formal verifcation argue that the technology has now reached a high level of maturity and applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Which of these two views is correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In other words, how realistic is the prospect of applying formal methods to production projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To help answer this question, it is important to have an objective basis: an assessment of existing atempts to apply formal methods in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te present survey provides such an assessment, by reviewing sofware systems fulflling two properties: they have actually been deployed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' and they were the subject, during their development, of formal verifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proft or commercial advantage and that copies bear this notice and the full citation on the frst page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Abstracting with credit is permited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specifc permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' © 2021 ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 0360-0300/2021/1-ART1 $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='00 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1145/3448975 ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='02206v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='SE] 5 Jan 2023 1:2 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Two or three decades ago, one could legitimately phrase the underlying question simply as: “Can formal verifcation be applied in industry?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In that form, it is no longer open: a number of well-publicized industrial projects have used formal verifcation, even if initially for relatively small programs in mission-critical and particularly life-critical areas such as transportation and defense, where the consequences of incorrectness in programs are so great as to justify any difculties and extra costs that formal verifcation might imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te contemporary version of the question does not ask any more about feasibility (which has been established) but about practical aspects, such as: How large a system can formal verifcation handle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' What special qualifcations or training does it require for the development team?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' What extra costs, if any, does it imply?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' While it is beyond the scope of this survey to provide defnitive answers to these and other questions on the practicality of formal verifcation, it introduces a factual basis for discussing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It analyzes a number of deployed, formally verifed systems according to a set of criteria listed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te lessons learned from this analysis appear in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Previous surveys have covered part of the scope of this article, not necessarily with a focus on actual deployment of the verifed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tey include the following: Surveys of systems verifed with a specifc approach: Event-B in [2], SPARK in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Surveys of verifed systems in a specifc application area or of a specifc kind: separation kernels [96];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' distributed systems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Surveys of approaches based on the concept of proof assistant [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te following references are more general: [88], from to 2009, presents a questionnaire-based summary of projects that had applied formal methods to some degree, not necessarily at the level of formally verifed code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' [92] and [91] are more recent and present verifed systems that the respective authors found most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tey capture the essence of the reviewed approaches, without going into technical details of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te present article, which does include a fair amount of technical detail, resulted from studying a signifcant set of formally verifed and deployed sofware systems of widely diferent kinds, extending across a variety of implementation languages and verifcation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te projects were selected using a combination of literature review and responses to a questionnaire widely circulated by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tis article helps answer the following questions : In what areas of the industry have formally verifed IT systems been deployed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' What are the properties of the formally verifed systems’ projects in terms of required initial developer expertise, learning efort and efect on the sofware process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' What approaches (programming languages, mathematical basis, verifcation techniques, verifcation tools, verifcation schemes) have been applied to verify deployed systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' What are the potential of and obstacles to generalizing the results to the sofware industry as a whole?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Are there specifc kinds of systems that do not lend themselves to formal verifcation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te rest of the article is structured as follows: Section 2 presents the selection criteria for the analyzed systems, and the analysis criteria for their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Sections 3 and 4 provide a short tutorial on formal verifcation, focusing on the methods actually used in the projects under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Section 5 is the core of the article;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' it describes and assesses the individual projects and their use of formal verifcation, according to the criteria of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:3 Section 6 draws the general lessons of the analysis and examines its limitations as well as its signifcance for the generalization of formal verifcation in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 2 SELECTED SYSTEMS Te sofware systems under review must be both “formally verifed” and “deployed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A system is formally verifed if it has been mathematically proven to possess properties specifed as part of its requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A system is deployed if it is either: Publicly available on the Web (in which case the authors of this article were able to use it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Not publicly available, but with strong evidence that it is used in production by at least several users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Omiting the second category would have excluded commercial, proprietary systems, which account for some of the most signifcant applications of formal verifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te downside is that analysis of these systems has to rely on available documents ofen writen by the authors of the respective systems, or interviews with these authors, rather than direct examination of the sofware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 Selection process Te selection of relevant systems used a combination of literature search and a questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te questionnaire, which remains available [70], was distributed to various relevant communities and Internet channels starting in December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It yielded responses on 20 systems, of which 10 were deemed relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Examination of the documentation on these systems, suggestions from various sources, a list of companies that use formal verifcation methods in sofware engineering [18], and the general literature on formal verifcation led, by transitive closure on the references, to the identifcation of 65 potentially relevant systems from December 2020 to October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Afer application of the selection criteria described below, thirty two systems remained (Ta- ble 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' For reasons of space, the present article focuses on eleven of them, selected for their representativeness of the various kinds of application areas and verifcation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 Criteria for the analysis Te description of the selected systems uses the following criteria: Scope: What system was verifed, in what application domain and for what purpose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Components: What are the verifed components and their roles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Verifed properties: What properties of the system were verifed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Project context: What is the motivation behind the project?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Decisions: What key decisions were made to help the verifcation process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tool stack: What tools were used for developing and verifying the sofware?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Style: What is the underlying approach to specifcation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Sofware characteristics: What results did the verifcation efort produce?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Project characteristics: What amount of resources was spent to verify the sofware?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons: What are the conclusions about the verifcation experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Blank entries express that the corresponding information was not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 3 VERIFICATION: BASIC CONCEPTS Te term “verifcation” covers techniques that ascertain the correctness of programs, subject to the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1:4 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Deployed and Verified Systems Surveyed Category Name Verifed properties Input PL Output PL Proof engine Spec/ Imp KLOC Efort (py) References Compiler CompCert semantics preservation Coq OCaml, C Coq 1 135 6 [53] [54] [43] CakeML semantics preservation CakeML CakeML HOL4 – 100 – [48] [78] [32] Vellvm semantics preservation Coq OCaml, C Coq 1 32 – [95] [93] [94] Vericert semantics preservation Coq OCaml Coq 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 [35] Vermillion functional correctness Coq OCaml Coq – 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='75 [49] Q*Cert functional correctness Coq OCaml Coq – – – [4] Library EifelBase2 functional correctness Eifel Eifel AutoProof, Boogie, Z3 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='37 6 [66] HACL* security, functional cor- rectness, memory safety F* C Z3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 31 < 1 [97] [33] DICE* security, functional cor- rectness, memory safety F* C Z3, Meta-F* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='7 29 [79] Signal* security, functional cor- rectness, memory safety F* WebAssembly Z3, ProVerif 4 [68] Amazon s2n functional correctness C C Coq, SAW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='7 > 3 [16] [73] OS seL4 functional correctness, security C C Isabelle, Z3, Sonolar 100 > 10 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 [29], [47] ProvenCore security C C ProvenTools – – 3 [55] Verve safety Beat C#, TAL Boogie, Z3 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='75 [89] mCertiKOS security, functional cor- rectness Coq ClightX, LAsm Coq 6 3 1 [30] [31] [21] mC2 security, functional cor- rectness Coq ClightX, LAsm Coq 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 2 [31] Hyper-V functional correctness, safety C C, Assem- bly Boogie, Z3 5 105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 [50] PikeOS functional correctness, security C C, Assem- bly Boogie, Z3 5 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 [8] ExpressOS security C#, Dafny C# Z3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='028 – – [59] Aeronautics SHOLIS functional correctness, security, timing/memory constraints SPARK SPARK Simplifer, Proof Checker 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='07 27 19 [45] Lockheed C130J functional correctness SPARK SPARK Simplifer, Proof Checker – 350 – [22] NATS iFACTS absence of run-time ex- ceptions, functional cor- rectness, memory safety SPARK SPARK Simplifer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 250 > 50 [14] [15] EuroFighter Typhoon functional correctness Ada Ada Supertac, ProofPower – 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 [77] [65] Transport Roissy Shuttle security B, Ada Ada EDiT B, Bertille 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='16 158 – [6] Dutch Tunnel CS safety and liveness mCRL2 Java mCRL2, Ver- Cors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='15 37 8 [64] Nuclear Power Sizewell B functional correctness B PL/M-86, Assembly MALPAS – 150 250 [82] Network Ironclad Apps functional correctness, security Dafny Dafny Boogie, Z3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 85 3 [27] Qark security Coq OCaml Coq, Ynot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='83 [41] [40] [69] CoCon security Isabelle/HOL Scala Isabelle/HOL BD-Security 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='67 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='25 [67] [42] CoSMed security Isabelle/HOL Scala Isabelle/HOL BD-Security 1 (ker- nel) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='33 (app) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='33 [7] DataBase Ynot functional correctness Coq OCaml Coq 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='03 8 [1] Verifcation Tool Flover soundness with respect to standard semantics Coq HOL4 Coq , HOL4 10 3 2 [9] [26] ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 Definitions Te meanings of “verifcation” has both broad and narrow meanings: In practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' the sofware industry mostly uses “verifcation” to mean testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In the programming research literature, the term is used instead to denote techniques for proving correctness mathematically, the meaning also retained in the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te sofware engineering literature sometimes distinguishes verifcation from validation, using “V & V” to cover their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In this terminology, verifcation is internal (checking consistency, as in “the system is doing things right” ) and validation external (checking the program against its specifcation, as in “checking that the program does the right things”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In keeping with the usual sense of “formal verifcation” in the programming research community, this article uses “verifcation” to cover both parts of “ V & V”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Verifcation is “dynamic” if it needs to execute the program, as in the case of testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It is “static” if it works only from the program text Tis article focuses on static techniques (which, since they do not perform any execution, require neither a compiler nor input data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Program proving is the most ambitious form of static verifcation, meant to ascertain that the program satisfes its specifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' If the specifcation covers all relevant aspects of the program behavior, the proof will demonstrate “full functional correctness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Other forms of static analysis only analyze the program for the presence or absence of specifc faults, such as deadlock or memory overfow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te following characteristics apply to both static and dynamic forms of verifcation: Verifcation is not debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It may fnd faults (“bugs”), but correcting them is not its job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te words “succeed” and “fail” mean something else for verifcation tools than for programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A verifcation tool succeeds if it either fnds faults or ascertains (mathematically in the case of formal techniques) that the program contains no faults of the specifed kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te tool fails if it is unable to decide either way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (In contrast, a program succeeds if it completes its job according to its specifcation, and fails otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Verifcation can succeed on a failing program — by identifying the fault — and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 Inherent theoretical and practical limitations It is well known that a passing test, or any number of them, do not demonstrate that the program is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (A non-passing test does prove that the program is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=') Tis observation is the basis for asserting the superiority of mathematical proofs (and static techniques in general) over tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Limitations remain, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' First, a failed proof (as defned above) does not indicate that the program is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It simply indicates that the proof tool is unable to decide either way – correct or incorrect program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tis frequently arising case is due to both theoretical and practical reasons: On the theoretical side, program correctness for any useful programming language is an undecidable problem in the sense that it is not possible to produce a tool that will always prove or disprove the correctness of any program in fnite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (Tis property does not preclude the production of tools that will yield proofs for some programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=') On the practical side, a failed proof atempt is not the end of the story but just a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (“Losing a batle, not the war”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=') Such failure is in fact the most common experience in the day-to-day practice of verifcation: try to verify the program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' fnd that the tool either reveals a fault or fails to produce a result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' in either case, tweak the program, the specifcation or both to try to correct the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1:6 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Such an iterative scheme is reminiscent of the “run a test, fx a bug, repeat” of traditional non-formal development using tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te diference is that in static approaches the basic iterative step involves analyzing the text of the program (rather than executing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' But in both cases the process is iterative (as opposed to an ideal two-step process of writing the program then running a tool to prove its correctness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As a result, even though modern verifcation tools are described as permiting “automatic” or “mechanical” verifcation, their actual use still involves human efort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' How much efort is an important practical factor to consider when assessing verifcation tools and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tis survey atempts, whenever information is available, to provide assessments of the efort that was involved in the verifcation of the reviewed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 The annotation issue It is only meaningful to verify a program against specifed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As noted above, these properties can specify the entire relevant behavior of the program (“full correctness”) or only some of its generic characteristics, for example that a square root computation will not produce an arithmetic overfow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te frst approach obviously yields more benefts, but it requires an extra annotation efort, since it needs, in addition to the program, a specifcation of the intended properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te programmer, or whoever is performing the verifcation, must equip the program of these properties and ofen, in practice, intermediate “proof obligations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te choice between the verifcation approaches reviewed next is in part a tradeof between the amount of annotation efort and the extent of properties proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4 VERIFICATION APPROACHES Te verifcation of the systems reviewed here relies on a variety of approaches and frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 Axiomatic semantics Axiomatic semantic (also called Hoare logic or Floyd-Hoare-Dijkstra semantics) is one of the most widespread formal frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It uses the most annotations and addresses full correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Axiomatic semantics assumes that the program is equipped with “verifcation conditions”, also called “assertions”, which may include routine preconditions and postconditions, loop invariants, class invariants (in object-oriented programming), as well as conditions included at arbitrary program places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A verifcation condition is a boolean-valued function on program states (or, in the case of postconditions, on two program states, initial and fnal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Proofs of termination of loops and recursive routines additionally require integer “variants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=" An example of a routine annotated with a precondition and a postcondition is, in the notation of the Eifel programming language 1 sqrt (x: REAL, epsilon: REAL): REAL 2 -- Non-negative square root of 'x' with precision 'epsilon' 3 require 4 x ≥ 0 5 do 6 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Algorithm to compute into Result the square root approximation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 7 ensure 8 Result ≥ 0 9 abs (Result ˆ2 − x) ≤ epsilon 10 end Te precondition (require) expresses a property of x in the original state, at the time of a call;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' the postcondition (ensure) expresses a property of the Result, in relation to the value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te combination of the precondition and postcondition of a routine are its specifcation, or “contract”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:7 Verifcation consists of proving that the implementation matches this specifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It relies on a set of rules (axioms and inference rules) associated with the programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' An example inference rule (with {P} A {Q} stating that if P, a verifcation condition, is satisfed prior to the execution of A, then Q will hold aferwards) characterizes the sequencing of instructions as usually represented by the “;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' symbol in programming languages: {𝑃}𝐴{𝑄}{𝑄}𝐵{𝑅} {𝑃}𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 𝐵{𝑅} (1) Te part above the line is a hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' if that hypothesis is satisfed, the inference rule makes it possible to infer the conclusion below the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te rule states that the sequence A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' B, assuming P on start, will produce R on end if there is an intermediate condition Q such that A ensures Q from P and B ensures R from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tis sequencing rule is typical of how the rules of axiomatic semantics enable reasoning formally about programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tey can be automated and fed into a proof tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tey make it possible to specify the efect of a program, typically through preconditions and postconditions, and to use the proof tool to prove that the program actually produces that efect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To achieve this result, it is necessary to provide enough verifcation conditions to guide the proof tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 Abstract interpretation Abstract interpretation is a mathematical framework for performing static analyses of programs, based on mapping concrete domains of execution values onto more abstract domains, so that analyses of important properties (such as arithmetic overfow, pointer nullness or safety constraints), which would be prohibitive on a concrete domain, can be performed on the abstract domain with its results still valid at the concrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Abstractions satisfying such properties are so-called Galois connections, enjoying conservation properties both ways (concrete to abstract and back).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Determining specifc properties of relevant program properties involves writing a set of equations applying on the variables of the program, based on its control fowgraph, data fow graph or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tese equations are mutually recursive, implying that the way to obtain a solution is to compute a fxpoint of the equations through an iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Such fxpoint computation, and prior to it the construction of the graph and its conversion into a set of equations, are usually impractical or even impossible for the program being verifed, if only because the “concrete domains” in which program variables take their values are very large or infnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Abstract interpretation will instead perform the computation on an abstracted version of the program, in an abstract domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To ensure that the fxpoint on the abstract domain is reached afer a fnite set of iterations, it may be necessary to simplify the abstracted computation further through narrowing and widening operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A simple example of abstraction could serve to determine whether a variable x can ever be zero at a certain program point where the instruction involves a division by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Assuming for simplicity that the original values (concrete domain) are integers, the abstract domain will only include fve values, representing zero (Z), positive (P), Negative (N), Botom (representing impossible values) and Top (representing all possible values), with a partial order relation “less defned than” defning a latice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te operations on numbers transpose in the abstract domain: for example Z + Z = Z - Z = Z, P + P = P - N = P, N + N = N - P = N, but P + N = N + P = N - N = P - P = Top (since the last operations may yield a result of any of the categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te static analysis in the abstract domain can determine whether the abstract value of x can ever be Z, much more easily than if we atempt to perform a similar analysis on the concrete program and domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Under the appropriate conditions, results obtained in the abstract domain can be mapped back to the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1:8 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 Model Checking Model checking is the result of a “why not?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' reaction to the accepted wisdom about the impossibility of certain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Exhaustive testing is well known to be impossible in practice, since the number of cases would be infnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' On further analysis, however: Computers are fnite automata;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' integers as represented by computers, for example, do not form an infnite set but one limited to (typically) 232 or 264 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Similarly, the number of times the body of an ordinary “while” loop can be executed is unbounded in principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' but in practice the number of iterations is, in any program execution, not only fnite but bounded (since a loop taking 100 years to execute would be of no interest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tese sizes are extraordinarily large at the human scale, making the number of possible states for any realistic program appear, at frst sight, intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' But modern computers are very powerful, executing billions of operations a second, which may make it possible to achieve the seemingly absurd goal of exhaustive state-space search, perhaps not for the program itself (except in elementary cases) but for a simplifed version known as a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' An elementary example of a model of a program is a “boolean model” which replaces every integer variable by a boolean variable, with False standing for 0 and True for any other integer value, dramatically reducing the number of possible states (since an integer variable now yields two states instead of, for example, 264).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Another example of a model-checking technique for fghting the phenomenon of “state explosion” is loop unfolding, which replaces every loop by a scheme executing the body 1 to N times, ofen for a small N (typically 2 or 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te model-checking algorithm will then verify a specifed property, such as liveness (non- deadlock) or non-starvation for a concurrent program, by constructing the set of all possible states of the model program and trying to fnd a “counter-example”: an execution path that leads to a state violating the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tis step may or may not be the end of the story: If there is a violation of the property (a fault) in the original program, it will (for the appropriate kinds of property) persist in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Ten if the state space exploration does not produce a counter-example, it defnitely proves that the original program is free from the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Finding the fault in a state of the model program is, however, not conclusive: the fault might be in the original, or it might be an artifact of the reduction to a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' For example, the property m ≠ n between integer variables is true if m = 1 and n = 2, but a model-checker using a boolean program will fnd a counter-example since both values map to True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In this case it is necessary to refne the model to fnd out more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='4 B Most approaches to verifcation analyze a program to determine whether it is correct, regardless of how it was produced in the frst place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' “Correctness by construction” denotes a diferent process: build sofware so that it is correct, intertwining the construction and verifcation eforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te B method [36] applies this idea, combined with the notion of refnement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Most systems using B rely on the “Event B” variant, which adds to the basic framework the notion of event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A refnement process starts with a very high-level view of the program, involving variables defning a state, abstract events afecting that state, and invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As an example, in a system controlling access to a road segment undergoing repair work, an invariant could state that all cars in the system are either stopped or traveling one-way (all East-West or all West-East, if these are the two directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te events might be “let a car enter East” , “Let a car exit East” and the same for West, “Car arrives East” and “Car arrives West” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Variables include: numbers 𝑤𝑒 and ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:9 𝑤𝑤 of cars waiting on the East and West sides, current direction 𝑑 of travel (boolean), number 𝑛 of cars currently traveling on the road segment, maximum number 𝑁 on that segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Each event is defned by its efect on the variables (and hence the state);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' for example each “Enter” event increases 𝑛 by one, decreases the respective waiting variable (𝑤𝑒 or 𝑤𝑤) by one, and leaves the other variables unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Invariants in this example include 𝑛 ≥ 0 and 𝑛 ≤ 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Events can have guards, meaning conditions that must be satisfed for an event to occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' for example the guard for “enter east” is that the direction of travel is East to West, 𝑤𝑒 > 0 and 𝑛 < 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te fundamental correctness rule is that every event, when executed with its guard satisfed as well as the invariants, must ensure that the invariant is satisfed again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te B method works by step-by-step refnement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Each step, refning an existing (“abstract”) model into a new (“concrete”) one, may introduce new events, new invariant properties, and more specifc versions of the abstract events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te refnement is correct if the new events and the event refnement preserve both the concrete and abstract invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' For example, we might refne the road construction model by introducing trafc lights on both ends, with events such as going from green to orange, orange to red etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' on either of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' New invariant properties appear (for example, if the East light is green the West light must be red, and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' We may refne the “let a car enter East” by including the change of light to green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' All new versions must satisfy the preceding properties as well as the new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Refnement proceeds until it has reached a level of detail where the result is explicit enough to be directly implemented in a programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' With the possible exception of this last translation step, the result is correct by construction, since every step has been proved correct in the sense of invariant preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' B-specifc proof tools support the method, to prove at each step that new events preserve invariants and that the new invariants imply those of the preceding refnement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 Proof techniques: SMT solvers One way to prove a “universal” property, stating that that all elements of a set 𝐸 satisfy a property 𝑃, is to prove the absence of a counter-example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' in other words, to prove that it is impossible to fnd an element of E that satisfes ¬𝑃, the negation of 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te properties 𝑃 of interest, and their negations, are boolean formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Disproving 𝑃 means fnding a variable assignment that satisfes ¬𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te boolean satisfability problem is NP-complete, potentially requiring unrealistic computation time, but for theories meeting specifc criteria, known as satisfability modulo theories (SMT), efective algorithms are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' SMT solvers apply this discovery and lie at the basis of many modern proof tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='6 Z Te Z specifcation language, a predecessor of B, is a formal specifcation language based on set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Z makes it possible to specify systems in terms of sets and operations on them represented by mathematical functions and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Associated tools support both consistency proofs of the specifcations themselves and proofs that program in specifc languages satisfy the specifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='7 HOL HOL (Higher-Order Logic) is a mathematical-logic framework underlying by two of the frameworks used by systems in this survey, HOL4 [37] and Isabelle/HOL [39, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As refected by the name, it can include logic of several increasing orders: propositional (zero-order), predicate calculus, second-order (with quantifcation over relations), third-order (with quantifcation over sets of sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It also supports typed 𝜆-calculus with object-level polymorphism [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1:10 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='8 Coq Te Coq framework [20] relies on a diferent logical basis: constructive intuitionistic type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A formal proof in Coq, according to the Curry–Howard correspondence, has an associated program with a suitable type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' to extract a program directly from the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='9 Labeled Transition Systems A Labeled Transition System (LTS) [54] is a mathematical relation 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑠𝑡𝑎𝑡𝑒 𝑡𝑟𝑎𝑐𝑒 −→ 𝑛𝑒𝑥𝑡 𝑠𝑡𝑎𝑡𝑒 that describes one step of execution of a program and its efect on the program state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te sequence of transitions from an initial state defnes the observable behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A fnite sequence describes a terminating program execution, where the program terminates either normally or with a run-time error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' An infnite sequence of transitions describes a program execution that runs forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5 THE SYSTEMS: DESCRIPTIONS Te present section describes the selected systems and reviews them through the ten criteria of the columns in table 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te criteria are mostly self-explanatory, but note the following: Verifed Properties: properties being verifed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' some properties only (termination, liveness…), or full functional correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Programming language(s): languages used for the implementation (not the specifcation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Specifcation/implementation (Spec/Imp) ratio : estimate of ratio between lines of specifca- tion/annotation and lines of implementation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Efort (py): estimate of development and verifcation efort, in person-years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 CompCert 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' CompCert is a compiler for the C programming language, intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='for compiling life- and mission-critical sofware that must meet high levels of assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te CompCert project focuses on compilation, excluding preprocessor, assembler, and linker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te later components are unverifed and come from a legacy compilation tool chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te compiler supports almost all of the C language (ISO C99) [54], generating code for the PowerPC, ARM, RISC-V and x86 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 Verified properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te compiler includes multiple passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It formally verifes semantic preservation between the input and output of every pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To that end it provides formal semantics for every source, intermediate and target language, from C to assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' such semantics defnes the set of all possible program behaviors [54], including termination (normal or abnormal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' CompCert guarantees that every compilation step preserves all behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tese formally verifed properties only apply to the correctness of the compiler itself, with no guarantee as to the correctness of the compiled sofware and the absence of harmful events such as null-pointer dereferencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' CompCert’s formal semantics for C and assembly is hand-crafed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' while there is a high degree of confdence that it faithfully describes these languages, no formal guarantee is possible since the languages, like most, are not formally defned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='4 Project context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te goal of CompCert is to avoid incorrect compilation, which would generate incorrect machine code from a correct source program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Many production compilers have bugs due to the complexity of code generation and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Ten even if the source code of a program has been proved correct, the version actually executed may produce the very condition violations that the program’s formal proof was supposed to have rooted out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 Decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te compiler is split into 20 passes using 11 intermediate languages [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te un- verifed parts deal with initial preprocessing, type refnement, and language-specifc simplifcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te formally verifed passes follow the conventional multi-pass compiler structure and include parsing, front-end compiler, back-end compiler and assembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te fnal assembling and linking steps involve standard non-verifed tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te part with most phases is the back-end compiler, with 12 passes implementing various optimizations and covering specifc target architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te staged implementation enables modular reasoning about every pass and postponing ver- ifcation of some passes (like parsing, assembling, and linking) to later development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As long as an intermediate language between two successive passes is the same, the proof of semantics preservation for both passes automatically guarantees semantics preservation for their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te verifed compiler code is extracted automatically from the proofs in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code of the extractor is not mechanically checked, though a proof on paper is available [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te Coq documentation explicitly states that there are cases when the translation from Coq to OCaml (used for code extraction) may be unsound (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=', due to the diference between the type systems), and CompCert developers are responsible for making sure no bad events (such as integer overfow or exceptions) occur when the compiler runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A validation tool called Valex compensates for current absence of formal proofs for assembling and linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' It reads and disassembles the generated executable, then compares it with the output produced by CompCert to ensure that there are no injections or other changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='6 Tool stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te verifed components of the compiler are writen in Coq to guarantee their correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Given the formal specifcation and the associated constructive proof (Coq works within the theory of the calculus of inductive constructions), Coq extracts a certifed program from the proof in OCaml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te extracted code becomes part of CompCert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='7 Style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te semantics of every source, intermediate and target language (from C to assembly) is specifed in small-step operational style as a labeled transition system (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Because the behavior of a program can be nondeterministic (due to multiple possible execution orders or undefned behavior of the source language), the specifcation relies on a refnement of the allowed behaviors, replacing every non-deterministic behavior with a more deterministic one, by proving 15 simulation diagrams for each intermediate translator independently and then composing them to establish semantic preservation for the whole compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te specifcations and proofs are writen in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te soundness theorem [53] states that if the source code 𝑆 satisfes the specifcation 𝑆𝑝𝑒𝑐 (writen 𝑆 |= 𝑆𝑝𝑒𝑐), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' all observable behaviors 𝐵 of 𝑆 satisfy 𝑆𝑝𝑒𝑐 (∀𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 𝑆 ⇓ 𝐵 =⇒ 𝑆𝑝𝑒𝑐(𝐵)), so does the compiled program 𝐶: 𝑆 |= 𝑆𝑝𝑒𝑐 =⇒ 𝐶 |= 𝑆𝑝𝑒𝑐 Tis result comes from two other theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' One states that every intermediate translator guarantees that the target program 𝑃𝑡𝑔𝑡 simulates the source program 𝑃𝑠𝑟𝑐: 𝑃𝑠𝑟𝑐 ≿ 𝑃𝑡𝑔𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Another one [76] shows that in this case, the behavior of the target program (the set of its execution traces) reproduces the one of the source program: Beh(𝑃𝑠𝑟𝑐) ⊇ Beh(𝑃𝑡𝑔𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='8 Sofware characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In 2016, authors reported [54] that the source code has 100 000 lines of Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' CompCert is empirically more reliable than GCC and LLVM: the test generation tool Csmith [90] found 79 bugs in GCC and 202 in LLVM, but none in the verifed parts of CompCert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='9 Project characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te project was started around 2005 [19] and resulted in more than 30 publications (conferences, journals and books), plus 3 PhD and 1 habilitation theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te frst public version of CompCert (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2) was released in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te frst commercial version has been available since 2015 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' As of 2016, the project had taken [54] 6 person-years (code + verifcation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1:12 Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='10 Lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te project demonstrates the feasibility of writing a formally verifed compiler for a popular programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te correctness comes at a price: the generated programs run 10–20% slower (depending on the CPU type and optimization level) compared to GCC 4 with optimizations turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Verifed compilation alone, however, turns out to be insufcient for industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Established tool chains require further extension of the compiler (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=', to produce debug information [54]), as well as refactoring of existing source code (to move compiler- dependent pragmas and hand-coded inline assembly code to the run-time environment [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te pipe-lined nature of compilation enables almost independent development and verifcation of more than 10 compilation passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Unverifed parts of the compiler rely on additional validation tools to guarantee correctness of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 HACL* 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1 Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' HACL* [97] is a verifed portable C library of cryptographic primitives for a new mandatory ciphersuite in TLS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 [71], also intended as the main cryptographic provider for the miTLS [11] verifed implementation and integrated in Mozilla’s NSS cryptographic library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2 Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' All cryptographic algorithms in HACL* are correct by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='3 Verified properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te developers of HACL* have verifed the following properties: Memory safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code never reads or writes memory at invalid locations, such as null or freed pointers, unallocated memory, or out-of-bounds of allocated memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Also, any locally allocated memory is eventually freed (exactly once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Functional correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code for each primitive conforms to its published standard specifcation on all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Mitigations against Side-Channel Atacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code does not reveal any secret inputs to an adversary, even if the adversary can observe low-level runtime behavior such as branching, memory access paterns, cache hits and misses, power consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Cryptographic security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code for each cryptographic construction implemented by the library is indistinguishable (with high probability) from some standard security defnition, under well-understood cryptographic assumptions on its underlying building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='4 Project context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te project’s primary goal was to build a reference implementation in C and prove that it conformed to computations described in the corresponding RFC standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='5 Decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te main implementation vehicles for HACL* are the F* functional language and Low*, an embedding in F* of a safe subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' HACL* code never allocates memory on the heap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' all temporary states are stored on the stack to simplify proofs of memory safety and avoid the need for explicit memory management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te Low* source code is broken into many small functions, in order to improve readability, modularity and code sharing, and to reduce the complexity of each proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Consequently, the default translation of this code to C would result in a set of small C functions, which can be overly verbose and hurts runtime performance with some compilers like CompCert (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To allow beter control over the generated code, the KreMLin compiler is sometimes directed (via program annotations) to inline certain functions and unroll certain loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' A large chunk of the bignum verifed code is shared across Poly1305, Curve25519 and Ed25519, meaning that this code is verifed once but used in three diferent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te sharing has no impact on the quality of the generated code because KreMLin inlines the generic code and specializes it for one particular set of bignum parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te net result is that Poly1305 and Curve25519 contain separate, specialized versions of the original Low* bignum library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' ACM Computing Surveys, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Publication date: January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Lessons from Formally Verified Deployed Sofware Systems 1:13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='6 Tool stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te developers frst write a high-level specifcation (Spec) for the primitive in a higher-order purely functional subset of F* (Pure F*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Tey then write an optimized implementation (Code) in Low*, a low-level subset of F* that can be efciently compiled to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te code is then verifed, using the F* Z3-based typechecker, for conformance to the Spec and to ensure that it respects the logical preconditions and type abstractions required by the F* standard library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' If type checking fails, there potentially may be a bug in the code, or it may be that the type checker requires more annotations to prove the code correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Finally, the Low* code for the primitive is translated via KreMLin to C code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' More precisely, it is translated to Clight, a subset of C that can be compiled with CompCert (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' In practice, however, the resulting Clight code is compiled with GCC due to perfromance considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content='7 Style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1a contains an example of a pure F* executable specifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Te meaning of this specifcation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' To implement prime feld arithmetic for the Poly1305 algorithm on 64-bit platforms, one strategy is to represent each 130-bit feld element as an array of three 64-bit limbs, where each limb uses 42 or 44 bits and so has room to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' When adding two such feld elements, one can simply add the arrays point-wise, and ignore carries, and the fsum function above does exactly that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1b contains a contract of the future implementation (fadd) of the abstraction (fsum) expressed in pure F* (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1 type limbs = b:buffer uint64_s{length b = 3} 2 let fsum (a:limbs) (b:limbs) = 3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (0ul) ← a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (0ul) + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (0ul);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (1ul) ← a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (1ul) + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (1ul);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (2ul) ← a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (2ul) + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' (2ul) (a) The pure F* abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' 1 val fsum: a:limbs → b:limbs → Stack unit 2 (requires (𝜆 h0 → live h0 a ∧ live h0 b 3 ∧ disjoint a b 4 ∧ index h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' [a] 0 + index h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E0T4oBgHgl3EQfRQDg/content/2301.02206v1.pdf'} +page_content=' [b] 0 q +when ˜θ > 0). In the opposite case, that is inside locally con- +tracting bulk flows, the local deceleration parameter becomes +smaller (i.e. ˜q < q for θ < 0). The latter case is clearly the most +intriguing, since it allows for the sign of the deceleration pa- +rameter to change, from positive to negative, when measured +by observers inside locally contracting bulk flows. Although +the sign-change of ˜q is simply an illusion and a local artefact +of the observer’s relative motion, the affected scales can be +large enough to make it look as a recent global event. If so, +an unsuspecting observer may be misled to believe that their +universe has recently entered a phase of accelerated expan- +sion. According to (7), the “transition scale”, where the local +deceleration parameter crosses the ˜q = 0 threshold is (Tsagas +2021) +λT = 1 +3 +√ +˜∣θ∣ +qH λH , +(8) +where q > 0 always (with q = 1/2 in the case of the Einstein-de +Sitter background). +Although theoretically the model outlined above is well +developed, it is not obvious yet how one should relate the +tilted cosmological scenario to the observations. Parametriz- +ing the deceleration function as ˜q = ˜q(z) and then using it +in (7), has led to a good fit with the Pantheon SNIA sam- +ple (Asvesta et al. 2022). Also, an apparent (Doppler-like) +dipole anisotropy is expected to appear in the observed dis- +tribution of the local deceleration parameter (˜q), due to the +bulk-flow motion relative to the CMB frame (Tsagas 2011). +However, which observational frame (heliocentric, geocen- +tric, galactic, or cosmological) should be employed and what +peculiar-velocity corrections should be applied to the data, in +order to observe the aforementioned dipolar anisotropy, are +the subjects of ongoing debate (Colin et al. 2019a,b; Rubin & +Heitlauf 2020; ?). In this paper, our aim is to study the dy- +namical structure of the peculiar velocity field directly from +data reconstruction. We choose the velocity field reconstruc- +tion of the 2M++ survey (Carrick et al. 2015), which has been +previously used in cosmology to correct the peculiar velocities +of the SNIA data in the last Pantheon+ compilation (Scolnic +et al. 2022; Carr et al. 2022). The methods used to character- +ize this peculiar velocity field and the procedures employed +to relate our results with those of the tilted cosmologies are +discussed in the next sections. +3 CLASSICAL FLUID APPROXIMATION +The kinematic analysis outlined in the previous section, is +straightforwardly adapted to the Newtonian framework as +well (e.g. see Ellis (1971, 1990) for further discussion and de- +tails). In so doing, one replaces the projector (hab), which +also acts as the metric tensor of the 3-space, with the Kro- +necker delta (δij). Also, time derivatives and 3-dimensional +covariant gradients are replaced by convective derivatives and +by ordinary partial derivatives respectively. Note that, given +the near spatial flatness of the observed universe, any cur- +vature corrections due to a nonzero connection (Γa +bc) will +be of the second perturbative order. Then, focusing on the +Figure 1. Peculiar velocities of the Pantheon+ SNIA compilation +peculiar-velocity field (v = vi), we have ˜θij = ∇v = ∂jvi and +˜θij = 1 +3 +˜θδij + ˜ςij + ˜ϖij . +(9) +Here, the (local) volume expansion/contraction scalar, the +shear tensor and the vorticity tensor of the bulk peculiar flow +are respectively defined as +˜θ += +∂ivi = δ +ij∂jvi , +(10) +˜ςij += +1 +2 (∂jvi + ∂ivj) − 1 +3 +˜θ δij, +(11) +˜ϖij += +1 +2 (∂jvi − ∂ivj) . +(12) +It ts possible to evaluate the gradient tensor of the local +peculiar-velocity field using this approach. In particular, the +gradient tensor can be reduced to the 3 × 3-matrix of the +partial derivatives of the ˜vi-field as: +˜θij = ∂jvi = +⎛ +⎜⎜⎜⎜⎜⎜ +⎝ +∂vx +∂x +∂vx +∂y +∂vx +∂y +∂vy +∂x +∂vy +∂y +∂vy +∂y +∂vz +∂x +∂vz +∂y +∂vz +∂y +⎞ +⎟⎟⎟⎟⎟⎟ +⎠ +, +(13) +directly relating ˜θij to the Jacobian tensor of the field. +4 THE 2M++ VELOCITY FIELD +RECONSTRUCTION +The +last +Supernovae +IA +compilation, +namely +Pan- +theon+ (Scolnic et al. 2022), was released in 2022 showing +a great improvement in the utility of data at low redshifts +for cosmological uses. Part of this improvement is due to +better corrections of the peculiar velocities of the SNIA +data (Carr et al. 2022) (see Figure 1). This was done by +using a velocity field reconstruction based on the 2M++ +galaxy survey (Carrick et al. 2015) (see Figure 2). The +reconstruction procedure can be summarized as follows. If +MNRAS 000, 1–?? (2023) + +400 +75 +50 +200 +25 +0 +-25 +-200 +-50 +75 +-400 +0 +50 +100 +150 +200 +250 +300 +350 +Ideg4 +E. Past´en et al. +Figure 2. Peculiar Velocity field reconstruction from the 2M++ density field in galactic coordinates. Visualizations in 3D, with (left) +and without (right) external dipole component. +δ(r) is the density contrast, then the peculiar velocity field +can be approximated as proportional to the gravitational +acceleration when the fluctuations are small: +v(r) = f(Ωm) +4π +∫ d +3r +′δ(r +′) r′ − r +∣r′ − r∣3 . +(14) +Here, f is the growth rate of cosmic structures defined as +f = Ωγ +m, where γ = 0.5 for ΛCDM cosmology. Also, r = HR +is measured in km/s where R, with R being the comoving +distance in Mpc and H the Hubble parameter. +Since the total density perturbation (δ) cannot be directly +observed, a bias parameter (b) has been introduced to relate +the observed density contrast (δg) with the real one: +δ = +δg +b , +(15) +at the linear level. Therefore, the important parameter in +evaluating the velocity field is the ratio β = f/b, since we can +write: +v(r) = β +4π ∫ d +3r +′δg(r +′) r′ − r +∣r′ − r∣3 , +(16) +to relate directly the peculiar velocity with the observed +galaxy density. Also, as the observations extend only up +to a maximum scale (Rmax), the contribution beyond this +length can be added as a constant external velocity parame- +ter (Vext), so that finally: +v(r) = β +4π ∫ +Rmax +d +3r +′δg(r +′) r′ − r +∣r′ − r∣3 + Vext , +(17) +where β and Vext are determined empirically from the recon- +struction of the density field. +We use the density contrast and the velocity field (see Fig- +ure 3)) given by (Carrick et al. 2015), which can be easily +downloaded from https://cosmicflows.iap.fr/. There, the au- +thors provide two useful data-cubes containing the density +contrast δ and the velocity field v using the best-fit param- +eters β = 0.431 ± 0.021 and Vext = (89 ± 21, −131 ± 23, 17 ± +26)km/s (with ∣Vext∣= 159±23km/s) in galactic Cartesian co- +ordinates. It is also important to note that a different value +of β, namely β = 0.341+0.031 +−0.047, was used to correct the Pan- +theon+ data (Said et al. 2020; Carr et al. 2022), claiming +that it gives a better fit when comparing the SDSS Funda- +mental Plane peculiar velocities to the predicted peculiar ve- +locity field. Overall, we can write v = βvrec, where vrec gives +the directions and relative magnitudes of the velocity field. +Then, it is easy to use both values and compare the results. +In order to apply the same corrections to Pantheon+, the +whole velocity field was approximated by a radially decaying +function along the direction of the bulk flow. The latter is a +200 Mpc sphere, composed by the sum of an external Vext +and a small average internal velocity v200. Interestingly, the +external dipole component does not contribute to the gradi- +ent as it is a constant.2 Therefore: +∇v = ∇(βvrec + Vext) = β∇vrec . +(18) +5 METHODS +5.1 Finite differences +We use the central finite difference method to compute +derivatives in each pixel of the data-cube as: +∇v = ∂jvi ≈ +vi(xj + s) − vi(xj − s) +2s +, +(19) +where xi is the central point of a pixel (I, J, K) of the array. +Also, s is the physical size of a pixel, so we can use directly the +2 Even a radially decaying Vext function, with a fixed direction, +does not affect the average divergence of the velocity field. +MNRAS 000, 1–?? (2023) + +150 +100 +Mpc) +50 +T-) +0 +-100 +-150 +-200 +200 +150 +100 +50 +-200 +-150 +-100 +-50 +0 +-100 +50 +-150 +100 +150 +200 +200150 +100 +Mpc) +50 +T-) +0 +-100 +-150 +-200 +200 +150 +100 +50 +-200 +-150 +-100 +-50 +-100 +50 +-150 +100 +150 +200 +200Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology +5 +Figure 3. The density contrast and the vector velocity field projected over the GZ = 0 and GZ = 50h−1Mpc galactic planes. +cube data to ensure the right conversion of a pixel to the phys- +ical value, which fortunately is the same for each coordinate. +For the case of the velocity field used here, with 2573 pixels +in a datacube of length 400/h (where h/100 km/secMpc is +the dimensionless normalised Hubble parameter), we have: +s = 400Mpc +257h +, +(20) +Neglecting the borders of the sample, leads to a 2553 array +containing each pixel in a 3 × 3 matrix that corresponds to +the gradient tensor of the peculiar velocity field. For a central +finite difference approximation of a function f, one may write: +f +′(x) = f(x + s) − f(x − s) +2s +− s2 +6 f +′′′(ξ) += f +′ +1(x) − ϵ , +(21) +for some ξ ∈ [x − s, x + s]. In the above, f ′ +1(x) is the cen- +tral approximation for the derivative and ϵ ∝ s2f ′′′(ξ) is the +truncation error. +MNRAS 000, 1–?? (2023) + +GZ= 0 (h-1 Mpc) +200 +150 - +10 +100 +GY (h-1 Mpc) +50 +0 +OS- +100 +150 +200 +200 +150 +100 +-50 +100 +150 +200 +GX (h-1 Mpc)GZ= 50 (h-1 Mpc) +200 +150 - +5 +100 +GY (h-1 Mpc) +50 +0 +-50 +-100 +150 +200 +200 +150 +100 +-50 +50 +100 +150 +200 +Gx (h-1 Mpc)GZ= 0 (h-1 Mpc) +200 - +150 +1000 +100 - + 800 +50 +GY (h-1 Mpc) +600 +-50 +400 +-100 - +200 +-150 +-200 - +-200 +-150 +-100 +-50 +50 +100 +150 +200 +GX (h-1 Mpc)GZ= 50 (h-1 Mpc) +200 - +150 +800 +100 - +50 + 600 +GY (h-1 Mpc) + 400 +-50 +-100 +200 +-150 +-200 - +-200 +-150 +-100 +-50 +50 +100 +150 +200 +GX (h-1 Mpc)6 +E. Past´en et al. +Figure 4. Graphic representation of the upper face of a box with side 2s enclosing the pixel [I, J, K]. The finite difference method +approximate the flux across this surface as simply the contribution of v[I, J, K + 1] over it (left). Meanwhile, the integral approximation +technique considers the contributions of the side and of the diagonal pixels as well (right). +5.2 Integral Approximations +A direct approximation of the divergence could be computed +recalling the definition of the operator: +∇ ⋅ v = lim +V →0 +1 +V ∯ +∂V v ⋅ dA . +(22) +In so doing, we choose a box-like volume of size 2s around +the central point of each pixel and compute the flux of the +velocity field through the box. Then, by dividing the flux +over the volume, we can extract an approximate value for +the divergence. Note the difference with the central finite dif- +ference method in equation (19), as this approach ignores the +contribution over diagonal pixels (see Figure 4) . +5.3 Theoretical estimation +Following equation (17) the divergence of the velocity field +and the density contrast are also related by the linear ex- +pression: +∇ ⋅ v(r) = β +4π ∫ +Rmax +d +3r +′δg(r +′)∇ ⋅ +r′ − r +∣r′ − r∣3 = −βδg(r) . +(23) +Note that, as the r coordinate is measured in km/s, to ex- +press the divergence in +km/s +Mpc/h units, we need to multiply this +quantity by a factor of 100h2. +6 RESULTS +We have computed the gradient matrix using the Numpy pack- +age from Python to manipulate matrix and arrays. The fol- +lowing three methods of estimating the volume scalar have +been used: +(i) A decomposition of the full gradient tensor of the ve- +locity field employing finite differences. +(ii) A integration approximation for the divergence using +a box volume around the pixels. +(iii) A theoretical estimation by means of relation (23). +The results obtained via these different methods are plot- +ted in Figure 5. To relate the latter with the tilted cosmology +scenario, we need to estimate an average value for ˜θ and thus +put the predictions of the tilted model to the test. In our +analysis, this corresponds to an average divergence of the +entire fluid, which is then averaged over a spherical volume +V = 4πλ3/3 as: +˜θ = 1 +V ∫ +V (∇ ⋅ v)dV ≈ s3 +V ∑ +i +(∇ ⋅ v)i , +(24) +where the sum is over the pixels that reside inside a sphere +of radius λ. +Assuming that λ = 200/h Mpc for example, which is the +radial scale of the survey, and setting β ≃ 0.43, as provided +by (Carrick et al. 2015), the finite difference method leads to: +˜θ ≈ −0.24 km/s +Mpc/h . +(25) +Surprisingly, we have a locally contracting peculiar veloc- +ity field, as the average divergence is negative. Alternatively, +if we use the value β ≃ 0.34 that was used to correct the Pan- +theon+ redshifts, we have: +˜θ ≈ −0.19 km/s +Mpc/h . +(26) +The corresponding values of the local divergence field +through the integral approximation method are: +˜θ ≈ −0.21 km/s +Mpc/h, +(27) +MNRAS 000, 1–?? (2023) + +v[I-1,J+1,K+1] +v[],J+1,K+1] +v[I+1,J+1,K+1] +v[I-1,J+1,K+1] +v[lJ+1,K+1] +v[I+1,J+1,K+1] +v[1-1,J,K+1] +v[I+1,J,K+1] +v[I-1,J,K+1] +v[l,J+1,K+1] +v[],J,K+1] +v[l,J,K+1] +v[I-1,J-1,K+1] +v[l,J-1,K+1] +v[I+1,J-1,K+1] +v[I-1,J-1,K+1] +v[I,J-1,K+1] +v[I+1,J-1,K+1] +zDivergence of the local large-scale structure velocity field and its implications for Tilted Cosmology +7 +Figure 5. Divergence of the velocity field in +km/s +Mpc/h using the finite difference, integral approximation and theoretical computation, +projected over GZ = 0 and GZ = 50h−1Mpc galactic planes. +MNRAS 000, 1–?? (2023) + +GZ= 50 (h-1 Mpc) +150 +100 +OS- +GY (h-1 Mpc) +50 +100 +50 +150 +-100 +200 +150 +-250 +150 +100 +-50 +50 +100 +150 +Gx (h-1 Mpc)GZ= 0 (h-1 Mpc) +150 +100 +-100 +GY (h-1 Mpc) +50 +0 +200 +-50 +O0E- +100 +150 +400 +150 +-100 +50 +100 +150 +Gx (h-1 Mpc)GZ= 50 (h-1 Mpc) +150 +100 +50 +GY (h-1 Mpc) +50 +0 +100 +-50 +150 +-100 +150 +200 +150 +-100 +50 +50 +100 +150 +Gx (h-1 Mpc)GZ= 0 (h-1 Mpc) +200 +150 - +100 +100 +GY (h-1 Mpc) +50 +200 +0 +0E- +-50 +400 +100 +500 +150 +200 +600 +-200 +150 +-100 +-50 +50 +100 +150 +200 +Gx (h-1 Mpc)GZ= 50 (h-1 Mpc) +200 +150 +100 +50 +GY (h-1 Mpc) +50 +100 +0 +150 +-50 +200 +-100 +250 +150 +00E- +200 +-200 +150 +-100 +-50 +0 +50 +100 +150 +200 +Gx (h-1 Mpc)GZ= 0 (h-1 Mpc) +150 +100 +100 +GY (h-1 Mpc) +50 +-200 +0 +50 +00E- +-100 +400 +150 +500 +150 +100 +-50 +100 +150 +Gx (h-1 Mpc)8 +E. Past´en et al. +Figure 6. Residual curl vector field in +km/s +Mpc/h units from finite difference method, projected over GZ = 0 and GZ = 50h−1Mpc +when β ≃ 0.43 and +˜θ ≈ −0.17 km/s +Mpc/h. +(28) +for β ≃ 0.34. +Finally, employing the theoretical estimation method, set- +ting h ≃ 0.7 and integrating the density contrast over the +same scale gives +˜θ ≈ −0.29 km/s +Mpc/h +(29) +and: +˜θ ≈ −0.23 km/s +Mpc/h. +(30) +respectively. Therefore, the theoretical estimation provides +the higher values for ˜θ, while the integral approximation gives +the lowest. What is most important, however, is that all three +methods are consistent both in the sign and in the magnitude +of ˜θ. +Substituting ˜θ into the right-hand side of equation (7), we +can compute representative estimates of the local decelera- +tion parameter (˜q) measured by the bulk-flow observers on +MNRAS 000, 1–?? (2023) + +GZ= (h-1 Mpc) +200 - +150 +20 +100 +15 +50 - +GY (h-1 Mpc) +0 +::::: +：： +::::::::: +10 +-50 - +. +-100 +5 +-150 - +-200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +GX (h-1 Mpc)GZ= 50 (h-1 Mpc) +200 - +150 +35 +100 - + 50 - +- 25 +GY (h-1 Mpc) +0 +20 +-50 - +15 +-100 +10 +-150 - +-200 +-200 +-150 +-100 +-50 +0 + 50 +100 +150 +200 +GX (h-1 Mpc)Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology +9 +Figure 7. Projections of the shear tensor in +km/s +Mpc/h units estimated with finite difference method over cartesian planes that pass on the +origin. +different scales (λ). The results, which assign negative values +to ˜q on scales up to 200 Mpc through all three estimation +methods, are summarized in Table 1. Note that we have set +h ≃ 0.7 in all cases. Also, although (7) holds in essentially +all tilted FRW models, here we have assumed an Einstein-de +Sitter background (with q = 0.5) for mathematical simplicity. +6.1 Uncertainties +We have identified both controlled and uncontrolled uncer- +tainties in our estimations. In the former group we have the fit +uncertainties for the reconstruction parameters β and Vext. +Of those two, we are mainly interested in β, given that Vext +does not enter the gradient calculation. Then, if we define the +divergence of the relative velocity field vrec as ˜θrec, we have: +˜θ = β˜θrec , +(31) +while the uncertainty in ˜θ due to the β parameter can be +written as: +∆˜θβ = ˜θrec∆β . +(32) +This is the uncertainty recorded in Table 1. +Turning to the uncontrolled uncertainties, we can group +different possible biases and systematic effects coming from +the reconstruction process, as well as errors between approx- +imations and real values. A detailed summary of the first +MNRAS 000, 1–?? (2023) + +GX= 0 (h-1 Mpc) +200 +150 - +70 +100 +- 60 +50 - +(h-1 Mpc) + 50 +0 +::::: + 40 +-50 - +30 +-100 +20 +-150 +10 +-200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +GY (h-1 Mpc)GY= 0 (h-1 Mpc) +200 - +100 +150 - +80 +100 +50 - +GX (h-1 Mpc) + 60 +0 +::::: +：： +-50 - + 40 +-100 +-20 +-150 - +-200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +GZ (h-1 Mpc)GZ= 0 (h-1 Mpc) +200 +150 +70 +100 +- 60 +50 - +GY (h-1 Mpc) + 50 +0 +::::: +： + 40 +-50 - +30 +-100 +20 +-150 +10 +-200 +-200 +-150 +-100 +-50 +0 +50 +100 +150 +200 +GX (h-1 Mpc)10 +E. Past´en et al. +type can be found in (Carrick et al. 2015). With regard to +the approximation errors, we can estimate the precision of +the estimation by comparing to the theoretical result. In this +respect, the finite difference method seems more precise than +the volume integration method, as it is closer to the theoret- +ically predicted values. Moreover, according to relation (14), +the velocity field should be irrotational as the field is propor- +tional to a Newtonian gravity potential in the linear regime. +However, when the anti-symmetric part of the gradient tensor +is computed we got a non-zero value, leading to a residual low +vorticity term that could be related with a deviation of the fi- +nite difference method with respect to theoretical estimation. +(potential velocity) A symmetric trace-less part of the gradi- +ent can also be computed via finite difference method. Resid- +ual Curl and projections of the estimated Shear are plotted +in Figures 6 and 7. +7 DISCUSSION +We have estimated the average volume scalar of the recon- +structed peculiar velocity of the local universe via different +methods. The volume scalar is related to the divergence of the +velocity field. This is so because the velocity divergence mea- +sures the change in the local volume of the associated bulk +flow and therefore its tendency to locally expand or contract. +Then, a positive divergence implies that the fluid tends to +expand locally, whereas a negative one indicates a contract- +ing region. We have plotted the divergence scalar for different +galactic planes in Figure 5. There, one can see that the pecu- +liar velocity divergence is highly negative in regions where the +density contrast is high, while it is positive in regions where +matter content is low. This is to be expected, of course, given +the attractive nature of gravity. At this point, it also helps +to recall the familiar divergence theorem: +∰ +V (∇ ⋅ v)dV = ∯ +∂V v ⋅ dA . +(33) +Integrating the divergence over the region V reveals whether +the latter contracts or expands, as the right-hand side of the +equation represents the fluid fraction that ”enters” or ”goes +out” of the volume surface ∂V over time. +Surprisingly, the values of the local volume scalar (˜θ) as- +sociated with the reconstructed peculiar velocity field, were +found negative over a range of scales and by means of different +estimation methods. This result has direct implications for +the tilted cosmological scenario (Tsagas 2011; Asvesta et al. +2022),. The latter predicts that observers living in contracting +bulk peculiar flows could measure a negative deceleration pa- +rameter locally, even when the universe is decelerating glob- +ally (Tsagas & Kadiltzoglou 2015; Tsagas 2021, 2022). Also, +as predicted, we found that the impact of the observer’s pecu- +liar motion becomes stronger on progressively smaller scales, +namely closer to the observer, while it decays away from them +(see Table 1). The transition length (λT ), that is the max- +imum scale where the local deceleration parameter appears +to cross the ˜q = 0 mark and turn negative, also depends on +the observer’s position inside the bulk flow. Following (8), +for observes residing within 70/h Mpc from the centre of +the bulk flow, we find λT ≳ 360 Mpc, λT ≳ 310 Mpc and +λT ≳ 390 Mpc, when adopting the Finite Difference method, +the Integral Approximation method and the Discrete Density +Integration method respectively. Overall, the closer the ob- +server is to the bulk-flow centre, the more negative the local +value of ˜ϑ and the larger the associated transition length. +As appealing these results may be, it is important to re- +main vigilant. It is possible, for example, that the values of +the average divergence could change, as more refined surveys +and models are developed. It is also still unknown whether +matter residing outside the survey range could impact the +mean divergence of the peculiar velocity field. Recall that in +the reconstruction used here this contribution was approxi- +mated by a constant velocity term. In addition, there have +been recent claims that we live in a large void extending up +to ∼ 300 Mpc. However, a negative expansion scalar is not +compatible with the idea of a large void, where one expects to +find an expanding bulk flow rather than a contracting one. +In this respect, our analysis does not seem to support the +presence of a large underdensity. +Finally, peculiar velocities seem unlikely to change the lo- +cal value of the Hubble parameter appreciably and therefore +to solve the H0 tension. One can immediately realise this by +looking at the linear relation (4a). Indeed, keeping in mind +that ∣˜θ∣/Θ = ∣˜θ∣/3H ≪ 1 on sufficiently large scales, the im- +pact of the observer’s relative motion on the Hubble param- +eter should be minimal.3 Instead, there might be other ex- +planations, such as systematics, the evolution of cosmological +parameters with redshift, etc (e.g. see Krishnan et al. (2020); +Colgain et al. (2022)). +8 CONCLUSIONS +We have computed the average divergence (˜θ) of the peculiar +velocity field reconstructed from the 2M++ survey, which was +used to correct cosmological redshifts in the last SNIA com- +pilation Pantheon+. In so doing, we employed three differ- +ent approximation methods, coming from standard numerical +analysis, the divergence theorem and from a linear theoretical +derivation of the peculiar velocity formulae. In all cases, the +resulting values of the velocity divergence were found neg- +ative over a range of scales, suggesting that we live inside +a contracting bulk flow. According to the tilted cosmologi- +cal scenario, the deceleration parameter measured locally by +observers residing in contracting bulk flows can be negative, +although the surrounding universe is globally decelerating. +Our numerical results support this scenario, thus allowing +for the recent accelerated expansion to be just an illusion +produced by our peculiar motion relative to the CMB rest +frame. Nevertheless, this possibility should be treated with +care, as the computed values are still representative of the +measurements a typical bulk-flow observer will make. There- +fore, better surveys with refined precision and broader range +are needed to improve the values computed here. In any case, +however, our results support the need for a deeper study and +for the proper understanding of the implications the observed +large-scale peculiar motions may have for our interpretation +of the cosmological parameters, +3 Recall that, although ∣˜θ∣/H ≪ 1 always during the linear regime, +this is not necessarily the case for the ratio ∣˜θ′∣/ ˙H. +MNRAS 000, 1–?? (2023) + +Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology +11 +λ( Mpc +h +) +˜θ( km/s +Mpc/h ) +˜q +Finite Difference +70 +−3.36+0.07 +−0.07 (−2.65+0.08 +−0.12) +-6.36 (-4.90) +100 +−2.77+0.06 +−0.06 (−2.19+0.10 +−0.07) +-2.27 (-1.70) +125 +−1.48+0.03 +−0.03 (−1.17+0.06 +−0.04) +-0.45 (-0.25) +150 +−0.65+0.01 +−0.01 (−0.51+0.02 +−0.02) ++0.21 (+0.27) +200 +−0.24+0.005 +−0.005 (−0.19+0.008 +−0.005) ++0.44 (+0.45) +Integral Approximation +70 +−2.45+0.05 +−0.05 (−1.94+0.09 +−0.06) +-4.5 (-3.45) +100 +−1.99+0.04 +−0.04 (−1.57+0.07 +−0.05) +-1.49 (-1.07) +125 +−1.13+0.02 +−0.02 (−0.90+0.04 +−0.03) +-0.22 (-0.07) +150 +−0.47+0.01 +−0.01 (−0.37+0.007 +−0.01 ) ++0.29 (+0.33) +200 +−0.21+0.004 +−0.004 (−0.17+0.008 +−0.005) ++0.45 (+0.46) +Discrete Density Integration +70 +−3.94+0.08 +−0.08 (−3.11+0.15 +−0.10) +-7.54 (-5.86) +100 +−3.17+0.07 +−0.07 (−2.50+0.12 +−0.08) +-2.67 (-2.00) +125 +−1.66+0.03 +−0.03 (−1.31+0.06 +−0.04) +-0.56 (-0.34) +150 +−0.76+0.02 +−0.02 (−0.59+0.03 +−0.02) ++0.16 (+0.23) +200 +−0.29+0.006 +−0.006 (−0.23 +0.01 +−0.007) ++0.42 (+0.44) +Table 1. Representative values for ˜q on different scales (λ), using β as it is in the datacube with the finite difference approximation, +integral approximation and theoretical estimation. In parenthesis are the values of ˜q obtained after using β from Pantheon+. Note that, +for numerical simplicity and demonstration purposes, we have set q = 0.5 in the CMB frame and h ≃ 0.7. Note that according to equation +7, the error propagation for ˜q estimations are negligible +ACKNOWLEDGMENTS +EP +acknowledges +support +from +the +graduate +scholar- +ship ANID-Subdirecci´on de Capital Humano/Doctorado +Nacional/2021-21210824. We also wish to thank Christos +Tsagas for his comments, which helped us understand fur- +ther the tilted cosmological scenario. +DATA AVAILABILITY +The data underlying this article, including the programs and +the results of gradient estimations, will be shared on reason- +able request to the corresponding author. +REFERENCES +Asvesta K., Kazantzidis L., Perivolaropoulos L., Tsagas C. G., +2022, Mon. Not. R. Astron. Soc., 513, 2394 +Carr A., Davis T. M., Scolnic D., Said K., Brout D., Peterson E. R., +Kessler R., 2022, Publications of the Astronomical Society of +Australia, 39 +Carrick J., Turnbull S. J., Lavaux G., Hudson M. J., 2015, Monthly +Notices of the Royal Astronomical Society, 450, 317 +Celerier M.-N., 2006, arXiv +Colgain E., Sheikh-Jabbari M. M., Solomon R., Dainotti M. 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(2023) + diff --git a/H9FIT4oBgHgl3EQfYCsV/content/tmp_files/load_file.txt b/H9FIT4oBgHgl3EQfYCsV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0463a600fd4d8db09c77877a42dbda7e86849ca --- /dev/null +++ b/H9FIT4oBgHgl3EQfYCsV/content/tmp_files/load_file.txt @@ -0,0 +1,933 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf,len=932 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) Preprint 27 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='0 Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology Erick Past´en1⋆ Sebasti´an G´alvez2† V´ıctor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' C´ardenas1‡ 1Instituto de F´ısica y Astronom´ıa, Universidad de Valpara´ıso, Gran Breta˜na 1111, Valpara´ıso, Chile 2Centro cient´ıfico tecnol´ogico de Valpara´ıso, Universidad Federico Santa Mar´ıa, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Espa˜na 1680, Valpara´ıso Chile Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We characterize the peculiar velocity field of the local large-scale structure reconstructed from the 2M + + survey, by treating it as a fluid, extracting the gradient and the divergence via different approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This reconstructed field is important for cosmology, since it was used to correct the peculiar redshifts of the last SNIA compilation Pantheon+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We conclude that the local velocity field can be described on average as a slightly contracting fluid, with intriguing implications for the “Tilted Cosmology” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We compute representative values of the apparent deceleration parameter (˜q) measured by observers inside the contracting region, in order to compare our results with the theoretical predictions of the tilted-universe scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' As predicted, the computed values are found to be negative on a range of averaged scales, allowing for a possible explanation of dark energy as an effect induced by our peculiar motion relative to the universal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Key words: dark energy – large-scale structure of Universe 1 INTRODUCTION Dark Energy (DE) is the usual explanation for the apparent universal acceleration implied by the SNIA data (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Perlmutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' However, the suggestion for the existence of dark energy is ultimately based on the cos- mological principle, that is on the assumption of a globally homogeneous and isotropic Friedmann universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The require- ment of an extra parameter ΩΛ is then necessary to explain the dimming of the supernovae magnitudes at large redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Nevertheless, new interesting ideas have emerged in recent years, putting in doubt the cosmological principle, the Fried- mann models and the existence of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' On sufficiently large scales the universe appears homo- geneous and isotropic, according to the Cosmic Microwave Background (CMB) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' On small scales, however, our cosmos is far from that, due to complex structures that produce overdensities/underdensities (Keenan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2013), fractal-like structures (Labini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Labini 2011) and bulk peculiar motions that are not at rest with respect to the Hubble flow (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Feindt 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Magoulas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' There have been many works claiming that some of these effects can mimic an apparent acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Possibly the combination of some (perhaps of all) of these contribu- tions may have an effect stronger than we have previously ⋆ E-mail: erick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='contreras@postgrado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='cl † E-mail: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='galvez@usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='cl ‡ E-mail: victor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='cardenas@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='cl thought (Celerier 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Enqvist 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Tsagas 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Cosmai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Asvesta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' One of the proposed scenarios is the “tilted cosmological model” (Tsagas 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The latter offers a natural environ- ment for the theoretical study of the observed large-scale peculiar motions, by allowing for two groups of relatively moving observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' One group is aligned with the reference frame of the cosmos, which is identified with the coordinate system of the CMB, where the associated dipole vanishes by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The second group are the real observers, living in typical galaxies like our Milky Way and moving relative to the CMB frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' see (Tsagas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Adopting a tilted almost-Friedmann universe and us- ing linear relativistic cosmological perturbation theory, it was shown that relative-motion effects can lead to an apparent change in the sign of the deceleration parameter inside lo- cally contracting bulk flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Although the effect is a local artefact of the observers’ peculiar motion, the affected scales can be large enough to have cosmological relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, observers inside (slightly) contracting bulk peculiar flows can be misled to believe that their universe recently entered a phase of accelerated expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Put another way, the unsus- pecting observers may misinterpret the local contraction of the bulk flow they live in, as global acceleration of the sur- rounding universe (see (Tsagas & Kadiltzoglou 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Tsagas 2021, 2022) for further discussion and details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Our aim is to investigate this possibility by comparing theory to observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We use the velocity field reconstruction from the 2M++ galaxy survey (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This reconstruction pro- © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='11246v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='CO] 26 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Past´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' vides a pair of data-cubes containing the density contrast and the velocity vectors in galactic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' One can use these data to apply basic calculus and also to perform corrections due to peculiar velocities in cosmological data, as it was done in the last Pantheon+ SNIA compilation (Scolnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In the present paper we estimate the av- erage volume scalar of this local velocity-field reconstruction by different methods and on different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In all cases, the local bulk flow is found to contract on average, leading to neg- ative values for the local deceleration parameter on a range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' These results seem to support the tilted cosmolog- ical scenario as an alternative natural explanation of the DE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In section 2 we provide a brief but concise description of tilted cosmological scenario, referring the reader to the re- lated literature for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In section 3 we discuss how to relate the parameters obtained from the velocity-field re- construction with the theory and in section 4 we present the data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Finally, we summarize the method and the results in sections 5 and 6 and discuss their implications for cosmol- ogy at the end of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In addition to cosmology, our analysis has potential applications to astrophysics and to the local structure dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2 TILTED COSMOLOGY MODEL Consider a perturbed Friedmann-Robertson-Walker (FRW) universe with two groups of relatively moving observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' As- suming that ua and ˜ua are the 4-velocities of these observers and va is the (non-relativistic) peculiar velocity of the latter group with respect to the former, we have ˜ua = ua + va , (1) to first approximation (with uava = 0 always).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Introducing two sets of observers means that (strictly speaking) there are two temporal directions (along ua and ˜ua) and two associated 3-spaces (orthogonal ua to and ˜ua).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, the corresponding (covariant) differential operators are ˙ = ua∇a and ′ = ˜ua∇a for the time derivatives, with Da = ha b∇b and ˜Da = ˜ha b∇b for the spatial gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, the tensors hab = gab + uaub and ˜hab = gab + ˜ua˜ub project orthogonal to ua and ˜ua respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The kinematic information of the observers’ motion is de- coded by decomposing the gradient of their 4-velocity field as follows ∇bua = 1 3 Θhab + σab + ωab − Aaub .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2) In the above, Θ is the volume expansion/contraction scalar (when positive/negative respectively), σab is the shear, ωab is the vorticity and Aa is the 4-acceleration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' see Tsagas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In an exactly analogous way, the ˜ua-field splits as ∇b˜ua = (˜Θ/3)˜hab + ˜σab + ˜ωab − ˜Aa˜ub, with the tildas denoting variables evaluated in the tilted frame of the bulk flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Relative to the same coordinate system, the peculiar-velocity field splits as ˜Db˜va = 1 3 ˜θ˜hab + ˜ςab + ˜ϖab , (3) where ˜θ, ˜ςab and ˜ϖab are the volume scalar, the shear and the vorticity of the bulk peculiar motion (Tsagas & Kadiltzoglou 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Of the last three variables, the most important for our purposes is the peculiar volume scalar (˜θ), which takes pos- itive/negative values in locally expanding/contracting bulk flows respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The three kinematic sets defined above are related by lengthy nonlinear expressions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' see Maartens (1998) for the full list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Assuming non-relativistic peculiar motions on an FRW background, we obtain the linear relations ˜Θ = Θ + ˜θ and ˜Θ ′ = ˙Θ + ˜θ ′ , (4) between the volume scalars and between their time deriva- tives evaluated in the two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' At this point, we note that Θ and ˜Θ monitor the expansion rate of the universe, namely the Hubble parameters, as measured in their corre- sponding frames (that is Θ = 3H and ˜Θ = 3 ˜H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, equa- tions (4a) and (4b) imply that the expansion and the accel- eration/deceleration rates measured in the tilted coordinate system differ from those measured in its CMB counterpart solely due to relative-motion effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In particular, recalling that q = −1 − 3 ˙Θ Θ2 and ˜q = −1 − 3˜Θ′ ˜Θ2 , (5) define the deceleration parameters in the CMB and the bulk- flow frames respectively, the following useful relation between ˜q and q can be obtained (Tsagas & Kadiltzoglou 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Tsagas 2021): ˜q = q + ˜θ′ 2 ˙H , (6) to first approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Recall that ˜Θ = Θ = 3H in the Fried- mann background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also note that, whereas ˜θ/H ≪ 1 at the linear level, the ratio ˜θ′/ ˙H of their time derivatives is not always small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Finally, using relativistic linear cosmological perturbation theory, we arrive at: ˜q = q + 1 9 (λH λ ) 2 ˜θ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (7) with λH = 1/H and λ representing the Hubble horizon and the scale of the bulk flow in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note that we have fo- cused on bulk peculiar flows with sizes considerably smaller than the Hubble length (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' λ ≪ λH – see (Tsagas & Kadilt- zoglou 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Tsagas 2021) for the full details of the deriva- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='1 Following (7), the deceleration parameter measured locally by the bulk flow observers (˜q) differs from that of the global universe, which by definition coincides with the deceleration parameter measured in the idealised CMB frame (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The difference is entirely due to the peculiar motion of the tilted observer, since ˜q = q when ˜θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, the “correction” term in (7) is scale-dependent and it gets stronger on progressively smaller scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' for λ ≪ λH), despite the fact that ˜θ/H ≪ 1 throughout the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Moreover, in accord with (7), the overall impact of relative motion on ˜q is also determined by the sign of the peculiar volume scalar (˜θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The latter is positive in locally expanding bulk flows, which means that 1 Expression (7) has been obtained on an Einstein-de Sitter back- ground, primarily for reasons of mathematical simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' It is fairly straightforward to show that the linear result (7) holds on essen- tially all FRW backgrounds, irrespective of their equation of state and spatial curvature (Tsagas 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology 3 the deceleration parameter measured by observers residing in them will be larger than that of the actual universe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' ˜q > q when ˜θ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In the opposite case, that is inside locally con- tracting bulk flows, the local deceleration parameter becomes smaller (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' ˜q < q for θ < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The latter case is clearly the most intriguing, since it allows for the sign of the deceleration pa- rameter to change, from positive to negative, when measured by observers inside locally contracting bulk flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Although the sign-change of ˜q is simply an illusion and a local artefact of the observer’s relative motion, the affected scales can be large enough to make it look as a recent global event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' If so, an unsuspecting observer may be misled to believe that their universe has recently entered a phase of accelerated expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' According to (7), the “transition scale”, where the local deceleration parameter crosses the ˜q = 0 threshold is (Tsagas 2021) λT = 1 3 √ ˜∣θ∣ qH λH , (8) where q > 0 always (with q = 1/2 in the case of the Einstein-de Sitter background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Although theoretically the model outlined above is well developed, it is not obvious yet how one should relate the tilted cosmological scenario to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Parametriz- ing the deceleration function as ˜q = ˜q(z) and then using it in (7), has led to a good fit with the Pantheon SNIA sam- ple (Asvesta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, an apparent (Doppler-like) dipole anisotropy is expected to appear in the observed dis- tribution of the local deceleration parameter (˜q), due to the bulk-flow motion relative to the CMB frame (Tsagas 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' However, which observational frame (heliocentric, geocen- tric, galactic, or cosmological) should be employed and what peculiar-velocity corrections should be applied to the data, in order to observe the aforementioned dipolar anisotropy, are the subjects of ongoing debate (Colin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Rubin & Heitlauf 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In this paper, our aim is to study the dy- namical structure of the peculiar velocity field directly from data reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We choose the velocity field reconstruc- tion of the 2M++ survey (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015), which has been previously used in cosmology to correct the peculiar velocities of the SNIA data in the last Pantheon+ compilation (Scolnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The methods used to character- ize this peculiar velocity field and the procedures employed to relate our results with those of the tilted cosmologies are discussed in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 3 CLASSICAL FLUID APPROXIMATION The kinematic analysis outlined in the previous section, is straightforwardly adapted to the Newtonian framework as well (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' see Ellis (1971, 1990) for further discussion and de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In so doing, one replaces the projector (hab), which also acts as the metric tensor of the 3-space, with the Kro- necker delta (δij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, time derivatives and 3-dimensional covariant gradients are replaced by convective derivatives and by ordinary partial derivatives respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note that, given the near spatial flatness of the observed universe, any cur- vature corrections due to a nonzero connection (Γa bc) will be of the second perturbative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, focusing on the Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Peculiar velocities of the Pantheon+ SNIA compilation peculiar-velocity field (v = vi), we have ˜θij = ∇v = ∂jvi and ˜θij = 1 3 ˜θδij + ˜ςij + ˜ϖij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (9) Here, the (local) volume expansion/contraction scalar, the shear tensor and the vorticity tensor of the bulk peculiar flow are respectively defined as ˜θ = ∂ivi = δ ij∂jvi , (10) ˜ςij = 1 2 (∂jvi + ∂ivj) − 1 3 ˜θ δij, (11) ˜ϖij = 1 2 (∂jvi − ∂ivj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (12) It ts possible to evaluate the gradient tensor of the local peculiar-velocity field using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In particular, the gradient tensor can be reduced to the 3 × 3-matrix of the partial derivatives of the ˜vi-field as: ˜θij = ∂jvi = ⎛ ⎜⎜⎜⎜⎜⎜ ⎝ ∂vx ∂x ∂vx ∂y ∂vx ∂y ∂vy ∂x ∂vy ∂y ∂vy ∂y ∂vz ∂x ∂vz ∂y ∂vz ∂y ⎞ ⎟⎟⎟⎟⎟⎟ ⎠ , (13) directly relating ˜θij to the Jacobian tensor of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 4 THE 2M++ VELOCITY FIELD RECONSTRUCTION The last Supernovae IA compilation, namely Pan- theon+ (Scolnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022), was released in 2022 showing a great improvement in the utility of data at low redshifts for cosmological uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Part of this improvement is due to better corrections of the peculiar velocities of the SNIA data (Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022) (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This was done by using a velocity field reconstruction based on the 2M++ galaxy survey (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015) (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The reconstruction procedure can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' If MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) 400 75 50 200 25 0 25 200 50 75 400 0 50 100 150 200 250 300 350 Ideg4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Past´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Peculiar Velocity field reconstruction from the 2M++ density field in galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Visualizations in 3D, with (left) and without (right) external dipole component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' δ(r) is the density contrast, then the peculiar velocity field can be approximated as proportional to the gravitational acceleration when the fluctuations are small: v(r) = f(Ωm) 4π ∫ d 3r ′δ(r ′) r′ − r ∣r′ − r∣3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (14) Here, f is the growth rate of cosmic structures defined as f = Ωγ m, where γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='5 for ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, r = HR is measured in km/s where R, with R being the comoving distance in Mpc and H the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Since the total density perturbation (δ) cannot be directly observed, a bias parameter (b) has been introduced to relate the observed density contrast (δg) with the real one: δ = δg b , (15) at the linear level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Therefore, the important parameter in evaluating the velocity field is the ratio β = f/b, since we can write: v(r) = β 4π ∫ d 3r ′δg(r ′) r′ − r ∣r′ − r∣3 , (16) to relate directly the peculiar velocity with the observed galaxy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, as the observations extend only up to a maximum scale (Rmax), the contribution beyond this length can be added as a constant external velocity parame- ter (Vext), so that finally: v(r) = β 4π ∫ Rmax d 3r ′δg(r ′) r′ − r ∣r′ − r∣3 + Vext , (17) where β and Vext are determined empirically from the recon- struction of the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We use the density contrast and the velocity field (see Fig- ure 3)) given by (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015), which can be easily downloaded from https://cosmicflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='iap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='fr/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' There, the au- thors provide two useful data-cubes containing the density contrast δ and the velocity field v using the best-fit param- eters β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='431 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='021 and Vext = (89 ± 21, −131 ± 23, 17 ± 26)km/s (with ∣Vext∣= 159±23km/s) in galactic Cartesian co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' It is also important to note that a different value of β, namely β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='341+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='031 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='047, was used to correct the Pan- theon+ data (Said et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022), claiming that it gives a better fit when comparing the SDSS Funda- mental Plane peculiar velocities to the predicted peculiar ve- locity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Overall, we can write v = βvrec, where vrec gives the directions and relative magnitudes of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, it is easy to use both values and compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In order to apply the same corrections to Pantheon+, the whole velocity field was approximated by a radially decaying function along the direction of the bulk flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The latter is a 200 Mpc sphere, composed by the sum of an external Vext and a small average internal velocity v200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Interestingly, the external dipole component does not contribute to the gradi- ent as it is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='2 Therefore: ∇v = ∇(βvrec + Vext) = β∇vrec .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (18) 5 METHODS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='1 Finite differences We use the central finite difference method to compute derivatives in each pixel of the data-cube as: ∇v = ∂jvi ≈ vi(xj + s) − vi(xj − s) 2s , (19) where xi is the central point of a pixel (I, J, K) of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, s is the physical size of a pixel, so we can use directly the 2 Even a radially decaying Vext function, with a fixed direction, does not affect the average divergence of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) 150 100 Mpc) 50 T-) 0 100 150 200 200 150 100 50 200 150 100 50 0 100 50 150 100 150 200 200150 100 Mpc) 50 T-) 0 100 150 200 200 150 100 50 200 150 100 50 100 50 150 100 150 200 200Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The density contrast and the vector velocity field projected over the GZ = 0 and GZ = 50h−1Mpc galactic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' cube data to ensure the right conversion of a pixel to the phys- ical value, which fortunately is the same for each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' For the case of the velocity field used here, with 2573 pixels in a datacube of length 400/h (where h/100 km/secMpc is the dimensionless normalised Hubble parameter), we have: s = 400Mpc 257h , (20) Neglecting the borders of the sample, leads to a 2553 array containing each pixel in a 3 × 3 matrix that corresponds to the gradient tensor of the peculiar velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' For a central finite difference approximation of a function f, one may write: f ′(x) = f(x + s) − f(x − s) 2s − s2 6 f ′′′(ξ) = f ′ 1(x) − ϵ , (21) for some ξ ∈ [x − s, x + s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In the above, f ′ 1(x) is the cen- tral approximation for the derivative and ϵ ∝ s2f ′′′(ξ) is the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GZ= 0 (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GY (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='OS- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GX (h-1 Mpc)GZ= 50 (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GY (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GX (h-1 Mpc)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Past´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Graphic representation of the upper face of a box with side 2s enclosing the pixel [I, J, K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The finite difference method approximate the flux across this surface as simply the contribution of v[I, J, K + 1] over it (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Meanwhile, the integral approximation technique considers the contributions of the side and of the diagonal pixels as well (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='2 Integral Approximations A direct approximation of the divergence could be computed recalling the definition of the operator: ∇ ⋅ v = lim V →0 1 V ∯ ∂V v ⋅ dA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (22) In so doing, we choose a box-like volume of size 2s around the central point of each pixel and compute the flux of the velocity field through the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, by dividing the flux over the volume, we can extract an approximate value for the divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note the difference with the central finite dif- ference method in equation (19), as this approach ignores the contribution over diagonal pixels (see Figure 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='3 Theoretical estimation Following equation (17) the divergence of the velocity field and the density contrast are also related by the linear ex- pression: ∇ ⋅ v(r) = β 4π ∫ Rmax d 3r ′δg(r ′)∇ ⋅ r′ − r ∣r′ − r∣3 = −βδg(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (23) Note that, as the r coordinate is measured in km/s, to ex- press the divergence in km/s Mpc/h units, we need to multiply this quantity by a factor of 100h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 6 RESULTS We have computed the gradient matrix using the Numpy pack- age from Python to manipulate matrix and arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The fol- lowing three methods of estimating the volume scalar have been used: (i) A decomposition of the full gradient tensor of the ve- locity field employing finite differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (ii) A integration approximation for the divergence using a box volume around the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (iii) A theoretical estimation by means of relation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The results obtained via these different methods are plot- ted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' To relate the latter with the tilted cosmology scenario, we need to estimate an average value for ˜θ and thus put the predictions of the tilted model to the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In our analysis, this corresponds to an average divergence of the entire fluid, which is then averaged over a spherical volume V = 4πλ3/3 as: ˜θ = 1 V ∫ V (∇ ⋅ v)dV ≈ s3 V ∑ i (∇ ⋅ v)i , (24) where the sum is over the pixels that reside inside a sphere of radius λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Assuming that λ = 200/h Mpc for example, which is the radial scale of the survey, and setting β ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='43, as provided by (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015), the finite difference method leads to: ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='24 km/s Mpc/h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (25) Surprisingly, we have a locally contracting peculiar veloc- ity field, as the average divergence is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Alternatively, if we use the value β ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='34 that was used to correct the Pan- theon+ redshifts, we have: ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='19 km/s Mpc/h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (26) The corresponding values of the local divergence field through the integral approximation method are: ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='21 km/s Mpc/h, (27) MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) v[I-1,J+1,K+1] v[],J+1,K+1] v[I+1,J+1,K+1] v[I-1,J+1,K+1] v[lJ+1,K+1] v[I+1,J+1,K+1] v[1-1,J,K+1] v[I+1,J,K+1] v[I-1,J,K+1] v[l,J+1,K+1] v[],J,K+1] v[l,J,K+1] v[I-1,J-1,K+1] v[l,J-1,K+1] v[I+1,J-1,K+1] v[I-1,J-1,K+1] v[I,J-1,K+1] v[I+1,J-1,K+1] zDivergence of the local large-scale structure velocity field and its implications for Tilted Cosmology 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Divergence of the velocity field in km/s Mpc/h using the finite difference, integral approximation and theoretical computation, projected over GZ = 0 and GZ = 50h−1Mpc galactic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GZ= 50 (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='OS- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GY (h-1 Mpc) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='Gx (h-1 Mpc)8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Past´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Residual curl vector field in km/s Mpc/h units from finite difference method, projected over GZ = 0 and GZ = 50h−1Mpc when β ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='43 and ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='17 km/s Mpc/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (28) for β ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Finally, employing the theoretical estimation method, set- ting h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='7 and integrating the density contrast over the same scale gives ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='29 km/s Mpc/h (29) and: ˜θ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='23 km/s Mpc/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (30) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Therefore, the theoretical estimation provides the higher values for ˜θ, while the integral approximation gives the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' What is most important, however, is that all three methods are consistent both in the sign and in the magnitude of ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Substituting ˜θ into the right-hand side of equation (7), we can compute representative estimates of the local decelera- tion parameter (˜q) measured by the bulk-flow observers on MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) GZ= (h-1 Mpc) 200 - 150 20 100 15 50 - GY (h-1 Mpc) 0 ::::: ：： ::::::::: 10 50 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 100 5 150 - 200 200 150 100 50 0 50 100 150 200 GX (h-1 Mpc)GZ= 50 (h-1 Mpc) 200 - 150 35 100 - 50 - 25 GY (h-1 Mpc) 0 20 50 - 15 100 10 150 - 200 200 150 100 50 0 50 100 150 200 GX (h-1 Mpc)Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology 9 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Projections of the shear tensor in km/s Mpc/h units estimated with finite difference method over cartesian planes that pass on the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' different scales (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The results, which assign negative values to ˜q on scales up to 200 Mpc through all three estimation methods, are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note that we have set h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='7 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, although (7) holds in essentially all tilted FRW models, here we have assumed an Einstein-de Sitter background (with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='5) for mathematical simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='1 Uncertainties We have identified both controlled and uncontrolled uncer- tainties in our estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In the former group we have the fit uncertainties for the reconstruction parameters β and Vext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Of those two, we are mainly interested in β, given that Vext does not enter the gradient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, if we define the divergence of the relative velocity field vrec as ˜θrec, we have: ˜θ = β˜θrec , (31) while the uncertainty in ˜θ due to the β parameter can be written as: ∆˜θβ = ˜θrec∆β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (32) This is the uncertainty recorded in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Turning to the uncontrolled uncertainties, we can group different possible biases and systematic effects coming from the reconstruction process, as well as errors between approx- imations and real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' A detailed summary of the first MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GX= 0 (h-1 Mpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='150 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='GX (h-1 Mpc)10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Past´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' type can be found in (Carrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' With regard to the approximation errors, we can estimate the precision of the estimation by comparing to the theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In this respect, the finite difference method seems more precise than the volume integration method, as it is closer to the theoret- ically predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Moreover, according to relation (14), the velocity field should be irrotational as the field is propor- tional to a Newtonian gravity potential in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' However, when the anti-symmetric part of the gradient tensor is computed we got a non-zero value, leading to a residual low vorticity term that could be related with a deviation of the fi- nite difference method with respect to theoretical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (potential velocity) A symmetric trace-less part of the gradi- ent can also be computed via finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Resid- ual Curl and projections of the estimated Shear are plotted in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 7 DISCUSSION We have estimated the average volume scalar of the recon- structed peculiar velocity of the local universe via different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The volume scalar is related to the divergence of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This is so because the velocity divergence mea- sures the change in the local volume of the associated bulk flow and therefore its tendency to locally expand or contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Then, a positive divergence implies that the fluid tends to expand locally, whereas a negative one indicates a contract- ing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We have plotted the divergence scalar for different galactic planes in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' There, one can see that the pecu- liar velocity divergence is highly negative in regions where the density contrast is high, while it is positive in regions where matter content is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This is to be expected, of course, given the attractive nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' At this point, it also helps to recall the familiar divergence theorem: ∰ V (∇ ⋅ v)dV = ∯ ∂V v ⋅ dA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (33) Integrating the divergence over the region V reveals whether the latter contracts or expands, as the right-hand side of the equation represents the fluid fraction that ”enters” or ”goes out” of the volume surface ∂V over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Surprisingly, the values of the local volume scalar (˜θ) as- sociated with the reconstructed peculiar velocity field, were found negative over a range of scales and by means of different estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' This result has direct implications for the tilted cosmological scenario (Tsagas 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Asvesta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 2022),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The latter predicts that observers living in contracting bulk peculiar flows could measure a negative deceleration pa- rameter locally, even when the universe is decelerating glob- ally (Tsagas & Kadiltzoglou 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Tsagas 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Also, as predicted, we found that the impact of the observer’s pecu- liar motion becomes stronger on progressively smaller scales, namely closer to the observer, while it decays away from them (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' The transition length (λT ), that is the max- imum scale where the local deceleration parameter appears to cross the ˜q = 0 mark and turn negative, also depends on the observer’s position inside the bulk flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Following (8), for observes residing within 70/h Mpc from the centre of the bulk flow, we find λT ≳ 360 Mpc, λT ≳ 310 Mpc and λT ≳ 390 Mpc, when adopting the Finite Difference method, the Integral Approximation method and the Discrete Density Integration method respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Overall, the closer the ob- server is to the bulk-flow centre, the more negative the local value of ˜ϑ and the larger the associated transition length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' As appealing these results may be, it is important to re- main vigilant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' It is possible, for example, that the values of the average divergence could change, as more refined surveys and models are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' It is also still unknown whether matter residing outside the survey range could impact the mean divergence of the peculiar velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Recall that in the reconstruction used here this contribution was approxi- mated by a constant velocity term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In addition, there have been recent claims that we live in a large void extending up to ∼ 300 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' However, a negative expansion scalar is not compatible with the idea of a large void, where one expects to find an expanding bulk flow rather than a contracting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In this respect, our analysis does not seem to support the presence of a large underdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Finally, peculiar velocities seem unlikely to change the lo- cal value of the Hubble parameter appreciably and therefore to solve the H0 tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' One can immediately realise this by looking at the linear relation (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Indeed, keeping in mind that ∣˜θ∣/Θ = ∣˜θ∣/3H ≪ 1 on sufficiently large scales, the im- pact of the observer’s relative motion on the Hubble param- eter should be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='3 Instead, there might be other ex- planations, such as systematics, the evolution of cosmological parameters with redshift, etc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' see Krishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Colgain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' 8 CONCLUSIONS We have computed the average divergence (˜θ) of the peculiar velocity field reconstructed from the 2M++ survey, which was used to correct cosmological redshifts in the last SNIA com- pilation Pantheon+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In so doing, we employed three differ- ent approximation methods, coming from standard numerical analysis, the divergence theorem and from a linear theoretical derivation of the peculiar velocity formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In all cases, the resulting values of the velocity divergence were found neg- ative over a range of scales, suggesting that we live inside a contracting bulk flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' According to the tilted cosmologi- cal scenario, the deceleration parameter measured locally by observers residing in contracting bulk flows can be negative, although the surrounding universe is globally decelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Our numerical results support this scenario, thus allowing for the recent accelerated expansion to be just an illusion produced by our peculiar motion relative to the CMB rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Nevertheless, this possibility should be treated with care, as the computed values are still representative of the measurements a typical bulk-flow observer will make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' There- fore, better surveys with refined precision and broader range are needed to improve the values computed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In any case, however, our results support the need for a deeper study and for the proper understanding of the implications the observed large-scale peculiar motions may have for our interpretation of the cosmological parameters, 3 Recall that, although ∣˜θ∣/H ≪ 1 always during the linear regime, this is not necessarily the case for the ratio ∣˜θ′∣/ ˙H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023) Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology 11 λ( Mpc h ) ˜θ( km/s Mpc/h ) ˜q Finite Difference 70 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='005) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='44 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='45) Integral Approximation 70 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='05 (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='06) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='5 (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='45) 100 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='04 (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='49 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07) 125 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='02 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='22 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07) 150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='01 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='01 ) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='29 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='33) 200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='004 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='005) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='45 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='46) Discrete Density Integration 70 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='08 (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='10) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='54 (-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='86) 100 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='07 (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='08) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='67 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='00) 125 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='03 (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='56 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='34) 150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='02 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='02) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='16 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='23) 200 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='006 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='23 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='007) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='42 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='44) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Representative values for ˜q on different scales (λ), using β as it is in the datacube with the finite difference approximation, integral approximation and theoretical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' In parenthesis are the values of ˜q obtained after using β from Pantheon+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note that, for numerical simplicity and demonstration purposes, we have set q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='5 in the CMB frame and h ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' Note that according to equation 7, the error propagation for ˜q estimations are negligible ACKNOWLEDGMENTS EP acknowledges support from the graduate scholar- ship ANID-Subdirecci´on de Capital Humano/Doctorado Nacional/2021-21210824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' We also wish to thank Christos Tsagas for his comments, which helped us understand fur- ther the tilted cosmological scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article, including the programs and the results of gradient estimations, will be shared on reason- able request to the corresponding author.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} +page_content=' (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FIT4oBgHgl3EQfYCsV/content/2301.11246v1.pdf'} diff --git a/HtE3T4oBgHgl3EQfWwou/content/tmp_files/2301.04471v1.pdf.txt b/HtE3T4oBgHgl3EQfWwou/content/tmp_files/2301.04471v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..67d99e9e9113e503cfb2ab8cf66e35d4694b03da --- /dev/null +++ b/HtE3T4oBgHgl3EQfWwou/content/tmp_files/2301.04471v1.pdf.txt @@ -0,0 +1,488 @@ +DATASET OF FLUORESCENCE SPECTRA AND CHEMICAL +PARAMETERS OF OLIVE OILS +AN OPEN SOURCE DATASET +Francesca Venturini ∗ +Institute of Applied Mathematics and Physics +Zurich University of Applied Sciences +Winterthur, Switzerland, vent@zhaw.ch +Artificial Intelligence Research and Development +TOELT LLC, Switzerland +Michela Sperti +PolitoBIOMed Lab +Department of Mechanical and +Aerospace Engineering +Politecnico di Torino, Turin, Italy +Umberto Michelucci +Artificial Intelligence Research and Development +TOELT LLC, Switzerland +umberto.michelucci@toelt.ai +Computer Science Department +Lucerne University of Applied Sciences and Arts +Lucerne, Switzerland +Arnaud Gucciardi +Artificial Intelligence Research and Development +TOELT LLC, Switzerland +arnaud.gucciardi@toelt.ai +Artificial Intelligence Laboratory +University of Ljubljana, Ljubljana, Slovenia +Vanessa M. Martos +Department of Plant Physiology +Faculty of Sciences +Biotechnology Institute +University of Granada, Spain +Marco A. Deriu +PolitoBIOMed Lab +Department of Mechanical and +Aerospace Engineering +Politecnico di Torino, Turin, Italy +January 12, 2023 +ABSTRACT +This dataset encompasses fluorescence spectra and chemical parameters of 24 olive oil samples from +the 2019–2020 harvest provided by the producer Conde de Benalúa, Granada, Spain. The oils are +characterized by different qualities: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and +6 lampante olive oil (LOO) samples. For each sample, the dataset includes fluorescence spectra +obtained with two excitation wavelengths, oil quality, and five chemical parameters necessary for +the quality assessment of olive oil. The fluorescence spectra were obtained by exciting the samples +at 365 nm and 395 nm under identical conditions. The dataset includes the values of the following +chemical parameters for each olive oil sample: acidity, peroxide value, K270, K232, ethyl esters, +and the quality of the samples (EVOO, VOO, or LOO). The dataset offers a unique possibility for +researchers in food technology to develop machine learning models based on fluorescence data for +the quality assessment of olive oil due to the availability of both spectroscopic and chemical data. +The dataset can be used, for example, to predict one or multiple chemical parameters or to classify +samples based on their quality from fluorescence spectra. +Keywords Fluorescence · Olive Oil · Chemical Parameters · Quality control +∗Contact email: vent@zhaw.ch +arXiv:2301.04471v1 [q-bio.QM] 10 Jan 2023 + +Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils +DATASET +1 +Summary +The dataset presented is a compilation of measurements of analytical chemistry and fluorescence spectroscopy. The +dataset includes fluorescence spectra and chemical parameters of 24 Spanish olive oils from the 2019–2020 harvest. The +24 samples were collected at SCA San Sebastián Puente del Ventorro, Benalua de las Villas, Spain. The data were later +measured at the Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Technikumstrasse +9, 8401 Winterthur, Switzerland. The fluorescence spectroscopy data was acquired by a miniature spectrometer with a +1024 element CCD array that acquires the entire spectrum in one single measurement. The dataset includes a total of +960 spectra (24 oil samples × 2 excitation wavelengths x 20 repeated measurements). Each of the 960 spectra is an +array of 1024 values whose elements are the intensity at the different pixel positions. The chemical parameters were +determined by accredited laboratories using the procedures described in the European Commission regulation and its +amendment Commission [2013, 1991]. These regulations control the methods for the quality assessment of olive oils +and provide a decision tree to verify whether an olive oil class is consistent with the declared quality. +The value of the dataset for research purposes is summarized in the points below. +• The data are useful for studying the link between optical properties (fluorescence and absorption spectroscopy), +chemical characteristics (such as oil acidity, peroxide value, and fatty acid content), and olive oil quality (extra +virgin, virgin, and lampante olive oil). +• This dataset is the first available that contains fluorescence spectra and chemical analysis obtained by accredited +laboratories on samples coming from a single producer. +• Many researchers can benefit from the data: computer scientists can use the data to develop machine learning +models that link optical to chemical properties; researchers in food technology that are interested in studying +chemical properties of olive oil samples of different qualities; engineers that want to develop new optical +analysis techniques alternative to the current expensive and time-consuming analytical chemistry methods. +• This dataset can be used to perform explainability analysis to identify spectral characteristics that are related +to different chemical properties (e.g., the acidity of the oil). An example is given in the paper Venturini et al. +[2023]. This will further advance the understanding of the complex chemical composition of olive oil and its +link to its quality and health benefits. +• This dataset can be used to develop instruments based on fluorescence spectroscopy for the rapid and cost- +effective quality assessment of olive oil. +2 +Data Description +The dataset consists of one CSV file that contains the columns described in Table 1. +A background file2 is also provided. The file contains 1024 values that correspond to the intensity measured by the +spectrometer without any light (dark counts). This spectrum can be subtracted from the raw fluorescence spectra to +remove the effect of the dark counts. The same file can be used for the spectra taken at both 365 nm and 395 nm. +The raw fluorescence spectra of selected oils obtained with excitation at 365 nm and 395 nm are shown in Fig. 1. +3 +Materials and methods +3.1 +Olive Oil Samples +The dataset contains the fluorescence spectra and the chemical parameters of 24 oils. The oils are characterized by +different quality categories: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and 6 lampante olive oil (LOO) +samples. All samples were provided by Conde de Benalúa, Granada, southern Spain, and were prepared from the +2019–2020 harvest. The properties and values of the chemical parameters of the oil samples are listed in Table 2. +For data acquisition, the samples were placed in commercial 4 ml clear glass vials, taking care that no headspace was +present to reduce oxidation. All oils were stored in the dark and at 20 °C during the entire time of the measurements. +3.2 +Fluorescence Data Acquisition +The fluorescence spectroscopy data were acquired using the portable sensor described in Venturini et al. [2021]. Since +already published, only the most relevant characteristics are reported here. The reader is referred to this publication +2Fluorescence_olive_oil_dataset_background.csv +2 + +Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils +DATASET +Feature +Datatype +Description +Sample +String +Oil sample name: the values are ’D03’,’D04’,’D05’, ’D06’, ’D07’ ,’D08’, ’D09’, +’D10’, ’D 19’, ’D20’, ’D35’, ’D38’, ’D45’, ’D46’, ’D47’, ’D49’, ’D51’, ’D52’, +’D53’, ’D64’, ’D77’, ’D81’, ’D92’,’D73’ +Repetition +Integer +Repetition number. There are 20 repetition for each oil and led: the iteration +number goes from 0 to 19) +Led +Integer +Excitation LED identifier: 1 (395 nm), 2 (365 nm) +Data +Float +The fluorescence spectra. The feature is a string composed of 1024 values given +between square brackets and seprated by a comma, as for example [1491.0, +1508.0, ..., 1545.0]. Each value is the raw intensity of the fluorescence signal at +the given pixel of the detector of the spectrometer. +Quality +String +Quality of the oil. Possible values are ‘EXTRA’, ‘VIRGIN’, ‘LAMPANTE’ +FAEES +Float +Fatty acid ethyl esters in mg/Kg: content of waxes, fatty acid methyl esters and +fatty acid ethyl esters +K232 +Float +UV Absorbance at 232 nm (K270) +K270 +Float +UV Absorbance at 270 nm (K232) +Acidity +Float +Acidity: expressed as percentage (%) of oleic acid +Peroxide Index +Float +Quantity of those substances in the sample, expressed in terms of milliequivalents +of active oxygen per kilogram (mEqO2/Kg), which oxidize potassium iodide. +Table 1: Information on each feature available in the dataset. +for more details. The schematic design of the spectrometer is sown in Fig. 2. The excitation light was provided by +two UV LEDs with emission at 365 nm and 395 nm driven by a current driver (MIC4801, Micrel Inc., San Jose, CA, +USA) to adjust the excitation intensity. The fluorescence signal was collected by a miniature spectrometer (STS-Vis, +Ocean Optics, Dunedin, FL, USA) with a 1024-element CCD array which acquires the entire spectrum in one single +measurement with a resolution of 16 nm. The spectrometer was placed at 90° with respect to the LEDs to avoid the +excitation light transmitted by the sample to reach the spectrometer. The sensor has a recess where standard 4 ml clear +glass vials with the sample can be inserted. +All spectra of the dataset were acquired on undiluted samples at room temperature under identical conditions (illumina- +tion intensity, integration time, and geometry) for a quantitative comparison. The integration time was 1 s. During the +measurements, the setup was kept in complete darkness to minimize the effect of stray light. +Each spectrum consists of an array of 1024 values (one for each pixel). The value corresponds to the intensity in counts +at the different positions of the pixels. To obtain the wavelength (in nanometers) corresponding to each pixel, the +following formula can be used: +i = a + b · i + c · i2 + d · i3 +(1) +where i indicates the pixel (i = 0, ..., 1023) and +a = 337.92288208 nm +b = 0.4470772743 nm +c = 3.55128 · 10−5 nm +d = −8.38601 · 10−9 nm +(2) +Calibration parameters were provided by the spectrometer manufacturer. All spectra correspond to the raw data without +any data processing (smoothing, background subtraction, or normalization). Since all the measurements were done +under identical conditions the intensities are directly comparable. +3.3 +Chemical Analysis +For each olive oil sample, the dataset includes the values of the following chemical parameters: acidity, peroxide value, +K270, K232, ethyl esters concentration and the samples quality class (EVOO, VOO, or LOO) (see Tab. 2). +The chemical parameters were determined by accredited laboratories using the procedures described in the European +Commission regulation and its amendment (Commission [2013, 1991]). +3 + +Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils +DATASET +Excitation 365 nm +EVOO +Excitation 395 nm +VOO +LOO +0 +4'000 +8'000 +Intensity (a.u.) +0 +4'000 +8'000 +Intensity (a.u.) +12'000 +0 +4'000 +8'000 +Intensity (a.u.) +678 nm +722 nm +678 nm +722 nm +500 +550 +600 +650 +700 +750 +500 +Wavelength (nm) +550 +600 +650 +700 +750 +800 +Wavelength (nm) +Figure 1: Fluorescence emission spectra of selected olive oils divided in the quality classes EVOO, VOO and LOO. On +the left: spectra obtained with excitation at 365 nm; on the right: spectra obtained with excitation at 395 nm. Each +curve shows a single spectrum without averaging or smoothing after the background subtraction. Reproduced from +Venturini et al. [2023]. +4 +Funding +This research was supported by the projects: “VIRTUOUS” funded by the European Union’s Horizon 2020 Project +H2020-MSCA-RISE-2019 Grant No. 872181; “SUSTAINABLE” funded by the European Union’s Horizon 2020 +Project H2020-MSCA-RISE-2020 Grant No. 101007702; “Project of Excellence” from Junta de Andalucia-FEDER- +Fondo de Desarrollo Europeo 2018. Ref. P18–H0-4700. +5 +Author Contributions +Conceptualization: Francesca Venturini and Umberto Michelucci; methodology: Francesca Venturini and Umberto +Michelucci; software, Michela Sperti and Arnaud Gucciardi; validation, Francesca Venturini and Umberto Michelucci; +formal analysis, Francesca Venturini and Umberto Michelucci; investigation, Francesca Venturini and Umberto +Michelucci; resources, Vanessa M. Martos; data curation, Michela Sperti and Arnaud Gucciardi; writing, original draft +preparation, Francesca Venturini and Umberto Michelucci; writing, review and editing, Francesca Venturini, Umberto +Michelucci, Arnaud Gucciardi and Marco A. Deriu; funding acquisition, Vanessa M. Martos and Marco A. Deriu. All +authors have read and agreed to the published version of the manuscript. +6 +Data Availability +The data presented in this study are openly available in Dataset of Fluorescence Spectra and Chemical Parameters of +Olive Oils at https://data.mendeley.com/datasets/thkcz3h6n6/6, DOI: 10.17632/thkcz3h6n6.6. +4 + +Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils +DATASET +Label +Acidity +Peroxide value +K270 +K232 +FAEES +Quality +(%) +(mEq O2/kg) +(mg/Kg) +D03 +0.35 +8.4 +0.123 +1.435 +26 +VOO +D04 +0.34 +8.6 +0.108 +1.403 +40 +VOO +D05 +0.36 +10.3 +0.112 +1.44 +18 +VOO +D06 +0.31 +9.2 +0.151 +1.484 +18 +VOO +D07 +0.50 +8.9 +0.150 +1.537 +47 +VOO +D08 +0.40 +8.5 +0.158 +1.546 +25 +VOO +D09 +- +- +- +- +- +LOO +D10 +- +- +- +- +- +LOO +D19 +0.25 +4.9 +0.13 +1.540 +10 +EVOO +D20 +0.26 +4.6 +0.14 +1.540 +10 +EVOO +D35 +0.17 +6.4 +0.12 +1.63 +8 +EVOO +D38 +0.16 +6.4 +0.12 +1.63 +9 +EVOO +D45 +0.17 +4.9 +0.12 +1.63 +7 +EVOO +D46 +0.18 +5.0 +0.13 +1.63 +8 +EVOO +D47 +0.18 +5.2 +0.13 +1.64 +16 +EVOO +D49 +0.9 +9.9 +- +- +- +LOO +D51 +2.16 +- +- +- +- +LOO +D52 +1.78 +22 +- +- +- +LOO +D53 +0.7 +8.7 +- +- +- +LOO +D64 +0.2 +7.1 +0.13 +1.63 +29 +VOO +D73 +0.2 +8.9 +0.14 +1.66 +15 +EVOO +D77 +0.24 +10.4 +0.13 +1.74 +26 +VOO +D81 +0.16 +4.9 +0.12 +1.63 +9 +EVOO +D92 +0.18 +5 +0.17 +1.91 +15 +EVOO +Table 2: List of olive oil samples and their physicochemical characteristics. FAEES: fatty acid ethyl esters, EVOO: +extra virgin olive oil, VOO: virgin olive oil, LOO: lampante olive oil. +7 +Ackowledgments +The authors would like to thank Michael Baumgartner and Ivo Herzig (Institute of Applied Mathematics and Physics, +Zurich University of Applied Sciences, Winterthur, Switzerland) for help for the realization of the sensor, and Josep +Palau Caballero and Arturo Jimenez (SCA San Sebastián Puente del Ventorro, s/n, 18566 Benalua de las Villas, Spain) +for providing the oil samples. +8 +Conflicts of Interest +The authors declare no conflicts of interest and no known competing financial interests or personal relationships that +could have appeared to influence the work reported in this paper. +9 +Abbreviations +The following abbreviations are used in this manuscript: +LOO +Lampante Olive Oil +EVOO +Extra Vigrin Olive Oil +VOO +Virgin Olive Oil +CCD +Charge-Coupled Device +LED +Light Emitting Diode +UV +Ultraviolet +FAEES +Fatty Acid Ethyl Ester +5 + +Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils +DATASET +LED +Driver +Spectrometer +Excitation +LED +Sample +Fluorescence +Raspberry Pi +Figure 2: Schematics of the portable fluorescence sensor. Blue: excitation light, red: fluorescence light. From Venturini +et al. [2021]. +References +European Commission. Commission implementing regulation no 1348/2013 of december 17 2013. Official Journal of +the European Union, 338:31–67, 2013. +European Commission. Commission regulation (eec) no. 2568/91 of 11 july 1991 on the characteristics of olive oil and +olive-residue oil and on the relevant methods of analysis official journal l 248, 5 september 1991. Offic. JL, 248:1–83, +1991. +Francesca Venturini, Michela Sperti, Umberto Michelucci, Arnaud Gucciardi, Vanessa M Martos, and Marco A Deriu. +Extraction of physicochemical properties from the fluorescence spectrum with 1d convolutional neural networks: +Application to olive oil. Journal of Food Engineering, 336:111198, 2023. +Francesca Venturini, Michela Sperti, Umberto Michelucci, Ivo Herzig, Michael Baumgartner, Josep Palau Caballero, +Arturo Jimenez, and Marco Agostino Deriu. Exploration of spanish olive oil quality with a miniaturized low-cost +fluorescence sensor and machine learning techniques. Foods, 10(5):1010, 2021. +6 + diff --git a/HtE3T4oBgHgl3EQfWwou/content/tmp_files/load_file.txt b/HtE3T4oBgHgl3EQfWwou/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65395f431724f7d73586d1969a55dd9d8bbe511c --- /dev/null +++ b/HtE3T4oBgHgl3EQfWwou/content/tmp_files/load_file.txt @@ -0,0 +1,242 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf,len=241 +page_content='DATASET OF FLUORESCENCE SPECTRA AND CHEMICAL PARAMETERS OF OLIVE OILS AN OPEN SOURCE DATASET Francesca Venturini ∗ Institute of Applied Mathematics and Physics Zurich University of Applied Sciences Winterthur, Switzerland, vent@zhaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='ch Artificial Intelligence Research and Development TOELT LLC, Switzerland Michela Sperti PolitoBIOMed Lab Department of Mechanical and Aerospace Engineering Politecnico di Torino, Turin, Italy Umberto Michelucci Artificial Intelligence Research and Development TOELT LLC, Switzerland umberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='michelucci@toelt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='ai Computer Science Department Lucerne University of Applied Sciences and Arts Lucerne, Switzerland Arnaud Gucciardi Artificial Intelligence Research and Development TOELT LLC, Switzerland arnaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='gucciardi@toelt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='ai Artificial Intelligence Laboratory University of Ljubljana, Ljubljana, Slovenia Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Martos Department of Plant Physiology Faculty of Sciences Biotechnology Institute University of Granada, Spain Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Deriu PolitoBIOMed Lab Department of Mechanical and Aerospace Engineering Politecnico di Torino, Turin, Italy January 12, 2023 ABSTRACT This dataset encompasses fluorescence spectra and chemical parameters of 24 olive oil samples from the 2019–2020 harvest provided by the producer Conde de Benalúa, Granada, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The oils are characterized by different qualities: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and 6 lampante olive oil (LOO) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' For each sample, the dataset includes fluorescence spectra obtained with two excitation wavelengths, oil quality, and five chemical parameters necessary for the quality assessment of olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The fluorescence spectra were obtained by exciting the samples at 365 nm and 395 nm under identical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The dataset includes the values of the following chemical parameters for each olive oil sample: acidity, peroxide value, K270, K232, ethyl esters, and the quality of the samples (EVOO, VOO, or LOO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The dataset offers a unique possibility for researchers in food technology to develop machine learning models based on fluorescence data for the quality assessment of olive oil due to the availability of both spectroscopic and chemical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The dataset can be used, for example, to predict one or multiple chemical parameters or to classify samples based on their quality from fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Keywords Fluorescence · Olive Oil · Chemical Parameters · Quality control ∗Contact email: vent@zhaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='ch arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='04471v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='QM] 10 Jan 2023 Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils DATASET 1 Summary The dataset presented is a compilation of measurements of analytical chemistry and fluorescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The dataset includes fluorescence spectra and chemical parameters of 24 Spanish olive oils from the 2019–2020 harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The 24 samples were collected at SCA San Sebastián Puente del Ventorro, Benalua de las Villas, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The data were later measured at the Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Technikumstrasse 9, 8401 Winterthur, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The fluorescence spectroscopy data was acquired by a miniature spectrometer with a 1024 element CCD array that acquires the entire spectrum in one single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The dataset includes a total of 960 spectra (24 oil samples × 2 excitation wavelengths x 20 repeated measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Each of the 960 spectra is an array of 1024 values whose elements are the intensity at the different pixel positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The chemical parameters were determined by accredited laboratories using the procedures described in the European Commission regulation and its amendment Commission [2013, 1991].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' These regulations control the methods for the quality assessment of olive oils and provide a decision tree to verify whether an olive oil class is consistent with the declared quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The value of the dataset for research purposes is summarized in the points below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The data are useful for studying the link between optical properties (fluorescence and absorption spectroscopy), chemical characteristics (such as oil acidity, peroxide value, and fatty acid content), and olive oil quality (extra virgin, virgin, and lampante olive oil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' This dataset is the first available that contains fluorescence spectra and chemical analysis obtained by accredited laboratories on samples coming from a single producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Many researchers can benefit from the data: computer scientists can use the data to develop machine learning models that link optical to chemical properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' researchers in food technology that are interested in studying chemical properties of olive oil samples of different qualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' engineers that want to develop new optical analysis techniques alternative to the current expensive and time-consuming analytical chemistry methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' This dataset can be used to perform explainability analysis to identify spectral characteristics that are related to different chemical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=', the acidity of the oil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' An example is given in the paper Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' This will further advance the understanding of the complex chemical composition of olive oil and its link to its quality and health benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' This dataset can be used to develop instruments based on fluorescence spectroscopy for the rapid and cost- effective quality assessment of olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 2 Data Description The dataset consists of one CSV file that contains the columns described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' A background file2 is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The file contains 1024 values that correspond to the intensity measured by the spectrometer without any light (dark counts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' This spectrum can be subtracted from the raw fluorescence spectra to remove the effect of the dark counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The same file can be used for the spectra taken at both 365 nm and 395 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The raw fluorescence spectra of selected oils obtained with excitation at 365 nm and 395 nm are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 3 Materials and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='1 Olive Oil Samples The dataset contains the fluorescence spectra and the chemical parameters of 24 oils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The oils are characterized by different quality categories: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and 6 lampante olive oil (LOO) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' All samples were provided by Conde de Benalúa, Granada, southern Spain, and were prepared from the 2019–2020 harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The properties and values of the chemical parameters of the oil samples are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' For data acquisition, the samples were placed in commercial 4 ml clear glass vials, taking care that no headspace was present to reduce oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' All oils were stored in the dark and at 20 °C during the entire time of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='2 Fluorescence Data Acquisition The fluorescence spectroscopy data were acquired using the portable sensor described in Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Since already published, only the most relevant characteristics are reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The reader is referred to this publication 2Fluorescence_olive_oil_dataset_background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='csv 2 Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils DATASET Feature Datatype Description Sample String Oil sample name: the values are ’D03’,’D04’,’D05’, ’D06’, ’D07’ ,’D08’, ’D09’, ’D10’, ’D 19’, ’D20’, ’D35’, ’D38’, ’D45’, ’D46’, ’D47’, ’D49’, ’D51’, ’D52’, ’D53’, ’D64’, ’D77’, ’D81’, ’D92’,’D73’ Repetition Integer Repetition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' There are 20 repetition for each oil and led: the iteration number goes from 0 to 19) Led Integer Excitation LED identifier: 1 (395 nm), 2 (365 nm) Data Float The fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The feature is a string composed of 1024 values given between square brackets and seprated by a comma, as for example [1491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='0, 1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=', 1545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Each value is the raw intensity of the fluorescence signal at the given pixel of the detector of the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Quality String Quality of the oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Possible values are ‘EXTRA’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' ‘VIRGIN’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' ‘LAMPANTE’ FAEES Float Fatty acid ethyl esters in mg/Kg: content of waxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' fatty acid methyl esters and fatty acid ethyl esters K232 Float UV Absorbance at 232 nm (K270) K270 Float UV Absorbance at 270 nm (K232) Acidity Float Acidity: expressed as percentage (%) of oleic acid Peroxide Index Float Quantity of those substances in the sample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' expressed in terms of milliequivalents of active oxygen per kilogram (mEqO2/Kg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' which oxidize potassium iodide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Table 1: Information on each feature available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The schematic design of the spectrometer is sown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The excitation light was provided by two UV LEDs with emission at 365 nm and 395 nm driven by a current driver (MIC4801, Micrel Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=', San Jose, CA, USA) to adjust the excitation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The fluorescence signal was collected by a miniature spectrometer (STS-Vis, Ocean Optics, Dunedin, FL, USA) with a 1024-element CCD array which acquires the entire spectrum in one single measurement with a resolution of 16 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The spectrometer was placed at 90° with respect to the LEDs to avoid the excitation light transmitted by the sample to reach the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The sensor has a recess where standard 4 ml clear glass vials with the sample can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' All spectra of the dataset were acquired on undiluted samples at room temperature under identical conditions (illumina- tion intensity, integration time, and geometry) for a quantitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The integration time was 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' During the measurements, the setup was kept in complete darkness to minimize the effect of stray light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Each spectrum consists of an array of 1024 values (one for each pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The value corresponds to the intensity in counts at the different positions of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' To obtain the wavelength (in nanometers) corresponding to each pixel, the following formula can be used: i = a + b · i + c · i2 + d · i3 (1) where i indicates the pixel (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=', 1023) and a = 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='92288208 nm b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='4470772743 nm c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='55128 · 10−5 nm d = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='38601 · 10−9 nm (2) Calibration parameters were provided by the spectrometer manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' All spectra correspond to the raw data without any data processing (smoothing, background subtraction, or normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Since all the measurements were done under identical conditions the intensities are directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='3 Chemical Analysis For each olive oil sample, the dataset includes the values of the following chemical parameters: acidity, peroxide value, K270, K232, ethyl esters concentration and the samples quality class (EVOO, VOO, or LOO) (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' The chemical parameters were determined by accredited laboratories using the procedures described in the European Commission regulation and its amendment (Commission [2013, 1991]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=" 3 Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils DATASET Excitation 365 nm EVOO Excitation 395 nm VOO LOO 0 4'000 8'000 Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=") 0 4'000 8'000 Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=") 12'000 0 4'000 8'000 Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=') 678 nm 722 nm 678 nm 722 nm 500 550 600 650 700 750 500 Wavelength (nm) 550 600 650 700 750 800 Wavelength (nm) Figure 1: Fluorescence emission spectra of selected olive oils divided in the quality classes EVOO, VOO and LOO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' On the left: spectra obtained with excitation at 365 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' on the right: spectra obtained with excitation at 395 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Each curve shows a single spectrum without averaging or smoothing after the background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Reproduced from Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 4 Funding This research was supported by the projects: “VIRTUOUS” funded by the European Union’s Horizon 2020 Project H2020-MSCA-RISE-2019 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 872181;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' “SUSTAINABLE” funded by the European Union’s Horizon 2020 Project H2020-MSCA-RISE-2020 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 101007702;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' “Project of Excellence” from Junta de Andalucia-FEDER- Fondo de Desarrollo Europeo 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' P18–H0-4700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 5 Author Contributions Conceptualization: Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' methodology: Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' software, Michela Sperti and Arnaud Gucciardi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' validation, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' formal analysis, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' investigation, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' resources, Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Martos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' data curation, Michela Sperti and Arnaud Gucciardi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' writing, original draft preparation, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' writing, review and editing, Francesca Venturini, Umberto Michelucci, Arnaud Gucciardi and Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Deriu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' funding acquisition, Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Martos and Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 6 Data Availability The data presented in this study are openly available in Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils at https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='mendeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='com/datasets/thkcz3h6n6/6, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='17632/thkcz3h6n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 4 Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils DATASET Label Acidity Peroxide value K270 K232 FAEES Quality (%) (mEq O2/kg) (mg/Kg) D03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='7 LOO D64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='63 29 VOO D73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='66 15 EVOO D77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='24 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='74 26 VOO D81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='63 9 EVOO D92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='18 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content='91 15 EVOO Table 2: List of olive oil samples and their physicochemical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' FAEES: fatty acid ethyl esters, EVOO: extra virgin olive oil, VOO: virgin olive oil, LOO: lampante olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 7 Ackowledgments The authors would like to thank Michael Baumgartner and Ivo Herzig (Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Winterthur, Switzerland) for help for the realization of the sensor, and Josep Palau Caballero and Arturo Jimenez (SCA San Sebastián Puente del Ventorro, s/n, 18566 Benalua de las Villas, Spain) for providing the oil samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 8 Conflicts of Interest The authors declare no conflicts of interest and no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 9 Abbreviations The following abbreviations are used in this manuscript: LOO Lampante Olive Oil EVOO Extra Vigrin Olive Oil VOO Virgin Olive Oil CCD Charge-Coupled Device LED Light Emitting Diode UV Ultraviolet FAEES Fatty Acid Ethyl Ester 5 Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils DATASET LED Driver Spectrometer Excitation LED Sample Fluorescence Raspberry Pi Figure 2: Schematics of the portable fluorescence sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Blue: excitation light, red: fluorescence light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' From Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' References European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Commission implementing regulation no 1348/2013 of december 17 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Official Journal of the European Union, 338:31–67, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Commission regulation (eec) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 2568/91 of 11 july 1991 on the characteristics of olive oil and olive-residue oil and on the relevant methods of analysis official journal l 248, 5 september 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Offic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' JL, 248:1–83, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Francesca Venturini, Michela Sperti, Umberto Michelucci, Arnaud Gucciardi, Vanessa M Martos, and Marco A Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Extraction of physicochemical properties from the fluorescence spectrum with 1d convolutional neural networks: Application to olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Journal of Food Engineering, 336:111198, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Francesca Venturini, Michela Sperti, Umberto Michelucci, Ivo Herzig, Michael Baumgartner, Josep Palau Caballero, Arturo Jimenez, and Marco Agostino Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Exploration of spanish olive oil quality with a miniaturized low-cost fluorescence sensor and machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' Foods, 10(5):1010, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'} diff --git a/INE0T4oBgHgl3EQfRwA9/content/2301.02211v1.pdf b/INE0T4oBgHgl3EQfRwA9/content/2301.02211v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..06ccf76cd1e49fb4205ae2b56a2b19dfc906dab2 --- /dev/null +++ b/INE0T4oBgHgl3EQfRwA9/content/2301.02211v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:951ba9adf4228aa649a3d483385f2aa5d1dbef3e29cd4076c1ce60f9f929ea02 +size 388257 diff --git a/INE0T4oBgHgl3EQfRwA9/vector_store/index.faiss b/INE0T4oBgHgl3EQfRwA9/vector_store/index.faiss new file mode 100644 index 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b/INFIT4oBgHgl3EQfYSsB/content/tmp_files/2301.11247v1.pdf.txt @@ -0,0 +1,1072 @@ +Prepared for submission to JINST +The EXTRA-BL4S experiment for the measurement of the +energy and angular distributions of transition radiation +X-rays +M. N. Mazziotta,𝑎,1 F. Loparco, 𝑎,𝑏,1 A. Anelli,𝑐 M. M. Belviso,𝑐 A. Buquicchio,𝑐 +E. V. Cassano,𝑐 M. De Cosmo,𝑐 P. Ginefra,𝑐 M. L. Martulli,𝑐 C. Picci,𝑐 D. Picicci,𝑐 +R. D. Soriano,𝑐 A. P. Tatulli,𝑐 G. Tripaldella,𝑐 V. M. Zupo,𝑐 M. F. Muscarella,𝑐 S. Turbacci,𝑐 +M. Boselli,𝑑 C. B. da Cruz E Silva,𝑑,2 M. Joos𝑑 and P. Schütze𝑒 +𝑎Istituto Nazionale di Fisica Nucleare, Sezione di Bari, +via Orabona 4, I-70126 Bari, Italy +𝑏Dipartimento di Fisica dell’Università e del Politecnico di Bari, +via Amendola 173, I-70126 Bari, Italy +𝑐The EXTRA Team Liceo Scientifico Statale "A. Scacchi", +Corso Cavour 241, I-70121 Bari, Italy +𝑑CERN, the European Organization for Nuclear Research, +Esplanade des Particules 1, 1211 Geneva, Switzerland +𝑒DESY, Notkestrasse 85, D-22607 Hamburg +E-mail: mazziotta@ba.infn.it, francesco.loparco@ba.infn.it +Abstract: We have designed and implemented an experiment to measure the angular distributions +and the energy spectra of the transition radiation X-rays emitted by fast electrons and positrons +crossing different radiators. Our experiment was selected among the proposals of the 2021 Beamline +for Schools contest, a competition for high-school students organized every year by CERN, and +was performed at the DESY II Test Beam facility area TB21, using a high-purity beam of electrons +or positrons with momenta in the range from 1 to 6 GeV/c. The measurements were performed +using a 100 𝜇m thick silicon pixel detector, with a pitch of 55 𝜇m. Our results are consistent with +the expectations from the theoretical models describing the production of transition radiation in +multilayer regular radiators. +Keywords: Transition radiation detectors; Particle identification methods +1Corresponding authors. +2Now at LIP - Laboratório de Instrumentação e Física Experimental de Partículas Avenida Prof. Gama Pinto 2, +Complexo Interdisciplinar (3is), 1649-003 Lisboa, Portugal +arXiv:2301.11247v1 [hep-ex] 26 Jan 2023 + +Contents +1 +Introduction +1 +2 +The BL4S competition +1 +3 +The EXTRA experiment +3 +4 +Data analysis +7 +4.1 +Conversion & Clustering +7 +4.2 +Detector alignment procedure +7 +4.3 +Data selection and analysis +8 +5 +Results +10 +6 +Conclusions +15 +1 +Introduction +In recent years, high-school physics curricula increasingly include topics related to modern high- +energy physics and particle detectors. Universities and research centers promote several programs +to bring high-school students in touch with modern physics and the scientific research. The Liceo +Scientifico “A. Scacchi” in Bari has taken part in such projects for years, and in 2021 the school +promoted the participation of a team of students of the 12𝑡ℎ and 13𝑡ℎ grade in the Beamline for +Schools (BL4S) competition. +BL4S is organized by CERN in collaboration with DESY, and offers to groups of high-school +students the unique opportunity to propose a scientific experiment at a particle accelerator facility +and to win a trip to perform it. Because of the maintenance of CERN accelerators, the experiment +was performed at the DESY II Beam Test facility in Hamburg. The students, coordinated by their +physics teachers and under the supervision of experienced researchers from the Physics Department +of the Bari University and from the INFN Unit in Bari, won the competition. +The goal of the experiment conceived by the team was to study the transition radiation emitted +by fast electrons and positrons crossing different kinds of radiators. This paper provides a short +presentation of the BL4S competition and presents the experiment and the result obtained by the +team during their beam time in Hamburg in September 2021. +2 +The BL4S competition +Beamline for Schools (BL4S) [1] is a physics competition organised by CERN and DESY, which +invites high-school students from all over the world to propose an experiment to be performed at a +– 1 – + +particle accelerator. Each team has to write an original scientific proposal, explaining the theoretical +background of the selected topic, and describing both the procedure to carry it out at a test beam +facility and the results that they expect to find. A jury of experts, including scientists of CERN +and DESY, review the proposal and select two teams (three from 2022 on) that win a trip to a fully +equipped beam line of a particle accelerator. +From 2014 to 2018 the winning experiments took place at the test beam area of the CERN +Proton Synchrotron (PS) accelerator. In 2019 the competition moved to the DESY II Test Beam +Facility (Hamburg) [2]. The partnership between CERN and the German laboratory allowed BL4S +to continue during the three-year long shutdown of the CERN accelerator complex for upgrade and +maintenance. +The competition is structured in several preparatory phases, which include conferences and +meetings with the organisers. Once the competition is announced, usually in Autumn, interested +teams start preparing their proposals. Teams can include students either from the same school or +from different schools. Having teams representing two schools or more is not unusual. During +the proposal preparation, students are involved in an intense research project. After the conception +and design of their experiment, the participants must write a well structured proposal and submit +it on time. The students are not alone in this process, but they are guided by their coaches, who +provide them with details on particle physics and teach them the necessary technical skills. Team +coaches can be teachers, parents or scientists of local universities. It is important that students are +well aware of each scientific detail of the proposed experiment, so that the theoretical background is +clear and solid. The students are required to write down in detail how they intend to use the particle +beam for their measurements and which equipment and detectors they need. Moreover, participants +often complement their theoretical hypothesis with computer simulations. In fact, it is fundamental +that students acquire the rudimentary programming skills that will be required in case of victory. +Lastly, conclusions must contain the team’s expectations and motivation, which play a significant +role in the jury’s decision. The BL4S organisers are always available to answer questions that the +teams might have during the preparation of their proposals. Many teams contact them to discuss +the feasibility of their experiments or practical problems that they encounter. +In the final phase of the competition, a jury consisting of more than 50 volunteers selects +the teams that are invited to a research institute to perform their experiment together with support +scientists. Prior to the visit, the winning teams work remotely with the BL4S scientists to refine +their experiments and perform a detailed planning. +The beam time of the winning teams usually happens just after the summer, and the students +have 12 full days of access to the experimental area to perform their measurements, supervised by +the support scientists. During their stay, they work as a team of professional scientist would do and +they complement their scientific experience with visits and lectures. +After taking the data at the beam line, the teams are encouraged to analyse their data to answer +the scientific question of the initial proposal, and to write a paper. During this phase, the team +members stay in close contact with the BL4S support scientists and the team coaches. +– 2 – + +Figure 1. Schematic view of the experimental setup. +3 +The EXTRA experiment +The EXTRA (Electron X-ray Transition RAdiation) experiment is designed to study the transition +radiation (TR) [3] emitted by fast electrons and positrons crossing different radiators. +Highly relativistic particles crossing the boundary between materials with different dielectric +constants can produce TR in the X-ray region. However, since the yield of TR photons emitted at +a single interface is considerably small (it is of the order of the fine structure constant 𝛼 ≈ 1/137), +multiple boundaries are needed to enhance the X-ray production. Periodic radiators, consisting +of stacks of thin foils of dielectric material separated by thicker air gaps, are commonly used in +transition radiation detectors (TRDs) [4]. +The main features of the TR emitted by a periodic radiator depend on the kinematic properties +of the radiating particles and on the radiator properties. They can be summarized as follows [5]: +1. The effective TR photon emission starts at a threshold Lorentz factor, which is given by +𝛾𝑡ℎ𝑟 = 𝑑1𝜔1/𝑐, where 𝑑1 is the thickness of the foils, while 𝜔1 is the plasma frequency of +the foil material. +2. The TR emission increases with the Lorentz factor 𝛾 until it reaches saturation at 𝛾𝑠𝑎𝑡 = +𝛾𝑡ℎ𝑟 +√︁ +𝑑2/𝑑1, where 𝑑2 is the thickness of the air gaps. +3. Most of the TR energy is emitted near the energy ℏ𝜔𝑚𝑎𝑥 = ℏ𝜔2 +1𝑑1/2𝜋𝑐. +4. The angular distribution of TR photons exhibits a few maxima and extends up to 𝜃𝑚𝑎𝑥 = +√︃ +1/𝛾2 + 𝜔2 +1/𝜔2. +We have designed an experimental setup to measure the energy spectra and the angular +distributions of the TR X-rays emitted by fast electrons and positrons crossing different radiators. +Similar measurements were performed in the past at the CERN SPS with beams of 20 GeV/c +electrons and of 120, 180 and 290 GeV/c muons, using silicon strip detectors [6], silicon pixel +– 3 – + +Beam +Radiator +Timepix3 +BeamTelescopeScintillatorsFigure 2. Pictures of the experimental setup. +– 4 – + +beamling +forschools +cern.ch/bl4sRadiator +Foil/gap material +d1 (𝜇m) +d2 (𝜇m) +N 𝑓 +EXTRA +polyethylene/air +23 +500 +150 +INFN +polyethylene/air +25 +300 +155 +CERN +polyethylene/air +25 +240 +190 +Table 1. Parameters of radiators used in the beam test: 𝑑1 and 𝑑2 are the thickness of the foils and the gap +respectively; 𝑁 𝑓 is the number of foils. +Radiator +distance ( cm) +Beam particle +Beam momenta ( GeV/c) +EXTRA +40.5 +𝑒− +1, 2, 3, 4, 5, 6 +88.0 +𝑒− +1, 2, 3, 4, 5, 6 +132.0 +𝑒− +1, 2, 3, 4, 5, 6 +INFN +88.9 +𝑒− +1, 2, 3, 4, 5, 6 +CERN +88.4 +𝑒−/𝑒+ +1, 2, 3, 4, 5 +Table 2. Summary of the data taking configurations. For each radiator the beam particle, their momenta and +the distance between the radiator and the X-ray detector are reported. +detectors [7, 8] and GaAs pixel detectors [8, 9]. Parallel to the measurements, an effort to develop +accurate Monte Carlo simulations of the TR process is being carried out [10]. One of the goals of +these activities is that of exploiting TR for the identification of charged hadrons in the TeV energy +region [11]. In this region all hadrons have Lorentz factor exceeding the typical threshold values +for TR production (usually 𝛾𝑡ℎ𝑟 ∼ 500 ÷ 1000), and the simultaneous measurement of the energies +and of the emission angles of TR X-rays can help to discriminate among different hadron species. +Our measurements were performed at the DESY II Test Beam Facility [2] area TB21, using +a beam of either electrons or positrons with momenta in the range from 1 to 6 GeV/c. A scheme +of the setup is shown in Fig. 1, while some pictures are shown in Fig. 2. The radiator is followed +by a Timepix3 assembly containing a thin silicon pixel sensor, which is used to detect the TR +X-rays. A downstream beam telescope, composed by an array of six silicon pixel detectors, is +used to reconstruct the tracks of the beam particles [12]. A set of two trigger scintillators, located +downstream of the last plane of the beam telescope, is used for triggering the data acquisition. +In our experiment we used three different radiators, which in the following will be labelled +as "EXTRA", "INFN" and "CERN" respectively. Their features are summarized in Tab. 1. In +particular, the EXTRA radiator was assembled for this measurement by the students at the Liceo +Scientifico "A. Scacchi" in Bari. Fig. 3 shows some picture taken during the assembly of this +radiator. The INFN and CERN radiators were borrowed from the Bari INFN Group and were used +in a beam test campaign performed in 2006 [13]. +With these radiators, several measurements were performed, changing the beam composition +and momentum, and the distance between the radiator and the X-ray detector. The different data +taking configurations are summarized in Tab. 2. +The TR X-rays were detected by a 100 𝜇m thick silicon sensor, bump-bonded to a Timepix3 +readout chip [14], consisting of a pixel matrix of 256×256 pixels with a pitch of 55 𝜇m. This silicon +– 5 – + +Figure 3. Assembly of the EXTRA radiator at the Liceo Scientifico "A. Scacchi". +detector assembly was placed such that the sensor faces the radiator to mitigate prior absorption in +the readout chip. The sensor of the assembly with the ID W5_E2 was operated at a bias voltage of +−21 V to ensure full depletion [15]. +The pixel pitch of the silicon sensor and its distance from the radiator determine the minimum +detectable angular separation 𝜃𝑚𝑖𝑛 of TR X-rays from the direction of the radiating particles, as +they should be separated of at least one pixel. Its value is in fact given by 𝜃𝑚𝑖𝑛 ≳ 𝑤/𝑑, where +𝑤 = 55 𝜇m is the pixel pitch and 𝑑 is the distance of the silicon detector from the radiator. The +configurations with larger distances allow to detect smaller angular separations; however, due to the +X-ray absorption in the radiator and in the air gap between the radiator and the sensor, the number +of detected TR X-rays will decrease with the distance from the radiator, and the angular resolution +will deteriorate due to multiple Coulomb scattering of the primary particles in air. +While the TR X-rays are likely absorbed by the front sensor, the radiating charged particles +traverse the detector and leave an ionization track in the detectors of the beam telescope, which +consists of an array of six regularly spaced silicon pixel detectors. In this configuration, scattering +in air is limited to a minimum, enabling a track resolution of a few 𝜇m extrapolated to the Timepix3 +detector [12], which is more than sufficient for an identification of the charged particle among two +or more clusters in the Timepix3 detector with cluster distances larger than a pixel pitch. +Finally, the two scintillators, approximately shadowing the size of the telescope sensor planes +and located at the end of the beam line, are used for triggering the data acquisition. +The data acquisition was performed using the software framework EUDAQ2 [16], which +– 6 – + +语integrates the control and readout of the Timepix3 assembly and the beam telescope, and features a +graphical user interface for the configuration of connected devices, starting and stopping runs and +data storage. An AIDA TLU [17] was used to form a trigger signal as a coincidence of the signals +from the two scintillators while enabling a busy-handshake with the detectors. +4 +Data analysis +4.1 +Conversion & Clustering +The raw data contains a collection of hit pixels per detector plane per trigger, which defines a +so-called "event", including the corresponding pixel addresses; for the data from the Timepix3 +assembly, the corresponding information on the energy deposit, in form of a digitised signal, is +also stored, while for the beam telescope no charge information is recorded. The collected data +are converted to ROOT TTree format [18] using the data analysis framework Corryvreckan [19]. +In addition, this software performs a clustering procedure, which identifies adjacent hit pixels and +connects them to form a so-called "cluster" under the hypothesis that pixel hits in one cluster are +caused by a single incident particle. The cluster center, as an estimation on the incidence position +of the particle, is calculated either as the center-of-gravity using the charge information, or as the +arithmetic mean of the pixel hit positions in case of binary hit information. +The energy calibration of the silicon pixel detector is performed assuming that the most probable +energy loss of 5 GeV/c electrons crossing a 100 𝜇m thick silicon layer is 25.41 keV. This value +has been calculated using a dedicated Monte Carlo simulation developed by H. Bichsel for the +calculation of the energy losses of charged particles in thin silicon absorbers [20]. +4.2 +Detector alignment procedure +The positions of the clusters in each silicon detector are evaluated in the local detector reference +frame, with the 𝑧-axis oriented along the beam direction and the 𝑥 − 𝑦 plane corresponding to +the detector plane, with the origin in the center of the detector. In the global reference frame the +𝑧-axis is also directed along the beam direction, and the detectors are disposed on planes parallel +to the 𝑥 − 𝑦 plane, with their centers at the coordinates (𝑥𝑖 +0, 𝑦𝑖 +0, 𝑧𝑖 +0). Due to mechanical tolerances +in the assembly of the detectors, the coordinates (𝑥𝑖 +0, 𝑦𝑖 +0) are slightly misaligned with respect to the +reference values (0, 0). +A dedicated alignment run has been therefore performed to evaluate the coordinates (𝑥𝑖 +0, 𝑦𝑖 +0) of +the centers of the silicon detectors (the index 𝑖 = 0 refers to the Timepix3 sensor, while the indices +𝑖 = 1 . . . 6 refer to the detectors of the beam telescope). The alignment run has been performed +removing the radiator from the beam line and using 5 GeV/c electrons. +We have implemented an iterative alignment procedure selecting a sample of events with only +one cluster in each silicon detector. This choice is aimed to select events with only one electron +track across all the detectors. In the first iteration we assume 𝑥𝑖 +0 = 0 and 𝑦𝑖 +0 = 0 for all detectors. We +fit all the tracks with a straight line and, for each track, we evaluate the residuals in each detector as +𝑟𝑖 +𝑥 = 𝑥𝑖 − 𝑥𝑖 +𝑓 𝑖𝑡 and 𝑟𝑖 +𝑦 = 𝑦𝑖 − 𝑦𝑖 +𝑓 𝑖𝑡, where (𝑥𝑖, 𝑦𝑖) and (𝑥𝑖 +𝑓 𝑖𝑡, 𝑦𝑖 +𝑓 𝑖𝑡) are respectively the true and fitted +positions of the cluster in the 𝑖-th detector. We then build the distributions of the residuals 𝑟𝑖 +𝑥 and 𝑟𝑖 +𝑦 +and, in the next iteration, we set 𝑥𝑖 +0 = −𝜇𝑖 +𝑥 and 𝑦𝑖 +0 = −𝜇𝑖 +𝑦, where 𝜇𝑖 +𝑥 and 𝜇𝑖 +𝑦 are the average values +– 7 – + +Figure 4. Distributions of the residuals in the silicon detector equipped with the TimePix3 chip after the +alignment procedure. +of these distributions. The iterative procedure is terminated when |𝜇𝑖 +𝑥| < 1 𝜇m and |𝜇𝑖 +𝑦| < 1 𝜇m +for all detectors. Convergence is reached after the second iteration. +Fig. 4 shows the distributions of the residuals in the silicon detector equipped with the Timepix3 +chip after the alignment procedure. The RMS of the residual distributions in both the 𝑥 and 𝑦 views +are of about 10 𝜇m. +Fig. 5 shows the distributions of the direction cosines of the electron tracks in the alignment +run. We see that the average values of the direction cosines 𝑐𝑥 and 𝑐𝑦 are slightly different from +zero. This result implies that the 𝑧-axis of our reference frame is not perfectly aligned with the +direction of the beam. The tilt angle can be estimated from the average value of 𝑐𝑧, and is of about +5 mrad. Finally, from the values of the RMS of the distributions of 𝑐𝑥 and 𝑐𝑦 we can deduce that +the beam divergence is of about 1 mrad in both the 𝑥 and 𝑦 directions. +4.3 +Data selection and analysis +As discussed in Sec. 3, several runs in different configurations have been taken, by changing the +beam composition and momentum, the radiator and its distance from the silicon pixel detector. +In each of these runs we have selected events with at least one cluster in the silicon pixel sensor +and at least 3 clusters in different detectors of the beam telescope. This choice is motivated by the +need of identifying, among the clusters in the silicon sensor, the one originated by the ionization +energy deposit of the beam particle and those eventually originated by the absorption of TR X-rays +produced in the upstream radiator. +Fig. 6 shows the distribution of the total number of clusters in the detectors of the beam +telescope for all the runs performed with electrons crossing the EXTRA radiator, which was placed +at a distance of 88.9 cm from the silicon pixel sensor. As expected, the distribution is peaked at 6 +clusters, corresponding to clean electron tracks, yielding one cluster in each detector. Events with +less than 6 clusters can be originated from inefficiencies of some detectors in the beam telescope +or from beam particles which do not cross all the telescope planes. Events with more than 6 +clusters can be originated from delta rays accompanying the primary electron track or from TR +X-rays passing through the upstream silicon sensor and being absorbed in any detector of the silicon +– 8 – + +X103 +Entries +542808 +1.2 +Mean +4.227e-04 +RMS +1.064e-02 +x? / ndf +1.047e+04/2389 +Constant +1.126e+03±1.961e+00 +Mean +4.303e-04±1.287e-05 +Sigma +9.346e-03±9.969e-06 +0.8 +ents +0.4 +0.2 +0 +/x10~3 +60 +40 +20 +0 +20 +40 +60 +Residualsinthex-view(mmX103 +Entries +542808 +1.2 +Mean +3.268e-04 +RMS +1.050e-02 +x? / ndf +1.044e+04/2379 +Constant +1.140e+03±1.981e+00 +Mean +3.287e-04± 1.271e-05 +Sigma +0.8 +9.229e-03±9.775e-06 +ents +0.4 +0.2 +/x10~3 +0 +60 +40 +20 +0 +20 +40 +60 +Residuals in the y-view (mm)Figure 5. Distributions of the direction cosines of the electron tracks in the silicon detector and in the beam +telescope in the alignment run. +telescope. We also see two peaks, at 12 and 18 clusters respectively, which include less than 1% of +the total number of events, and which likely correspond to double and triple electron tracks. +The clusters in the detectors of the beam telescope are used to reconstruct the tracks of the +beam particles in the telescope. To select events with single electron (positron) tracks, we require +less than 10 clusters in the beam telescope. Candidate tracks are built by selecting all the possible +cluster combinations with only one cluster per plane of the telescope. The clusters of each candidate +track are then fitted with a straight line and the 𝜒2 of the fit is evaluated. The track with the best 𝜒2 +is then selected. +Once the track of the radiating particle in the beam telescope is reconstructed, we evaluate the +coordinates (𝑥𝑡𝑟𝑎𝑐𝑘, 𝑦𝑡𝑟𝑎𝑐𝑘) of its intersection with the upstream silicon pixel sensor. Then, if more +clusters are found in the sensor, the cluster nearest to the track is associated to the particle ("particle +cluster"), while other clusters are associated to possible TR X-rays ("X-ray clusters"). Clearly, if +only one cluster is found in the silicon pixel sensor, it is associated to the particle and no X-rays are +detected. +– 9 – + +X103 +Entries +542808 +Mean 4.254e-03 +25 +RMS +9.283e-04 +20 +Events +15 +10 +L0 +×10~3 +-10 +-8 +-6 +4 +-2 +0 +2 +4 +6 +80 +10 +CxX103 +Entries +542808 +30 +Mean -1.903e-03 +RMS +8.606e-04 +25 +20 +Events +15 +10 +LO +0 +/×10~3 +-10 +-8 +-6 +-4 +2 +0 +2 +4 +9 +8 +10 +CyX103 +Entries +542808 +Mean 1.160e-05 +30 +RMS4.175e-06 +25 +20 +Events +15 +10 +L0 +0 +×10~6 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +1-CzFigure 6. Distribution of the total number of clusters in the beam telescope for all the runs performed with +electrons crossing the EXTRA radiator, placed at a distance of 88.9 cm from the silicon pixel sensor. +5 +Results +In Figs. 8, 9 and 10 the results obtained in the runs with the EXTRA radiator are summarized. The +plots in each figure correspond to the configurations with the silicon detector placed at the distances +of 40.5 cm, 88 cm and 132 cm from the radiator respectively. The plots are built selecting events +with the particle cluster inside a square of a 3 × 3 mm2 area, in the centre of the TimePix3 detector. +All the distributions shown in the above plots are normalized to the total number of selected events. +The top panels of each figure show the distributions of the relative positions of the TR X- +rays (evaluated from the X-ray clusters) with respect to the radiating electron (evaluated from the +particle cluster). As expected, TR photons tend to accumulate in rings centered on the position of +the radiating particle and the number of photons per electron increases with the beam momentum +(and consequently with the Lorentz factor of the radiating particles). +The central panels show the distributions of the TR X-ray energies as a function of their +angular separation from the radiating particle. Most X-rays are emitted at angles 𝜃 ≲ 2 mrad from +the radiating particle, with energies peaked at energies < 10 keV. A second peak of X-rays emitted +at angles ∼ 3.5 mrad and with the same energies as the first peak can also be seen, and it becomes +more evident as the beam momentum increases. +Finally, the bottom panels show the energy distributions of the absorbed TR X-rays compared +with the distributions of the energies deposited by the parent electrons in the TimePix3 detector. +As discussed in Sec. 4.1, the energy losses of the electrons follow Landau distributions with a most +probable value of 25.4 keV, while X-ray energies are peaked at less than 10 keV. We see that the +area of the X-ray energy spectra increases with increasing electron momentum. This behaviour is +– 10 – + +Entries7541667 +Mean +6.177 +RMS +1.049 +10 +10~3 +10-4 +0 +20 +Numberofclusters inthebeamtelescopeFigure 7. Distribution of the distances of the "particle clusters" from the track in the silicon sensor for the +runs performed with electrons crossing the EXTRA radiator, placed at a distance of 88.9 cm from the silicon +pixel sensor. +expected since the spectra are normalized to the total number of electrons and the TR yield increases +with the Lorentz factor of the radiating particle. +A summary of the results obtained in all the configurations explored is shown in Fig. 11. +The average number of detected TR X-rays per electron is shown as a function of the beam +momentum. We see that for all configurations the number of detected photons increases with the +beam momentum and saturates above 4 GeV/c. This behavior is expected, since the threshold +Lorentz factor for all radiators is 𝛾𝑡ℎ𝑟 ≃ 103 and the saturation Lorentz factors are in the range +4÷5×103. Comparing the results obtained with the EXTRA radiator in the different configurations +we see that the average number of detected TR X-rays decreases when the radiator-detector distance +is increased. The increase of the distance causes an increase of the X-ray absorption in the air gap +between the radiator and the detector, which is not compensated by the lower minimum detectable +angle between the photons and the radiating particles. We remark here that the results shown in +Fig. 11 referred to the CERN radiator have been obtained from a joint analysis of the data samples +collected with both the electron and positron beams (see Tab. 2). This choice is motivated by the +fact that the separate analyses of the electron and positron data samples yield the same results. This +feature was expected, since the properties of TR are independent of the sign of the charge of the +radiating particle. +The experimental results shown in Fig. 11 are compared with the predictions obtained by +folding the TR yield, evaluated with the theoretical formulae for regular radiators [4, 21] with the +X-ray absorption probabilities in the air gap between the radiator and the TimePix3 detector and in +– 11 – + +X10~3 +Entries7541668 +25 +Mean +0.0408 +RMS +0.0645 +20 +Fractionofevents +15 +10 +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Distance betweenthe track and theparticle cluster (mm)Figure 8. +Summary of the results obtained in the runs with the EXTRA radiator at 40.5 cm from the +TimpePix3 detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with +respect to the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their +angular separation from the electrons; bottom panel: electron and X-ray energy distributions. +– 12 – + +EXTRA radiator, d = 40.5 cm +10 +Ypart(mm) +> +6Gevc +2 +0 +X- Xpart (mm)EXTRA radiator,d = 40.5 cm +50 +Gevic +2Gevic +=3GeV/c +40 +30 +/Electron +(keV) +20 +Energy +Photons/ +10 +0 +50 +Photon +5Gev +=6Gev +40 +30 +20 +10 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +Electron-Photonangularseparation(mradEXTRA radiator, d = 40.5 cm +0.12 +p = 1 GeV/c +p = 2 GeV/c +p = 3 GeV/c +Electron +Electron +Electron +0.10 +Photons +Photons +Photons +0.08 +0.06 +Entries +0.04 +0.02 +of +Fraction +0.00 +0.12 +p = 4 GeV/c +p = 5 GeV/c +p = 6 GeV/c +Electron +Electron +Electron +0.10 +Photons +Photons +Photons +0.08 +0.06 +0.04 +0.02 +0.00 +0 +20 +40 +60 +80 +1000 +20 +40 +60 +80 +100 0 +20 +40 +60 +80 +100 +Energy (keV)Figure 9. Summary of the results obtained in the runs with the EXTRA radiator at 88 cm from the TimpePix3 +detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with respect to +the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their angular +separation from the electrons; bottom panel: electron and X-ray energy distributions. +– 13 – + +EXTRA radiator, d = 88 cm += +Gevlo += +2GeVIc +3GeVIc +10-3 +Photons/Electron +2 +=4 GeV/c += +5GeVic +6GeV/c +- +10-5 +0 +2 +2 +0 +1 +2 +-2 +2 +1 +2 +X - Xpart (mm)EXTRA radiator, d = 88 cm +50 +1 GeV/o +2Gev/o +=3GeV/o +40 +30 +10 +lectror +20 +10 +- +E +Energy +10 +itons/ +0 +Phot +50 +Photon +p=4GeV/c +of +40 +10-5 +er +lumbe +30 +10-6之 +20 +10 +0 +2 +2 +6 +8 +0 +4 +6 +8EXTRA radiator, d = 88 cm +0.12 +p = 1 GeV/c +p = 2 GeV/c +p = 3 GeV/c +Electron +Electron +Electron +0.10 +Photons +Photons +Photons +0.08 +0.06 +Entries +0.04 +0.02 +of +Fraction +0.00 +0.12 +p = 4 GeV/c +p= 5 GeV/c +p = 6 GeV/c +Electron +Electron +Electron +0.10 +Photons +Photons +Photons +0.08 +0.06 +0.04 +0.02 +0.00 +0 +20 +40 +60 +80 +1000 +20 +40 +60 +80 +100 0 +20 +40 +60 +80 +100 +Energy (keV)Figure 10. Summary of the results obtained in the runs with the EXTRA radiator at 132 cm from the +TimpePix3 detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with +respect to the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their +angular separation from the electrons; bottom panel: electron and X-ray energy distributions. +– 14 – + +EXTRA radiator, d = 132 cm +Gew +p = 2 GeV/c +=3GeV/C +10 +()d +> +=4GeV/C +p=5GeV/c +- +0 +0 +1 +0 +1 +2 +-2 +-1 +0 +1 +2 +X - Xpart (mm)EXTRA radiator, d = 132 cm +50 +=2GeV/o +3Gev/c +40 +30 +10 +lectron +20 +Energy +10 +Photons +S +50 +Photon +40 +Jumber +30 +10~5之 +20 +10 +1 +0 +2 +4 +6 +8 +0 +2 +4 +6 +0 +2 +4 +8 +Electron-Photonangularseparation(mradEXTRA radiator, d = 132 cm +0.12 +p = 1 GeV/c +p = 2 GeV/c +p = 3 GeV/c +Electron +Electron +Electron +0.10 +Photons +Photons +Photons +0.08 +0.06 +Entries +0.04 +0.02 +of +Fraction +0.00 +0.12 +p = 4 GeV/c +p= 5 GeV/c +Electron +Electron +0.10 +Photons +Photons +0.08 +0.06 +0.04 +0.02 +0.00 +0 +20 +40 +60 +80 +1000 +20 +40 +60 +80 +100 0 +20 +40 +60 +80 +100 +Energy (keV)Figure 11. Average number of TR photons as a function of electron beam momentum for the three radiator +types and for the different distances from the TimpePix3 detector. The dashed lines show the predictions +for the different configurations. The results obtained in a run without radiator and in two runs with dummy +radiators are also shown. +the silicon layer of the detector. The theoretical curves seem to be in a reasonable agreement with +the experimental results. +Finally, we have performed some control runs to check our results. +A run with 5 GeV/c +electrons without any radiator was performed to evaluate the possible contamination to the detected +TR signal from bremsstrahlung photons produced in the upstream materials and accompanying +the beam particles and the possible contamination from noisy pixels. In this run we found about +0.03 X-rays per electron; in addition, since all X-ray clusters are found very close to the particle +cluster, the contamination from noisy pixels can be considered negligible. We also performed two +additional runs with 6 GeV/c electrons, in which we replaced the radiator with some "dummy" +radiators: in particular, we used a set of paper towels, which were arranged in a stack simulating +a regular radiator, and a piece of sponge, which simulates an irregular radiator1. In both cases we +observed a TR signal of about 0.17 X-rays per electron. +6 +Conclusions +In the framework of the BL4S competition we have designed and implemented an experiment to +measure the TR emitted by fast electrons and positrons crossing different kind of radiators. The +1Irregular radiators made of foams or fiber mats are sometimes used in TRDs. +– 15 – + +1.2 +Photons +EXTRA 40.5 cm +CERN 88.4 cm +EXTRA 88.0 cm +No Radiator +1.0 +EXTRA132cm +Towels 88.4cm +INFN88.9cm +Sponge 88.4 cm +TR +0.8 +0.6 +0.4 +Average +0.2 +0.0 +0 +1 +2 +3 +4 +5 +6 +Electronmomentum(GeY/c)measurement has been performed at the DESY II Test Beam Facility area TB21, using electrons and +positron beams with momenta up to 6 GeV/c. We have measured the energy spectra and the angular +distribution of the TR X-rays using a 100 𝜇m thick pixel silicon detector, with a pitch of 55 𝜇m. +The experimental results are well reproduced by the theoretical curves obtained from standard TR +models. +BL4S has offered the students the chance to be actively involved in all the aspects of an +experimental research: during the preparation of the proposal, they have learned how to design +an experimental setup, optimizing the detectors available for the measurement; after their proposal +was selected, they have been involved in the design and in the assembly of their own radiator; then, +at DESY, they had the chance to run a real beam test; finally, they have taken part to the analysis of +the data collected in the test. However, the most important educational result of this experience is +that the students learned how to apply the scientific approach not only in the field of research, but +also to the solution of everyday life challenges. +Acknowledgments +The members of the EXTRA team thank the CERN and DESY support scientists, the beamline +scientists, the volunteers and the BL4S organisers who helped them during the preparation and the +implementation of their experiments. All the scientists involved in the competition dedicated a lot +of their time to answer all the questions the students had, giving them precious advises for their +future career. The team was really pleased to find such wonderful people, who showed them what +unconditional love for science really means. +A big thank to the Teomizli team from Mexico, the other winning team of the BL4S 2021. +Meeting peers from the other side of the world and work with them as a unique team of scientists +has been an enriching opportunity. +Beamline for Schools is an education and outreach project funded by the CERN & Society +Foundation and supported by individual donors, foundations and companies. In 2021, the project +was funded by the Wilhelm and Else Heraeus Foundation. Additional contributions have been +received from the Arconic Foundation, Amgen Switzerland AG, and the Ernest Solvay Fund +managed by the King Baudouin Foundation. +The EXTRA team also acknowledges financial support from CERN and DESY for their +participation to the beam test campaign. +The EXTRA team thanks B. Fanti, I. Iusco, D. Ricchiuti and all the personnel of the Liceo +Scientifico “A. 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Corryvreckan: a modular 4D track +reconstruction and analysis software for test beam data. Journal of Instrumentation, 16(03):P03008, +mar 2021. +– 17 – + +[20] Hans Bichsel. Straggling in Thin Silicon Detectors. Rev. Mod. Phys., 60:663–699, 1988. +[21] C. W. Fabjan and W. Struczinski. Coherent Emission of Transition Radiation in Periodic Radiators. +Phys. Lett. B, 57:483–486, 1975. +– 18 – + diff --git a/INFIT4oBgHgl3EQfYSsB/content/tmp_files/load_file.txt b/INFIT4oBgHgl3EQfYSsB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32c0c6c61963a211fa04aeb793e84d53dd25329e --- /dev/null +++ b/INFIT4oBgHgl3EQfYSsB/content/tmp_files/load_file.txt @@ -0,0 +1,575 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf,len=574 +page_content='Prepared for submission to JINST The EXTRA-BL4S experiment for the measurement of the energy and angular distributions of transition radiation X-rays M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Mazziotta,𝑎,1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Loparco, 𝑎,𝑏,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Anelli,𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Belviso,𝑐 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Buquicchio,𝑐 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Cassano,𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' De Cosmo,𝑐 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Ginefra,𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Martulli,𝑐 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Picci,𝑐 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Picicci,𝑐 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Soriano,𝑐 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Tatulli,𝑐 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Tripaldella,𝑐 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Zupo,𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Muscarella,𝑐 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Turbacci,𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Boselli,𝑑 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' da Cruz E Silva,𝑑,2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Joos𝑑 and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Schütze𝑒 𝑎Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via Orabona 4, I-70126 Bari, Italy 𝑏Dipartimento di Fisica dell’Università e del Politecnico di Bari, via Amendola 173, I-70126 Bari, Italy 𝑐The EXTRA Team Liceo Scientifico Statale "A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Scacchi", Corso Cavour 241, I-70121 Bari, Italy 𝑑CERN, the European Organization for Nuclear Research, Esplanade des Particules 1, 1211 Geneva, Switzerland 𝑒DESY, Notkestrasse 85, D-22607 Hamburg E-mail: mazziotta@ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='it, francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='loparco@ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='it Abstract: We have designed and implemented an experiment to measure the angular distributions and the energy spectra of the transition radiation X-rays emitted by fast electrons and positrons crossing different radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Our experiment was selected among the proposals of the 2021 Beamline for Schools contest, a competition for high-school students organized every year by CERN, and was performed at the DESY II Test Beam facility area TB21, using a high-purity beam of electrons or positrons with momenta in the range from 1 to 6 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The measurements were performed using a 100 𝜇m thick silicon pixel detector, with a pitch of 55 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Our results are consistent with the expectations from the theoretical models describing the production of transition radiation in multilayer regular radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Keywords: Transition radiation detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Particle identification methods 1Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2Now at LIP - Laboratório de Instrumentação e Física Experimental de Partículas Avenida Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Gama Pinto 2, Complexo Interdisciplinar (3is), 1649-003 Lisboa, Portugal arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='11247v1 [hep-ex] 26 Jan 2023 Contents 1 Introduction 1 2 The BL4S competition 1 3 The EXTRA experiment 3 4 Data analysis 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='1 Conversion & Clustering 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 Detector alignment procedure 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='3 Data selection and analysis 8 5 Results 10 6 Conclusions 15 1 Introduction In recent years, high-school physics curricula increasingly include topics related to modern high- energy physics and particle detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Universities and research centers promote several programs to bring high-school students in touch with modern physics and the scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The Liceo Scientifico “A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Scacchi” in Bari has taken part in such projects for years, and in 2021 the school promoted the participation of a team of students of the 12𝑡ℎ and 13𝑡ℎ grade in the Beamline for Schools (BL4S) competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' BL4S is organized by CERN in collaboration with DESY, and offers to groups of high-school students the unique opportunity to propose a scientific experiment at a particle accelerator facility and to win a trip to perform it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Because of the maintenance of CERN accelerators, the experiment was performed at the DESY II Beam Test facility in Hamburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The students, coordinated by their physics teachers and under the supervision of experienced researchers from the Physics Department of the Bari University and from the INFN Unit in Bari, won the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The goal of the experiment conceived by the team was to study the transition radiation emitted by fast electrons and positrons crossing different kinds of radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This paper provides a short presentation of the BL4S competition and presents the experiment and the result obtained by the team during their beam time in Hamburg in September 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2 The BL4S competition Beamline for Schools (BL4S) [1] is a physics competition organised by CERN and DESY, which invites high-school students from all over the world to propose an experiment to be performed at a – 1 – particle accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Each team has to write an original scientific proposal, explaining the theoretical background of the selected topic, and describing both the procedure to carry it out at a test beam facility and the results that they expect to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A jury of experts, including scientists of CERN and DESY, review the proposal and select two teams (three from 2022 on) that win a trip to a fully equipped beam line of a particle accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' From 2014 to 2018 the winning experiments took place at the test beam area of the CERN Proton Synchrotron (PS) accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In 2019 the competition moved to the DESY II Test Beam Facility (Hamburg) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The partnership between CERN and the German laboratory allowed BL4S to continue during the three-year long shutdown of the CERN accelerator complex for upgrade and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The competition is structured in several preparatory phases, which include conferences and meetings with the organisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Once the competition is announced, usually in Autumn, interested teams start preparing their proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Teams can include students either from the same school or from different schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Having teams representing two schools or more is not unusual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' During the proposal preparation, students are involved in an intense research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' After the conception and design of their experiment, the participants must write a well structured proposal and submit it on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The students are not alone in this process, but they are guided by their coaches, who provide them with details on particle physics and teach them the necessary technical skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Team coaches can be teachers, parents or scientists of local universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' It is important that students are well aware of each scientific detail of the proposed experiment, so that the theoretical background is clear and solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The students are required to write down in detail how they intend to use the particle beam for their measurements and which equipment and detectors they need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Moreover, participants often complement their theoretical hypothesis with computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In fact, it is fundamental that students acquire the rudimentary programming skills that will be required in case of victory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Lastly, conclusions must contain the team’s expectations and motivation, which play a significant role in the jury’s decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The BL4S organisers are always available to answer questions that the teams might have during the preparation of their proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Many teams contact them to discuss the feasibility of their experiments or practical problems that they encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In the final phase of the competition, a jury consisting of more than 50 volunteers selects the teams that are invited to a research institute to perform their experiment together with support scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Prior to the visit, the winning teams work remotely with the BL4S scientists to refine their experiments and perform a detailed planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The beam time of the winning teams usually happens just after the summer, and the students have 12 full days of access to the experimental area to perform their measurements, supervised by the support scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' During their stay, they work as a team of professional scientist would do and they complement their scientific experience with visits and lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' After taking the data at the beam line, the teams are encouraged to analyse their data to answer the scientific question of the initial proposal, and to write a paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' During this phase, the team members stay in close contact with the BL4S support scientists and the team coaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 2 – Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Schematic view of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 3 The EXTRA experiment The EXTRA (Electron X-ray Transition RAdiation) experiment is designed to study the transition radiation (TR) [3] emitted by fast electrons and positrons crossing different radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Highly relativistic particles crossing the boundary between materials with different dielectric constants can produce TR in the X-ray region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' However, since the yield of TR photons emitted at a single interface is considerably small (it is of the order of the fine structure constant 𝛼 ≈ 1/137), multiple boundaries are needed to enhance the X-ray production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Periodic radiators, consisting of stacks of thin foils of dielectric material separated by thicker air gaps, are commonly used in transition radiation detectors (TRDs) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The main features of the TR emitted by a periodic radiator depend on the kinematic properties of the radiating particles and on the radiator properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' They can be summarized as follows [5]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The effective TR photon emission starts at a threshold Lorentz factor, which is given by 𝛾𝑡ℎ𝑟 = 𝑑1𝜔1/𝑐, where 𝑑1 is the thickness of the foils, while 𝜔1 is the plasma frequency of the foil material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The TR emission increases with the Lorentz factor 𝛾 until it reaches saturation at 𝛾𝑠𝑎𝑡 = 𝛾𝑡ℎ𝑟 √︁ 𝑑2/𝑑1, where 𝑑2 is the thickness of the air gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Most of the TR energy is emitted near the energy ℏ𝜔𝑚𝑎𝑥 = ℏ𝜔2 1𝑑1/2𝜋𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The angular distribution of TR photons exhibits a few maxima and extends up to 𝜃𝑚𝑎𝑥 = √︃ 1/𝛾2 + 𝜔2 1/𝜔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We have designed an experimental setup to measure the energy spectra and the angular distributions of the TR X-rays emitted by fast electrons and positrons crossing different radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Similar measurements were performed in the past at the CERN SPS with beams of 20 GeV/c electrons and of 120, 180 and 290 GeV/c muons, using silicon strip detectors [6], silicon pixel – 3 – Beam Radiator Timepix3 BeamTelescopeScintillatorsFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Pictures of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 4 – beamling forschools cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='ch/bl4sRadiator Foil/gap material d1 (𝜇m) d2 (𝜇m) N 𝑓 EXTRA polyethylene/air 23 500 150 INFN polyethylene/air 25 300 155 CERN polyethylene/air 25 240 190 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Parameters of radiators used in the beam test: 𝑑1 and 𝑑2 are the thickness of the foils and the gap respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 𝑁 𝑓 is the number of foils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Radiator distance ( cm) Beam particle Beam momenta ( GeV/c) EXTRA 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 𝑒− 1, 2, 3, 4, 5, 6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0 𝑒− 1, 2, 3, 4, 5, 6 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0 𝑒− 1, 2, 3, 4, 5, 6 INFN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9 𝑒− 1, 2, 3, 4, 5, 6 CERN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 𝑒−/𝑒+ 1, 2, 3, 4, 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Summary of the data taking configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' For each radiator the beam particle, their momenta and the distance between the radiator and the X-ray detector are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' detectors [7, 8] and GaAs pixel detectors [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Parallel to the measurements, an effort to develop accurate Monte Carlo simulations of the TR process is being carried out [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' One of the goals of these activities is that of exploiting TR for the identification of charged hadrons in the TeV energy region [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In this region all hadrons have Lorentz factor exceeding the typical threshold values for TR production (usually 𝛾𝑡ℎ𝑟 ∼ 500 ÷ 1000), and the simultaneous measurement of the energies and of the emission angles of TR X-rays can help to discriminate among different hadron species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Our measurements were performed at the DESY II Test Beam Facility [2] area TB21, using a beam of either electrons or positrons with momenta in the range from 1 to 6 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A scheme of the setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 1, while some pictures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The radiator is followed by a Timepix3 assembly containing a thin silicon pixel sensor, which is used to detect the TR X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A downstream beam telescope, composed by an array of six silicon pixel detectors, is used to reconstruct the tracks of the beam particles [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A set of two trigger scintillators, located downstream of the last plane of the beam telescope, is used for triggering the data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In our experiment we used three different radiators, which in the following will be labelled as "EXTRA", "INFN" and "CERN" respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Their features are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In particular, the EXTRA radiator was assembled for this measurement by the students at the Liceo Scientifico "A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Scacchi" in Bari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 3 shows some picture taken during the assembly of this radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The INFN and CERN radiators were borrowed from the Bari INFN Group and were used in a beam test campaign performed in 2006 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' With these radiators, several measurements were performed, changing the beam composition and momentum, and the distance between the radiator and the X-ray detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The different data taking configurations are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The TR X-rays were detected by a 100 𝜇m thick silicon sensor, bump-bonded to a Timepix3 readout chip [14], consisting of a pixel matrix of 256×256 pixels with a pitch of 55 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This silicon – 5 – Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Assembly of the EXTRA radiator at the Liceo Scientifico "A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Scacchi".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' detector assembly was placed such that the sensor faces the radiator to mitigate prior absorption in the readout chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The sensor of the assembly with the ID W5_E2 was operated at a bias voltage of −21 V to ensure full depletion [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The pixel pitch of the silicon sensor and its distance from the radiator determine the minimum detectable angular separation 𝜃𝑚𝑖𝑛 of TR X-rays from the direction of the radiating particles, as they should be separated of at least one pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Its value is in fact given by 𝜃𝑚𝑖𝑛 ≳ 𝑤/𝑑, where 𝑤 = 55 𝜇m is the pixel pitch and 𝑑 is the distance of the silicon detector from the radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The configurations with larger distances allow to detect smaller angular separations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' however, due to the X-ray absorption in the radiator and in the air gap between the radiator and the sensor, the number of detected TR X-rays will decrease with the distance from the radiator, and the angular resolution will deteriorate due to multiple Coulomb scattering of the primary particles in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' While the TR X-rays are likely absorbed by the front sensor, the radiating charged particles traverse the detector and leave an ionization track in the detectors of the beam telescope, which consists of an array of six regularly spaced silicon pixel detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In this configuration, scattering in air is limited to a minimum, enabling a track resolution of a few 𝜇m extrapolated to the Timepix3 detector [12], which is more than sufficient for an identification of the charged particle among two or more clusters in the Timepix3 detector with cluster distances larger than a pixel pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Finally, the two scintillators, approximately shadowing the size of the telescope sensor planes and located at the end of the beam line, are used for triggering the data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The data acquisition was performed using the software framework EUDAQ2 [16], which – 6 – 语integrates the control and readout of the Timepix3 assembly and the beam telescope, and features a graphical user interface for the configuration of connected devices, starting and stopping runs and data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' An AIDA TLU [17] was used to form a trigger signal as a coincidence of the signals from the two scintillators while enabling a busy-handshake with the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4 Data analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='1 Conversion & Clustering The raw data contains a collection of hit pixels per detector plane per trigger, which defines a so-called "event", including the corresponding pixel addresses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' for the data from the Timepix3 assembly, the corresponding information on the energy deposit, in form of a digitised signal, is also stored, while for the beam telescope no charge information is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The collected data are converted to ROOT TTree format [18] using the data analysis framework Corryvreckan [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In addition, this software performs a clustering procedure, which identifies adjacent hit pixels and connects them to form a so-called "cluster" under the hypothesis that pixel hits in one cluster are caused by a single incident particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The cluster center, as an estimation on the incidence position of the particle, is calculated either as the center-of-gravity using the charge information, or as the arithmetic mean of the pixel hit positions in case of binary hit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The energy calibration of the silicon pixel detector is performed assuming that the most probable energy loss of 5 GeV/c electrons crossing a 100 𝜇m thick silicon layer is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='41 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This value has been calculated using a dedicated Monte Carlo simulation developed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Bichsel for the calculation of the energy losses of charged particles in thin silicon absorbers [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 Detector alignment procedure The positions of the clusters in each silicon detector are evaluated in the local detector reference frame, with the 𝑧-axis oriented along the beam direction and the 𝑥 − 𝑦 plane corresponding to the detector plane, with the origin in the center of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In the global reference frame the 𝑧-axis is also directed along the beam direction, and the detectors are disposed on planes parallel to the 𝑥 − 𝑦 plane, with their centers at the coordinates (𝑥𝑖 0, 𝑦𝑖 0, 𝑧𝑖 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Due to mechanical tolerances in the assembly of the detectors, the coordinates (𝑥𝑖 0, 𝑦𝑖 0) are slightly misaligned with respect to the reference values (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A dedicated alignment run has been therefore performed to evaluate the coordinates (𝑥𝑖 0, 𝑦𝑖 0) of the centers of the silicon detectors (the index 𝑖 = 0 refers to the Timepix3 sensor, while the indices 𝑖 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 6 refer to the detectors of the beam telescope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The alignment run has been performed removing the radiator from the beam line and using 5 GeV/c electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We have implemented an iterative alignment procedure selecting a sample of events with only one cluster in each silicon detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This choice is aimed to select events with only one electron track across all the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In the first iteration we assume 𝑥𝑖 0 = 0 and 𝑦𝑖 0 = 0 for all detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We fit all the tracks with a straight line and, for each track, we evaluate the residuals in each detector as 𝑟𝑖 𝑥 = 𝑥𝑖 − 𝑥𝑖 𝑓 𝑖𝑡 and 𝑟𝑖 𝑦 = 𝑦𝑖 − 𝑦𝑖 𝑓 𝑖𝑡, where (𝑥𝑖, 𝑦𝑖) and (𝑥𝑖 𝑓 𝑖𝑡, 𝑦𝑖 𝑓 𝑖𝑡) are respectively the true and fitted positions of the cluster in the 𝑖-th detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We then build the distributions of the residuals 𝑟𝑖 𝑥 and 𝑟𝑖 𝑦 and, in the next iteration, we set 𝑥𝑖 0 = −𝜇𝑖 𝑥 and 𝑦𝑖 0 = −𝜇𝑖 𝑦, where 𝜇𝑖 𝑥 and 𝜇𝑖 𝑦 are the average values – 7 – Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Distributions of the residuals in the silicon detector equipped with the TimePix3 chip after the alignment procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' of these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The iterative procedure is terminated when |𝜇𝑖 𝑥| < 1 𝜇m and |𝜇𝑖 𝑦| < 1 𝜇m for all detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Convergence is reached after the second iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4 shows the distributions of the residuals in the silicon detector equipped with the Timepix3 chip after the alignment procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The RMS of the residual distributions in both the 𝑥 and 𝑦 views are of about 10 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 5 shows the distributions of the direction cosines of the electron tracks in the alignment run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We see that the average values of the direction cosines 𝑐𝑥 and 𝑐𝑦 are slightly different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This result implies that the 𝑧-axis of our reference frame is not perfectly aligned with the direction of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The tilt angle can be estimated from the average value of 𝑐𝑧, and is of about 5 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Finally, from the values of the RMS of the distributions of 𝑐𝑥 and 𝑐𝑦 we can deduce that the beam divergence is of about 1 mrad in both the 𝑥 and 𝑦 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='3 Data selection and analysis As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 3, several runs in different configurations have been taken, by changing the beam composition and momentum, the radiator and its distance from the silicon pixel detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In each of these runs we have selected events with at least one cluster in the silicon pixel sensor and at least 3 clusters in different detectors of the beam telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This choice is motivated by the need of identifying, among the clusters in the silicon sensor, the one originated by the ionization energy deposit of the beam particle and those eventually originated by the absorption of TR X-rays produced in the upstream radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 6 shows the distribution of the total number of clusters in the detectors of the beam telescope for all the runs performed with electrons crossing the EXTRA radiator, which was placed at a distance of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9 cm from the silicon pixel sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' As expected, the distribution is peaked at 6 clusters, corresponding to clean electron tracks, yielding one cluster in each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Events with less than 6 clusters can be originated from inefficiencies of some detectors in the beam telescope or from beam particles which do not cross all the telescope planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Events with more than 6 clusters can be originated from delta rays accompanying the primary electron track or from TR X-rays passing through the upstream silicon sensor and being absorbed in any detector of the silicon – 8 – X103 Entries 542808 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 Mean 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='227e-04 RMS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='064e-02 x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' / ndf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='047e+04/2389 Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='126e+03±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='961e+00 Mean 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='303e-04±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='287e-05 Sigma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='346e-03±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='969e-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='8 ents 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 0 /x10~3 60 40 20 0 20 40 60 Residualsinthex-view(mmX103 Entries 542808 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 Mean 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='268e-04 RMS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='050e-02 x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' / ndf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='044e+04/2379 Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='140e+03±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='981e+00 Mean 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='287e-04± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='271e-05 Sigma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='229e-03±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='775e-06 ents 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 /x10~3 0 60 40 20 0 20 40 60 Residuals in the y-view (mm)Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Distributions of the direction cosines of the electron tracks in the silicon detector and in the beam telescope in the alignment run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We also see two peaks, at 12 and 18 clusters respectively, which include less than 1% of the total number of events, and which likely correspond to double and triple electron tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The clusters in the detectors of the beam telescope are used to reconstruct the tracks of the beam particles in the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' To select events with single electron (positron) tracks, we require less than 10 clusters in the beam telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Candidate tracks are built by selecting all the possible cluster combinations with only one cluster per plane of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The clusters of each candidate track are then fitted with a straight line and the 𝜒2 of the fit is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The track with the best 𝜒2 is then selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Once the track of the radiating particle in the beam telescope is reconstructed, we evaluate the coordinates (𝑥𝑡𝑟𝑎𝑐𝑘, 𝑦𝑡𝑟𝑎𝑐𝑘) of its intersection with the upstream silicon pixel sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Then, if more clusters are found in the sensor, the cluster nearest to the track is associated to the particle ("particle cluster"), while other clusters are associated to possible TR X-rays ("X-ray clusters").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Clearly, if only one cluster is found in the silicon pixel sensor, it is associated to the particle and no X-rays are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 9 – X103 Entries 542808 Mean 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='254e-03 25 RMS 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='283e-04 20 Events 15 10 L0 ×10~3 10 8 6 4 2 0 2 4 6 80 10 CxX103 Entries 542808 30 Mean -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='903e-03 RMS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='606e-04 25 20 Events 15 10 LO 0 /×10~3 10 8 6 4 2 0 2 4 9 8 10 CyX103 Entries 542808 Mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='160e-05 30 RMS4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='175e-06 25 20 Events 15 10 L0 0 ×10~6 0 5 10 15 20 25 30 35 40 45 50 1-CzFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Distribution of the total number of clusters in the beam telescope for all the runs performed with electrons crossing the EXTRA radiator, placed at a distance of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9 cm from the silicon pixel sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 5 Results In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 8, 9 and 10 the results obtained in the runs with the EXTRA radiator are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The plots in each figure correspond to the configurations with the silicon detector placed at the distances of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm, 88 cm and 132 cm from the radiator respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The plots are built selecting events with the particle cluster inside a square of a 3 × 3 mm2 area, in the centre of the TimePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' All the distributions shown in the above plots are normalized to the total number of selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The top panels of each figure show the distributions of the relative positions of the TR X- rays (evaluated from the X-ray clusters) with respect to the radiating electron (evaluated from the particle cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' As expected, TR photons tend to accumulate in rings centered on the position of the radiating particle and the number of photons per electron increases with the beam momentum (and consequently with the Lorentz factor of the radiating particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The central panels show the distributions of the TR X-ray energies as a function of their angular separation from the radiating particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Most X-rays are emitted at angles 𝜃 ≲ 2 mrad from the radiating particle, with energies peaked at energies < 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A second peak of X-rays emitted at angles ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 mrad and with the same energies as the first peak can also be seen, and it becomes more evident as the beam momentum increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Finally, the bottom panels show the energy distributions of the absorbed TR X-rays compared with the distributions of the energies deposited by the parent electrons in the TimePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='1, the energy losses of the electrons follow Landau distributions with a most probable value of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 keV, while X-ray energies are peaked at less than 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We see that the area of the X-ray energy spectra increases with increasing electron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This behaviour is – 10 – Entries7541667 Mean 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='177 RMS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='049 10 10~3 10-4 0 20 Numberofclusters inthebeamtelescopeFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Distribution of the distances of the "particle clusters" from the track in the silicon sensor for the runs performed with electrons crossing the EXTRA radiator, placed at a distance of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9 cm from the silicon pixel sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' expected since the spectra are normalized to the total number of electrons and the TR yield increases with the Lorentz factor of the radiating particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A summary of the results obtained in all the configurations explored is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The average number of detected TR X-rays per electron is shown as a function of the beam momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We see that for all configurations the number of detected photons increases with the beam momentum and saturates above 4 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This behavior is expected, since the threshold Lorentz factor for all radiators is 𝛾𝑡ℎ𝑟 ≃ 103 and the saturation Lorentz factors are in the range 4÷5×103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Comparing the results obtained with the EXTRA radiator in the different configurations we see that the average number of detected TR X-rays decreases when the radiator-detector distance is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The increase of the distance causes an increase of the X-ray absorption in the air gap between the radiator and the detector, which is not compensated by the lower minimum detectable angle between the photons and the radiating particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We remark here that the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 11 referred to the CERN radiator have been obtained from a joint analysis of the data samples collected with both the electron and positron beams (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This choice is motivated by the fact that the separate analyses of the electron and positron data samples yield the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' This feature was expected, since the properties of TR are independent of the sign of the charge of the radiating particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The experimental results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 11 are compared with the predictions obtained by folding the TR yield, evaluated with the theoretical formulae for regular radiators [4, 21] with the X-ray absorption probabilities in the air gap between the radiator and the TimePix3 detector and in – 11 – X10~3 Entries7541668 25 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0408 RMS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0645 20 Fractionofevents 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9 1 Distance betweenthe track and theparticle cluster (mm)Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Summary of the results obtained in the runs with the EXTRA radiator at 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm from the TimpePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with respect to the electrons (particle clusters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' middle panel: distribution of X-ray energies as a function of their angular separation from the electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' bottom panel: electron and X-ray energy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 12 – EXTRA radiator, d = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm 10 Ypart(mm) > 6Gevc 2 0 X- Xpart (mm)EXTRA radiator,d = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm 50 Gevic 2Gevic =3GeV/c 40 30 /Electron (keV) 20 Energy Photons/ 10 0 50 Photon 5Gev =6Gev 40 30 20 10 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Electron-Photonangularseparation(mradEXTRA radiator, d = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 1 GeV/c p = 2 GeV/c p = 3 GeV/c Electron Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 Entries 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 of Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 4 GeV/c p = 5 GeV/c p = 6 GeV/c Electron Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0 20 40 60 80 1000 20 40 60 80 100 0 20 40 60 80 100 Energy (keV)Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Summary of the results obtained in the runs with the EXTRA radiator at 88 cm from the TimpePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with respect to the electrons (particle clusters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' middle panel: distribution of X-ray energies as a function of their angular separation from the electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' bottom panel: electron and X-ray energy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 13 – EXTRA radiator, d = 88 cm = Gevlo = 2GeVIc 3GeVIc 10-3 Photons/Electron 2 =4 GeV/c = 5GeVic 6GeV/c 10-5 0 2 2 0 1 2 2 2 1 2 X - Xpart (mm)EXTRA radiator, d = 88 cm 50 1 GeV/o 2Gev/o =3GeV/o 40 30 10 lectror 20 10 E Energy 10 itons/ 0 Phot 50 Photon p=4GeV/c of 40 10-5 er lumbe 30 10-6之 20 10 0 2 2 6 8 0 4 6 8EXTRA radiator, d = 88 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 1 GeV/c p = 2 GeV/c p = 3 GeV/c Electron Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 Entries 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 of Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 4 GeV/c p= 5 GeV/c p = 6 GeV/c Electron Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0 20 40 60 80 1000 20 40 60 80 100 0 20 40 60 80 100 Energy (keV)Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Summary of the results obtained in the runs with the EXTRA radiator at 132 cm from the TimpePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with respect to the electrons (particle clusters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' middle panel: distribution of X-ray energies as a function of their angular separation from the electrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' bottom panel: electron and X-ray energy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 14 – EXTRA radiator, d = 132 cm Gew p = 2 GeV/c =3GeV/C 10 ()d > =4GeV/C p=5GeV/c 0 0 1 0 1 2 2 1 0 1 2 X - Xpart (mm)EXTRA radiator, d = 132 cm 50 =2GeV/o 3Gev/c 40 30 10 lectron 20 Energy 10 Photons S 50 Photon 40 Jumber 30 10~5之 20 10 1 0 2 4 6 8 0 2 4 6 0 2 4 8 Electron-Photonangularseparation(mradEXTRA radiator, d = 132 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 1 GeV/c p = 2 GeV/c p = 3 GeV/c Electron Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 Entries 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 of Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='12 p = 4 GeV/c p= 5 GeV/c Electron Electron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='10 Photons Photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='00 0 20 40 60 80 1000 20 40 60 80 100 0 20 40 60 80 100 Energy (keV)Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Average number of TR photons as a function of electron beam momentum for the three radiator types and for the different distances from the TimpePix3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The dashed lines show the predictions for the different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The results obtained in a run without radiator and in two runs with dummy radiators are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' the silicon layer of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The theoretical curves seem to be in a reasonable agreement with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Finally, we have performed some control runs to check our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A run with 5 GeV/c electrons without any radiator was performed to evaluate the possible contamination to the detected TR signal from bremsstrahlung photons produced in the upstream materials and accompanying the beam particles and the possible contamination from noisy pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In this run we found about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='03 X-rays per electron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' in addition, since all X-ray clusters are found very close to the particle cluster, the contamination from noisy pixels can be considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We also performed two additional runs with 6 GeV/c electrons, in which we replaced the radiator with some "dummy" radiators: in particular, we used a set of paper towels, which were arranged in a stack simulating a regular radiator, and a piece of sponge, which simulates an irregular radiator1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In both cases we observed a TR signal of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='17 X-rays per electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' 6 Conclusions In the framework of the BL4S competition we have designed and implemented an experiment to measure the TR emitted by fast electrons and positrons crossing different kind of radiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The 1Irregular radiators made of foams or fiber mats are sometimes used in TRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' – 15 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 Photons EXTRA 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='5 cm CERN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 cm EXTRA 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0 cm No Radiator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0 EXTRA132cm Towels 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4cm INFN88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='9cm Sponge 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 cm TR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='4 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content='0 0 1 2 3 4 5 6 Electronmomentum(GeY/c)measurement has been performed at the DESY II Test Beam Facility area TB21, using electrons and positron beams with momenta up to 6 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' We have measured the energy spectra and the angular distribution of the TR X-rays using a 100 𝜇m thick pixel silicon detector, with a pitch of 55 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The experimental results are well reproduced by the theoretical curves obtained from standard TR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' BL4S has offered the students the chance to be actively involved in all the aspects of an experimental research: during the preparation of the proposal, they have learned how to design an experimental setup, optimizing the detectors available for the measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' after their proposal was selected, they have been involved in the design and in the assembly of their own radiator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' then, at DESY, they had the chance to run a real beam test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' finally, they have taken part to the analysis of the data collected in the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' However, the most important educational result of this experience is that the students learned how to apply the scientific approach not only in the field of research, but also to the solution of everyday life challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Acknowledgments The members of the EXTRA team thank the CERN and DESY support scientists, the beamline scientists, the volunteers and the BL4S organisers who helped them during the preparation and the implementation of their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' All the scientists involved in the competition dedicated a lot of their time to answer all the questions the students had, giving them precious advises for their future career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The team was really pleased to find such wonderful people, who showed them what unconditional love for science really means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' A big thank to the Teomizli team from Mexico, the other winning team of the BL4S 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Meeting peers from the other side of the world and work with them as a unique team of scientists has been an enriching opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Beamline for Schools is an education and outreach project funded by the CERN & Society Foundation and supported by individual donors, foundations and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' In 2021, the project was funded by the Wilhelm and Else Heraeus Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Additional contributions have been received from the Arconic Foundation, Amgen Switzerland AG, and the Ernest Solvay Fund managed by the King Baudouin Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The EXTRA team also acknowledges financial support from CERN and DESY for their participation to the beam test campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' The EXTRA team thanks B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Fanti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFIT4oBgHgl3EQfYSsB/content/2301.11247v1.pdf'} +page_content=' Iusco, D.' metadata={'source': 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b/K9AyT4oBgHgl3EQf6foW/content/tmp_files/2301.00821v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0c79a1fea5ce0e8edab6006ed55f4ed0b3ea656 --- /dev/null +++ b/K9AyT4oBgHgl3EQf6foW/content/tmp_files/2301.00821v1.pdf.txt @@ -0,0 +1,4266 @@ +CERN-TH-2023-001 +Anomalies From the Covariant Derivative Expansion +Timothy Cohen,1,2,3 Xiaochuan Lu,4,3 and Zhengkang Zhang5 +1 Theoretical Physics Department, CERN, 1211 Geneva, Switzerland +2 Theoretical Particle Physics Laboratory, EPFL, 1015 Lausanne, Switzerland +3 Institute for Fundamental Science, University of Oregon, Eugene, OR 97403, USA +4 Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA +5 Department of Physics, University of California, Santa Barbara, CA 91106, USA +E-mail: tim.cohen@cern.ch, xil224@ucsd.edu, zkzhang@ucsb.edu +Abstract: We revisit the calculation of anomalies for global and gauge symmetries +in the framework of the Covariant Derivative Expansion (CDE). Due to the presence +of UV divergences, the result is an ambiguous quantity that depends on the regular- +ization procedure and the renormalization scheme. We introduce a class of regulators +that facilitate a straightforward evaluation of the anomaly exclusively in d = 4 space- +time dimensions using the CDE methodology. We derive a master formula for the +anomaly that integrates various known results into a unified framework. +arXiv:2301.00821v1 [hep-ph] 2 Jan 2023 + +Contents +1 +Introduction +3 +2 +Anomalies in the Functional Formalism +4 +2.1 +Defining the Anomaly +4 +2.2 +Connection to the Path Integral Measure +7 +2.3 +Connection to Ward Identities +8 +3 +Regularizing the Anomaly +9 +3.1 +What is Regularization? +10 +3.2 +Anomaly as a Regulated Functional Trace +12 +3.3 +Connection to Other Regularization Prescriptions +17 +3.4 +Consistency With the Wess-Zumino Condition +18 +4 +Master Formula for the Anomaly From CDE +21 +4.1 +Evaluating the Dirac Traces +23 +4.2 +The Evaluated Master Formula +24 +5 +Implications of the Master Formula +27 +5.1 +Simple Non-Abelian Group +28 +5.2 +Product of Non-Abelian Sectors +30 +5.3 +Product of Abelian Sectors +30 +5.4 +Product of Abelian and Non-Abelian Sectors +34 +6 +Discussion and Future Directions +36 +Acknowledgments +37 +Appendix +37 +A +Comments on Cyclic Permutation +37 +A.1 +Internal Trace Notation +38 +A.2 +Conditions for Cyclic Permutations in Internal Traces +41 +References +44 +– 2 – + +1 +Introduction +It is well known that symmetries of the classical action can be broken by quantum +effects. This so-called anomaly has far-reaching consequences, from explaining the +neutral pion decay to providing critical consistency checks on gauge theories with chi- +ral fermions. Well-established techniques exist for computing anomalies both using +Feynman diagrams, and also directly from the path integral. +They can be com- +puted for global and gauged symmetries, Abelian and non-Abelian groups, and take +‘consistent’ and/or ‘covariant’ forms. The results generally depend on the choice of +regulator, and consist of a relevant piece that reflects the IR properties of the theory, +and an irrelevant piece that can be absorbed by varying the renormalization scheme.1 +In this paper, we revisit the calculation of anomalies from the path integral us- +ing an approach known as the Covariant Derivative Expansion (CDE). This allows +us to derive a unified framework that incorporates various types of anomalies into +one master formula. The CDE was originally invented in the mid-1980s [15–17] to +facilitate one-loop calculations of correlation functions purely in terms of functional +traces, avoiding the introduction of Feynman diagrams. In recent years, the method +has been applied in a variety of new settings, which has led to significant theoretical +developments. These include the discovery of a variation on the framework, ‘sim- +plified CDE’ [18, 19], the incorporation of the method of regions [20], organizing +schemes using diagrammatic frameworks [21, 22], as well as techniques that yield +effective actions that include all orders in the fields [23]. With these developments, +the power and efficiency of CDE has been demonstrated for connecting the UV with +the IR, i.e., computing low-energy Effective Field Theories (EFTs) from integrating +out heavy states in a perturbative UV model; see e.g. Ref. [24] for a review. We +now know how to use CDE to perform matching calculations across a mass thresh- +old, as well as to extract the renormalization group evolution equations for the EFT +couplings. The CDE has become such a well-developed tool that there now exist +packages which automate these calculations [25–27]. +The practical success of CDE in connecting UV and IR descriptions of quantum +field theories motivates applying it to compute the anomaly. The approach taken +here will be to work exclusively in d = 4 spacetime dimensions, which allows us to +avoid any of the complications that arise when attempting to define Weyl fermions +in dimensional regularization.2 In this paper, we generalize the classic Fujikawa ap- +1There is of course a vast literature on the anomaly, including the excellent reviews Refs. [1, 2]. +The story began with its discovery in 1969 by Adler [3] and by Bell and Jackiw [4]. It was soon +after understood to be one-loop exact [5]. The connection between the anomaly and the topological +winding number of the gauge field was discovered in Refs. [6–9]. +Of great importance to the +approach taken here is Fujikawa’s derivation of the anomaly from the non-invariance of the path +integral measure [10–14]. +2It is well known that handling the γ5 matrix in d ̸= 4 spacetime dimensions is a nontrivial task +[28–31]. See Refs. [32, 33] for recent CDE calculations of anomalies with dimensional regularization. +– 3 – + +proach by expressing the anomaly as a functional trace, which must be regularized to +be well-defined. We introduce a novel regularization prescription, with a set of reg- +ulators parameterized by a set of numbers collectively denoted by β, which one can +choose based on which symmetries one wishes to preserve. We emphasize that our +regularization yields unambiguous evaluation results once the values of β are speci- +fied. We derive a master formula for the anomaly using CDE, whose explicit forms +are given by Eqs. (4.15) and (4.17). This master formula encodes a variety of known +results for anomaly calculations. In particular, we examine all possible combinations +of continuous symmetry groups, and show in each case how our master formula re- +produces the known (relevant) anomaly results, as well as the anomaly cancellation +conditions. This establishes that the CDE can accommodate this important effect +in perturbative quantum field theory, and sets the stage for its applications to EFT +matching across anomalous thresholds. +The rest of this paper is organized as follows. We first review the functional +formalism in Sec. 2, with an emphasis on the definition of the anomaly and its con- +nections to the fermionic path integral measure and the anomalous Ward identities. +In Sec. 3, we isolate the functional trace that encodes the anomalies and introduce +our novel regularization prescription to make it well-defined. We discuss the relation +between our regulator and some similar approaches in the literature, and also a suf- +ficient condition for it to be consistent with the Wess-Zumino condition. In Sec. 4, +we carry out the CDE evaluation to obtain our master formula for the anomaly. We +then demonstrate in Sec. 5 that this master formula reproduces various known results +regarding anomalies by examining all possible combinations of continuous symme- +try groups. Some future directions are discussed in Sec. 6. A technical clarification +regarding CDE manipulations is provided in App. A. +2 +Anomalies in the Functional Formalism +In this section, we briefly review the well-known functional formalism for anomalies, +which also serves the purpose of introducing our notation. Much of this section is +drawn from the review article by Bilal [2]. Our discussion here crucially relies on the +famous connection between anomalies and the path integral measure first discovered +by Fujikawa [10]. +2.1 +Defining the Anomaly +We begin with the definition of the anomaly. Consider a general gauge theory coupled +to a set of left-handed Weyl fermions collectively denoted by χ: +L = − 1 +4g2F a +µνF aµν + χ†¯σµPµχ , +(2.1) +– 4 – + +where we have defined the Hermitian covariant derivative +Pµ ≡ iDµ = i∂µ + Gµ = i∂µ + Ga +µta , +(2.2) +where ta are the (Hermitian) gauge group generators. The gauge field strength is +given by +Fµν = F a +µν ta = −i [Pµ, Pν] = (∂µGν) − (∂νGµ) − i [Gµ, Gν] . +(2.3) +The kinetic term for the gauge fields in Eq. (2.1) should be read as a sum over terms +normalized with different gauge couplings in the case of a product gauge group. +A gauge transformation can be parameterized by the matrix +Uα = eiα = eiαata , +(2.4) +where the transformation parameters αa = αa(x) are functions of spacetime. Under +Eq. (2.4), the building blocks of our theory transform as +χ +→ +χα = Uαχ , +(2.5a) +χ† +→ +χ† +α = χ†U † +α , +(2.5b) +P µ +→ +P µ +α = UαP µU † +α , +(2.5c) +Gµ +→ +Gµ +α = UαGµU † +α + Uα +� +i∂µU † +α +� +. +(2.5d) +We will use δα to denote the first-order (in α) gauge variation; for example, +δαGµ ≡ (Gµ +α − Gµ) +�� +O(α) = Dµα = ∂µα − i +� +Gµ, α +� +. +(2.6) +The Lagrangian in Eq. (2.1) defines an action that is gauge invariant at the +classical level. However, quantum effects can spoil gauge invariance. If this happens, +we say that the theory has an anomaly. +To define the anomaly, we consider the bosonic effective action W[G], computed +from the path integral by integrating out the fermions, while treating the gauge field +as a classical background: +eiW[G] ≡ +� +DχDχ† eiSf[χ,χ†,G] , +(2.7) +where Sf ≡ +� +d4x χ†¯σµPµχ is the fermion bilinear part of the classical action. If +we would also like to treat the gauge field Gµ as a dynamical quantum field by +– 5 – + +performing its path integral, +� +DGDχDχ† ei +� +d4x L = +� +DG e +i +� +− +1 +4g2 +� +d4x F a +µνF aµν+W[G] +� +, +(2.8) +we need W[G] to be gauge invariant (upon regularization and renormalization). +Gauge invariance of the classical action Sf does not guarantee that of W[G], since +quantum effects (due to the fermionic path integral measure) can break gauge in- +variance. +The anomaly functional A[α], which we also simply refer to as the anomaly, can +be defined by taking the gauge variation of the bosonic effective action W[G]:3 +A[α] ≡ +� +d4x αa(x)Aa(x) ≡ δαW[G] . +(2.9) +If A[α] = 0, the theory is anomaly-free and the path integral in Eq. (2.8) yields a well- +behaved quantum theory. If A[α] ̸= 0 but is equal to the gauge variation of a local +action, A[α] = δα(− +� +d4x Lct), it is called an irrelevant anomaly and can be removed +by renormalization, i.e., by adding local counterterms Lct to the Lagrangian (see e.g. +Ref. [34] for a systematic study of such counterterms); in this case the (renormalized) +quantum theory is also well-behaved. On the other hand, a nonzero A[α] that cannot +be written as the gauge variation of a local action, called a relevant anomaly, implies +that the gauge theory is not well-defined at the quantum level; in this case, the +anomaly is an IR effect and cannot be removed by renormalization. +The definition Eq. (2.9) we adopt here is known as the consistent anomaly, in the +sense that it should – if properly regularized – satisfy the Wess-Zumino consistency +condition [35]: +δα1A[α2] − δα2A[α1] = A +� +−i[α1, α2] +� +, +(2.10) +which is a direct consequence of the Lie algebra: +(δα1δα2 − δα2δα1)W[G] = δ−i[α1,α2]W[G] . +(2.11) +The Wess-Zumino consistency condition is also equivalent to the statement that the +anomaly is BRST-closed when α is replaced by the ghost field ω = ωata: +A[ω] = δBRSTW[G] +=⇒ +δBRSTA[ω] = 0 , +(2.12) +which follows from the nil-potency of the BRST transformation, δ2 +BRST = 0. However, +since the bosonic effective action W[G] is not a local functional of the gauge field +Gµ, the fact that A[ω] = δBRSTW[G] does not mean that the anomaly is BRST-exact +3Note that in such variations, we restrict to the set of α(x) that fall off fast enough at infinity +such that one can always use integration by parts (see e.g. Eq. (2.17) below). In particular, a +constant α(x) does not belong to this set. +– 6 – + +on the space of local functionals. Anomalies that are BRST-exact on this space can +be absorbed by local counterterms, A[ω] = δBRST(− +� +d4x Lct), and are the irrelevant +anomalies, while the relevant anomalies are given by nontrivial BRST cohomology +classes (closed but not exact) on this space. +Finally, we note that while we have focused on gauge symmetries in the discussion +above, anomalies of global symmetries can be treated in the same framework by +artificially gauging all the (classical) global symmetries of interest. Concretely, we +introduce auxiliary gauge fields for all the global symmetries as part of Gµ, and +take Uα to also include local transformations associated with the global symmetry +generators. Then A[α] as defined above will also contain anomalies of the global +symmetries, and a nonzero value of A[α] implies that the classical global symmetry +cannot be gauged in the quantum theory. +In what follows, we will assume this +artificial gauging has been done for all the classical global symmetries of interest, +and will not distinguish between global and gauge symmetries. +2.2 +Connection to the Path Integral Measure +As explained above, the classical action Sf in Eq. (2.7) is gauge invariant, so the only +possible source of the anomaly is the path integral measure over the fermionic fields. +Specifically, performing the transformation in Eq. (2.5) changes the measure by a +Jacobian factor: +DχαDχ† +α = J −1 +α DχDχ† . +(2.13) +Therefore, we have +eiW[Gα] = +� +DχDχ† eiSf[χ,χ†,Gα] = +� +DχαDχ† +α eiSf[χα,χ† +α,Gα] += +� +J −1 +α DχDχ† eiSf[χ,χ†,G] += eiW[G] +� +J −1 +α DχDχ† eiSf[χ,χ†,G] +� +DχDχ† eiSf[χ,χ†,G] += eiW[G] � +J −1 +α +� +G . +(2.14) +In the first line, we just relabeled the dummy integration variables, χ → χα; in +the second line, we used Eq. (2.13) and the gauge invariance of the classical ac- +tion Sf; in the last line, we multiplied and divided the expression by eiW[G] = +� +DχDχ† eiSf[χ,χ†,G]. +Taking the logarithm of Eq. (2.14), we arrive at a relation +between the Jacobian factor4 and the anomaly: +− i log +� +J −1 +α +� +G = W[Gα] − W[G] = A[α] + O(α2) . +(2.15) +4We note that sometimes in the literature, +� +J −1 +α +� +G is simply written as J −1 +α (G) or just J −1 +α . +This might give an impression that it does not depend on the details of the action Sf. Throughout +this paper, we manifestly write it as an expectation value +� +J −1 +α +� +G to emphasize that it is a quantum +expectation value and a priori may depend on what interactions are included in the action. +– 7 – + +We see that when the quantum expectation value of the Jacobian factor is trivial, +there is no anomaly +� +J −1 +α +� +G = 1 +=⇒ +A[α] = 0 , +(2.16) +while anomalies are associated with the quantum breaking of classical symmetries. +2.3 +Connection to Ward Identities +The connection between the anomaly and the Ward identities can be made by noting +δαW[G] = +� +d4x +� +δαGa +µ(x) +� +δW +δGa +µ(x) = +� +d4x (Dµα)a +δW +δGa +µ(x) += − +� +d4x αa(x) +� +Dµ +δW +δGµ(x) +�a +. +(2.17) +Comparing this to Eq. (2.9), we get +� +Dµ +δW +δGµ(x) +�a += −Aa(x) . +(2.18) +Meanwhile, since Ga +µ acts as a source for the fermion current Jaµ = χ†¯σµtaχ, we have +δW +δGa +µ(x) = ⟨Jaµ⟩G . +(2.19) +Together, these imply +(Dµ ⟨Jµ⟩G)a = −Aa(x) , +(2.20) +i.e. the fermion current is covariant up to the anomaly. The BRST symmetry that is +critical to the quantization of gauge theory requires (Dµ ⟨Jµ⟩G)a = 0. This makes the +connection between anomaly cancellation and consistency of gauge theory precise. +We can use Eq. (2.20), or equivalently Eq. (2.18), as a generating functional for +the Ward identities. First, let us explicitly write out the left-hand side of Eq. (2.18): +∂µ +� δW +δGa +µ +� ++ f abc Gb +µ +δW +δGc +µ += −Aa(x) . +(2.21) +Now taking the kth functional derivative, we get +∂µ +� +δk+1W +δGa +µδGb1 +µ1 · · · δGbk +µk +������ +G=0 ++ +k +� +i=1 +f abic +δkW +δGb1 +µ1 · · · δGc +µi · · · δGbk +µk +����� +G=0 += − +δkAa(x) +δGb1 +µ1 · · · δGbk +µk +����� +G=0 +. +(2.22) +– 8 – + +These are the anomalous Ward identities, and are often written in terms of the +connected correlation functions of the fermion currents: +∂µ⟨Jµ,aJµ1,b1 · · · Jµk,bk⟩conn + +k +� +i=1 +f abic⟨Jµ1,b1 · · · Jµi,c · · · Jµkbk⟩conn += − +δkAa(x) +δGb1 +µ1 · · · δGbk +µk +����� +G=0 +. +(2.23) +We see that a Gk term in Aa(x) corresponds to a mismatch between the (k+1)-point +and k-point correlation functions of the fermion currents. Eq. (2.23) is sometimes +taken as a definition of the anomaly in renormalized perturbation theory. In the +case of an irrelevant anomaly, one can add local counterterms which give additional +contributions to the left-hand side of Eq. (2.22) and correspond to choosing a different +renormalization scheme for the current correlators in Eq. (2.23). A relevant anomaly, +on the other hand, constitutes a genuine violation of the classical Ward identities that +cannot be remedied by renormalization. It is also worth noting that Eq. (2.23) can +be used to prove that Aa(x) truncates at a finite power of the gauge field Gµ. +3 +Regularizing the Anomaly +The definition in Eq. (2.9) does not fully specify the value of the anomaly, because +(the gauge variation of) the bosonic effective action W[G] is not well-defined in the +absence of a regulator. In this section, we introduce our regularization prescription. +Then the CDE evaluation of the regularized anomaly will be presented in Sec. 4. +Before discussing the case of anomalies, we first review the basic idea of reg- +ularization and illustrate the role of regularization prescriptions in Sec. 3.1 using +some simple toy series. (Experts can safely skip this subsection.) Then in Sec. 3.2, +we introduce our regularization prescription for the anomaly, motivated by its conve- +nience for evaluating the functional trace using CDE. Specifically, we will be working +in strictly d = 4 spacetime dimensions, i.e., we will not be using dimensional reg- +ularization. Instead, we will insert a damping factor into the functional trace, in a +similar spirit to heat kernel regularization. In fact, we will introduce a class of such +damping factors parameterized by a set of numbers β; different choices of these β +parameters correspond to different regularization schemes and will lead to different +results. In Sec. 3.3, we comment on the connection between our regularization pre- +scription and some familiar approaches in the literature. In particular, we will see +that both the heat kernel and Pauli-Villars regulators can be viewed as specific in- +carnations of our approach. Finally, we check our regularization prescription against +the Wess-Zumino consistency condition in Sec. 3.4, and show how it may be satisfied +or violated depending on the choice of β values. +– 9 – + +3.1 +What is Regularization? +In this subsection, we illustrate the role of regularization with some simple toy series. +In particular, we demonstrate how different regularization prescriptions correspond to +different definitions for a non-converging series and hence generically lead to different +results upon evaluation. We will also clarify the allowed manipulations for a non- +converging series. +When we encounter a series that is not convergent, its sum does not have a +well-defined value. However, it is often useful to promote such a series into a ‘func- +tion series,’ where the summands are functions; these functions must reproduce the +original series term by term when their argument takes a particular value (or limit). +Then we can define the sum through analytic continuation: we first sum the func- +tion series inside its convergence region to obtain an analytic function, and then take +the limit corresponding to the original series to define the value of the latter. This +regularization procedure leads to a regulated (finite) series. +Let us explain how this works using a simple example. Consider the series +s1 = +∞ +� +k=0 +2k = 1 + 2 + 4 + 8 + · · · . +(3.1) +Clearly, this is a non-converging series. +However, we could associate it with the +function series +s1 +⇐⇒ +� ∞ +� +k=0 +xk +������ +x=2 +regularization +−−−−−−−−→ +1 +1 − x +���� +x=2 += −1 . +(3.2) +This function series converges to f1(x) = +1 +1−x within the disk |x| < 1, but not +at x = 2. But we can take f1(x = 2) as the definition for the sum s1. This is +what we mean by a regulated series. Another example is the famous zeta function +regularization originally used by Euler: the diverging series +s2 = +∞ +� +k=1 +k = 1 + 2 + 3 + 4 + · · · +(3.3) +can be regularized as +s2 +⇐⇒ +� ∞ +� +k=1 +1 +ks +������ +s=−1 +regularization +−−−−−−−−→ +ζ(s) +�� +s=−1 = − 1 +12 . +(3.4) +As mentioned above, when we promote a non-converging series into a function +series, we require that the function series reproduces the original series term by term +when evaluated at a certain point. Clearly, this does not uniquely specify the choice: +– 10 – + +given a non-converging series, one can usually promote it into many different function +series. These correspond to different regularization schemes and serve as different +definitions of the sum of the original series. To see this concretely, let us consider +the following toy series +s0 = +∞ +� +k=0 +(−1)k = 1 − 1 + 1 − 1 + 1 − 1 + · · · . +(3.5) +To regularize this series, we could choose to promote it to any of the following set of +function series parameterized by a number β: +fβ(τ) = τ 0 − τ 1+β + τ 2 − τ 3+β + τ 4 − τ 5+β + · · · . +(3.6) +Then we have +s0 +⇐⇒ +fβ(τ) +�� +τ→1 +regularization +−−−−−−−−→ +1 − τ 1+β +1 − τ 2 +����� +τ→1 += 1 + β +2 +. +(3.7) +We see that with different values for β, the original non-converging series s0 can be +defined/regularized to take different values. +If we are going to regularize a non-converging series with a function series that is +absolutely convergent (in its convergence region), then one can shuffle and/or group +terms in the latter without changing its analytic continuation. Alternatively, one +could shuffle and/or group terms first in the original non-converging series, and then +regularize the new expression with an absolutely convergent function series. This +second way will lead to the same result upon evaluation, and it is sometimes more +convenient because the series is easier to massage before promoting it into a function +series. However, when we shuffle and/or group terms in the original non-converging +series to go from one expression to another, we have to remember that none of these +expressions is well-defined yet, so it is not appropriate to say that they are equal +(‘=’). Instead, they are just ‘equivalent’ in the sense that they would be equal if one +were to regularize them with the same absolutely convergent function series (with +the same shuffling and/or grouping of terms). In this paper, we use the symbol ‘≃’ to +denote this equivalence relation between non-converging series (see equations below +starting from Eq. (3.14)). +Let us take the same toy series example s0 to illustrate this point, as well as the +use of the ‘≃’ notation. Since the function series Eq. (3.6) is absolutely convergent +within the disk |τ| < 1, we can group its terms to get another series: +fβ(τ) +group terms +−−−−−−−→ �fβ(τ) ≡ +∞ +� +k=0 +� +τ 2k − τ 2k+1+β� +analytic continuation +−−−−−−−−−−−−→ 1 − τ 1+β +1 − τ 2 , +(3.8) +– 11 – + +which has the same analytic continuation. Alternatively, one could first group terms +in the original non-converging series: +s0 ≃ �s0 ≡ +∞ +� +k=0 +(1 − 1) = 0 + 0 + 0 + · · · . +(3.9) +Note that we have used the ‘≃’ sign here between s0 and �s0. The new series �s0 is a +converging series and does have a default definition, so a regularization for �s0 is not +mandatory. However, one could still use the function series �fβ(τ) to regularize it, +because +� +τ 2k − τ 2k+1+β���� +τ=1 = 0 +(3.10) +would also reproduce the series �s0 term by term. With this regularization, one would +then get the same evaluation result 1+β +2 +as in Eq. (3.7). Our use of the ‘≃’ sign here +is emphasizing this: s0 and �s0 are equal only when we use the same regularization +prescription for them (although one of them has a different default definition in the +absence of regularization). +We note in particular that performing cyclic permutations inside a trace is a +typical type of shuffling and/or grouping of terms: +tr (AB) = +� +i +�� +a +AiaBai +� +, +(3.11a) +tr (BA) = +� +a +�� +i +BaiAia +� += +� +a +�� +i +AiaBai +� +. +(3.11b) +The two traces are related by a change of summation order. When the matrices A +and B are infinite dimensional, such as in the case of functional traces, each trace is +a sum over a (double) series. If the series is not convergent and needs regularization +to be well-defined, then it is not appropriate to claim that the two traces are equal, +as we have just explained. Instead, we should use the ‘≃’ sign: +tr (AB) ≃ tr (BA) , +(3.12) +to emphasize that they would be equal when we use the same absolutely convergent +function series to regulate them. +3.2 +Anomaly as a Regulated Functional Trace +Let us now turn to the case of interest in this paper, the anomaly functional A[α] +defined in Eq. (2.9). First, we would like to isolate the functional trace that encodes +the anomalies. We start with the definition of W[Gα], Eq. (2.7) with Gµ replaced by +– 12 – + +Gµ +α according to Eq. (2.5d). It can be formally written as a functional determinant: +eiW[Gα] = +� +DχDχ† eiSf[χ,χ†,Gα] = det +� +Uα ¯σµPµ U † +α +� +. +(3.13) +Taking the logarithm and expanding in α, we get +W[Gα] = −i log det +� +Uα ¯σµPµ U † +α +� +≃ −i log det (¯σµPµ + iα ¯σµPµ − ¯σµPµ iα) + O(α2) +≃ −i log det (¯σµPµ) − i log det +� +1 + +1 +¯σνPν +(iα ¯σµPµ − ¯σµPµ iα) +� ++ O(α2) +≃ W[G] − i Tr log +� +1 + +1 +¯σνPν +(iα ¯σµPµ − ¯σµPµ iα) +� ++ O(α2) +≃ W[G] + Tr +� +1 +¯σνPν +(α ¯σµPµ − ¯σµPµ α) +� ++ O(α2) . +(3.14) +According to the definition in Eq. (2.9), the leading order contribution to the differ- +ence W[Gα] − W[G] gives the anomaly. Therefore, we obtain +A[α] ≃ Tr +� +1 +¯σνPν +� +α ¯σµPµ − ¯σµPµ α +�� +. +(3.15) +At this point, we have formally written the anomaly as a functional trace. How- +ever, we emphasize that the functional trace in Eq. (3.15) is the sum of a series that +is not convergent, so it does not have a definite value and requires regularization to +become well-defined. As elaborated in Sec. 3.1, different regularization prescriptions +can yield different results upon evaluation. In fact, the same is true for the expression +in each line of Eq. (3.14). Therefore, we have used the notation ‘≃’ to emphasize +that these expressions are not exactly equal ‘=’ unless they are regularized in the +same way. +One may also attempt to perform a cyclic permutation within the functional +trace in Eq. (3.15), so the two terms appear to cancel. However, as explained in +Sec. 3.1, such a cyclic permutation amounts to shuffling and/or grouping terms in +the original non-converging series to obtain a new series: +A[α] ≃ Tr[0] = 0 + 0 + 0 + · · · . +(3.16) +Although this new series is zero term by term, which is convergent and hence has +a default definition without a regulator, this does not contradict the statement that +regularization prescriptions exist that yield a nonzero value for this series. As em- +phasized by the ‘≃’ sign, the two expressions above would only be equal under the +– 13 – + +same regularization prescription. The default definition of the right-hand side (which +gives zero) corresponds to one particular choice of regularization (a trivial one), so +its evaluation result would not be equal to that of the left-hand side if a different +regularization prescription is chosen for the latter. +To motivate our regulator, let us first check what would happen if we go ahead +and evaluate the functional trace in Eq. (3.15) with CDE. Focusing on the first term, +we have +Tr +� +1 +¯σνPν +α ¯σµPµ +� +≃ +� +d4x +� +d4q +(2π)4 tr +� +1 +¯σν(qν + Pν) α ¯σµ(qµ + Pµ) +� +≃ +� +d4x +� +d4q +(2π)4 tr +� ∞ +� +k=0 +� +−σλqλ +q2 +¯στPτ +�k σνqν +q2 +α ¯σµ(qµ + Pµ) +� +≃ +� +d4x +� +d4q +(2π)4 tr +� ∞ +� +k=0 +� +− /q +q2 /P +�k /q +q2 α (/q + /P) 1 − γ5 +2 +� +≃ +� +d4x +� +d4q +(2π)4 tr +� ∞ +� +k=0 +� +− /q +q2 �Pβ +�k /q +q2 α (/q + �Pβ) 1 − γ5 +2 +� +≃ +� +d4x +� +d4q +(2π)4 tr +� +1 +/q + �Pβ +α (/q + �Pβ) 1 − γ5 +2 +� +≃ Tr +� +1 +�Pβ +α �Pβ +1 − γ5 +2 +� +. +(3.17) +The first line above simply follows from the definition of the functional trace.5 To +obtain the second line, we have performed a Taylor expansion in terms of the Hermi- +tian covariant derivative Pµ, an operation called the Covariant Derivative Expansion +(CDE) in the literature.6 To get the third line, we used the following identity between +Pauli matrices and the Dirac gamma matrices: +tr +� +(σµ1¯σν1) · · · (σµk¯σνk) +� += tr +� +(γµ1γν1) · · · (γµkγνk) 1 − γ5 +2 +� +. +(3.18) +5See Eq. (A.12) for a more detailed explanation of the shift Pµ → qµ + Pµ. We note that the +internal traces ‘tr’ from here on in the main text are actually what we denote by ‘trx’ in App. A. See +App. A.1, especially the discussion around Eq. (A.18) for a careful clarification on this notation. +6More precisely, the operation here is called ‘simplified CDE’ [18], in which one makes a Taylor +expansion directly in terms of the ‘open’ covariant derivatives. This is different from the ‘original +CDE’ [15–17] where one inserts additional factors to ‘close’ the covariant derivatives (i.e. put them +into commutators) before performing the Taylor expansion. See the discussion around Eq. (A.28) +for an elaboration on open vs. closed derivatives in functional operators, and App. B of Ref. [19] +for a detailed discussion on simplified vs. original CDE. +– 14 – + +Starting from the fourth line of (3.17), we have introduced the β-modified covariant +derivative:7 +�Pβ ≡ i/∂ + /G +�1 − γ5 +2 ++ β 1 + γ5 +2 +� +≡ i/∂ + +� +a +/G +ata +�1 − γ5 +2 ++ βa +1 + γ5 +2 +� +. +(3.19) +Finally, in the last line of Eq. (3.17), we rewrote the result as a functional trace. +Note that we take the βa parameters to be degenerate within each simple gauge +group sector so that βata (no sum over a) satisfy the same Lie algebra as ta. Here +and in what follows, we explicitly write out the summation over adjoint indices when +the presence of βa results in more than two identical adjoint indices in an expression. +The identity in Eq. (3.18) has allowed us to convert the two-component ex- +pression (left-hand side of Eq. (3.17)) into a four-component expression (last line in +Eq. (3.17)) with an insertion of the projector operator 1−γ5 +2 . The same procedure +goes through when both terms in Eq. (3.15) are present, so we have +A[α] ≃ Tr +� +1 +�Pβ +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� +. +(3.20) +At this stage, it seems that the β parameters could take arbitrary values without +affecting the value of the expression, because it comes with the factor 1+γ5 +2 +which +will get annihilated by the projector 1−γ5 +2 +at the end of the expression. However, we +stress that this β-parameterized functional trace is still the sum of a non-converging +series, so we need to introduce a regulator to make it well-defined. As we will see +below, once we regulate this expression, different β’s will define different values for +the functional trace. +Motivated by the form of the expression in Eq. (3.20), we choose to insert a +damping factor to define the regularized anomaly: +AΛ +β[α] ≡ Tr +� +f +� +− +�P 2 +β +Λ2 +� +1 +�Pβ +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� +, +(3.21) +where the function f(u) satisfies the following conditions: +f(0) = 1 , +f(+∞) = 0 , +� ∞ +0 +duf(u) +well-defined , +(3.22a) +7Note that when β ̸= 1, the operator �Pβ is not gauge covariant. This is the reason why we will +not always get a covariant anomaly; see discussion in Sec. 3.4 for more details. +– 15 – + +undnf +dun +���� +u=0 += undnf +dun +���� +u→+∞ += 0 +for +n ≥ 1 . +(3.22b) +Typical examples of such functions are +f(u) = e−u , +and +f(u) = +2 +(1 + u)(2 + u) . +(3.23) +The renormalized anomaly is then given by +Aβ[α] ≡ lim +Λ→∞ +� +AΛ +β[α] + δα +� +d4x LΛ +ct +� +, +(3.24) +where LΛ +ct is the local counterterm Lagrangian. Note in particular that the regularized +anomaly AΛ +β[α] generically contains an O(Λ2) piece that is irrelevant for β values +satisfying the Wess-Zumino consistency condition, in which case we should include +operators with appropriate O(Λ2) coefficients in LΛ +ct to obtain a finite result for the +renormalized anomaly Aβ[α]. LΛ +ct can also contain O(Λ0) counterterms, and their +coefficients specify the renormalization scheme. +Having included the damping factor f +� +− �P 2 +β/Λ2� +, the functional trace AΛ +β[α] +is now the sum of an absolutely convergent series. So at this point one is free to +manipulate this expression, e.g. perform cyclic permutations while maintaining a +genuine ‘=’ sign. Our regularization prescription Eq. (3.21) is designed to facilitate +the evaluation with CDE. In particular, the damping factor inserted commutes with +the β-modified covariant derivative: +� +f +� +− +�P 2 +β +Λ2 +� +, �Pβ +� += 0 . +(3.25) +Also note from the definition of �Pβ in Eq. (3.19) that it anticommutes with γ5: +�Pβγ5 = −γ5 �Pβ . +(3.26) +Making use of these identities, we can simplify Eq. (3.21) to +AΛ +β[α] = Tr +� +f +� +− +�P 2 +β +Λ2 +� +α γ5 +� +, +(3.27) +from which it is clear that the evaluation result will depend on the parameters β. +One interpretation of this regulator is that the β parameters determine the com- +bination of background fields that are turned on when computing the anomaly. This +effectively forces the path integral measure DχDχ† to be organized according to the +– 16 – + +eigenmodes of the operator �Pβ.8 +We will proceed with the evaluation of AΛ +β[α] in Sec. 4, after the next two sub- +sections which discuss how our prescription connects to other familiar regularization +approaches, and how the Wess-Zumino consistency condition is satisfied or violated +by different choices of β. +3.3 +Connection to Other Regularization Prescriptions +Let us make a few comments on the connection between our regularization prescrip- +tion Eq. (3.21) and some approaches that often appear in the literature, in particular, +heat kernel regularization and Pauli-Villars regularization. +Regularizing Eq. (3.20) with a heat kernel regulator, one obtains +AHK +β +≡ Tr +� +e +�P 2 +β/Λ2 1 +�Pβ +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� +. +(3.28) +Comparing with Eq. (3.21), we see that the heat kernel regularization amounts to +choosing the damping function to be +Heat kernel : +f(u) = e−u . +(3.29) +Alternatively, regularizing Eq. (3.20) with one Pauli-Villars field, one obtains +APV,1 +β +≡ Tr +�� +1 +�Pβ +− +1 +�Pβ − Λ +� � +α �Pβ − �Pβ α +� 1 − γ5 +2 +� += Tr +� +−Λ +�Pβ( �Pβ − Λ) +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� += Tr +� +−Λ +�Pβ − Λ +α γ5 +� += Tr +� +−Λ2 +�P 2 +β − Λ2 α γ5 +� += Tr +� +−Λ2 +�P 2 +β − Λ2 +1 +�Pβ +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� +. +(3.30) +Comparing with Eq. (3.21), we see that this amounts to choosing the damping func- +tion to be +Pauli-Villars with one regulator field : +f(u) = +1 +1 + u . +(3.31) +Note that this damping factor does not satisfy all the conditions listed in Eq. (3.22), +and hence would not regulate all the divergences. This motivates considering Pauli- +Villars regularization with three regulator fields, for which one obtains the regularized +8We leave implicit possible analytic continuations needed to make �Pβ a Hermitian operator that +has a well-defined eigenvalue problem. +– 17 – + +anomaly as +APV,3 +β +≡ Tr +�� +1 +�Pβ +− +1 +�Pβ − M1 ++ +1 +�Pβ − M2 +− +1 +�Pβ − M3 +� � +α �Pβ − �Pβ α +� 1 − γ5 +2 +� += Tr +� +−(M1 − M2 + M3) �P 2 +β + 2M1M3 �Pβ − M1M2M3 +( �Pβ − M1)( �Pβ − M2)( �Pβ − M3) +α γ5 +� += Tr +� +M 2 +1M 2 +3 (2 �P 2 +β − M 2 +2) +( �P 2 +β − M 2 +1)( �P 2 +β − M 2 +2)( �P 2 +β − M 2 +3) +α γ5 +� +, +(3.32) +where we have assumed the relation M 2 +1 − M 2 +2 + M 2 +3 = 0. If we now take +M 2 +1 = M 2 +3 = Λ2 , +and +M 2 +2 = 2Λ2 , +(3.33) +this simplifies to +APV,3 +β += Tr +� +2Λ4 +( �P 2 +β − Λ2)( �P 2 +β − 2Λ2) +1 +�Pβ +� +α �Pβ − �Pβ α +� 1 − γ5 +2 +� +. +(3.34) +Comparing with Eq. (3.21), we see that this amounts to choosing the damping func- +tion to be +Pauli-Villars with three regulator fields : +f(u) = +2 +(1 + u)(2 + u) . +(3.35) +This damping function does satisfy all the conditions listed in Eq. (3.22), and will +successfully regularize all the divergences. +3.4 +Consistency With the Wess-Zumino Condition +Since we have adopted the definition of anomaly as the gauge variation of the bosonic +effective action: +A[α] ≡ δαW[G] = (W[Gα] − W[G])|O(α) , +(3.36) +we expect it to satisfy the Wess-Zumino consistency condition, as reviewed in Sec. 2.1. +However, an implicit assumption behind this expectation is that there is a well- +defined W[G]. Importantly, our regularization prescription presented in Sec. 3.2 is +directly applied to δαW[G], instead of W[G]. In this case, the Wess-Zumino consis- +tency condition may not be satisfied, because generic β values may not correspond to +applying the same (or ‘consistent’) regularization prescription to W[Gα] and W[G]. +In this subsection, we check the Wess-Zumino consistency condition for the regular- +ized anomaly AΛ +β[α] at the level of Eq. (3.27), and give a partial but general answer +– 18 – + +to the question of what β values lead to a Wess-Zumino consistent anomaly: +δα1AΛ +β[α2] − δα2AΛ +β[α1] +?= AΛ +β +� +−i[α1, α2] +� +. +(3.37) +We will revisit this question in Sec. 5 after evaluating Eq. (3.27) in Sec. 4. +Using the expression of AΛ +β[α] in Eq. (3.27), we can write the first term in +Eq. (3.37) as (it is understood that we will be dropping terms of order O(α2 +1, α2 +2) +throughout this subsection): +δα1AΛ +β[α2] = Tr +� +f +� +− +�P 2 +β[α1] +Λ2 +� +γ5α2 +� +− Tr +� +f +� +− +�P 2 +β +Λ2 +� +γ5α2 +� +, +(3.38) +where �Pβ[α1] denotes the gauge transformation of �Pβ: +�Pβ[α1] ≡ i/∂ + /Gα1 +�1 − γ5 +2 ++ β 1 + γ5 +2 +� += i/∂ + +� +Uα1 /GU † +α1 + Uα1 +� +i/∂U † +α1 +� � �1 − γ5 +2 ++ β 1 + γ5 +2 +� +. +(3.39) +We note that when β = 1, Eq. (3.38) is quite easy to calculate because �Pβ=1 = /P +transforms covariantly and so does the damping factor: +�Pβ=1[α1] = Uα1 �Pβ=1U † +α1 +=⇒ +f +� +− +�P 2 +β=1[α1] +Λ2 +� += Uα1f +� +− +�P 2 +β=1 +Λ2 +� +U † +α1 . +(3.40) +This leads us to the so-called covariant anomaly that satisfies +δα1AΛ +β=1[α2] = Tr +� +f +� +− +�P 2 +β=1 +Λ2 +� +γ5 � +U † +α1 α2 Uα1 − α2 +� +� += AΛ +β=1 +� +−i[α1, α2] +� +. +(3.41) +We see that this covariant anomaly generically would not satisfy the Wess-Zumino +consistency condition; it is off by a factor of two compared to Eq. (3.37): +δα1AΛ +β=1[α2] − δα2AΛ +β=1[α1] = 2 AΛ +β=1 +� +−i[α1, α2] +� +̸= +AΛ +β=1 +� +−i[α1, α2] +� +. +(3.42) +The only exceptions are when the anomaly itself vanishes AΛ +β=1[α] = 0 (once summed +over fermion species) or when the two gauge transformations under consideration +commute, [α1, α2] = 0. In these cases, the Wess-Zumino consistency condition itself +is trivial, and the covariant anomaly is also a consistent anomaly. +For general β values, �Pβ does not transform covariantly, and calculating Eq. (3.38) +– 19 – + +is more tedious. It is useful to write �Pβ in terms of its chirality components: +�Pβ = /P 1 − γ5 +2 ++ /P β +1 + γ5 +2 +, +(3.43) +with +/P = i/∂ + /G = i/∂ + +� +a +/G +ata , +(3.44a) +/P β ≡ i/∂ + β /G = i/∂ + +� +a +βa /G +ata . +(3.44b) +The left-handed component is gauge covariant, but the right-handed component +transforms in a complicated manner for general β values. To proceed, let us rewrite +Eq. (3.39) also in terms of its chirality components: +�Pβ +−→ +�Pβ[α1] = /Lα1 +1 − γ5 +2 ++ /Rα1 +1 + γ5 +2 +, +(3.45) +with +/P +−→ +/Lα1 ≡ Uα1 /PU † +α1 , +(3.46a) +/P β +−→ +/Rα1 ≡ i/∂ + β +� +Uα1 /GU † +α1 + Uα1 +� +i/∂U † +α1 +� � +. +(3.46b) +To check the Wess-Zumino consistency condition Eq. (3.37), we can Taylor expand +the damping factors in Eq. (3.38) and examine a general kth power term therein. We +have +Tr +� � +�P 2 +β[α1] +�k +γ5α2 +� += Tr +�� �/Rα1 /Lα1 +�k 1−γ5 +2 ++ +�/Lα1 /Rα1 +�k 1+γ5 +2 +� +γ5α2 +� += Tr +� +1+γ5 +2 +�/PU † +α1 /Rα1Uα1 +�k−1 /P U † +α1 +�/Rα1, α2 +� +Uα1 +� +, +(3.47a) +Tr +� � +�P 2 +β +�k +γ5α2 +� += Tr +� +1+γ5 +2 +�/P /P β +�k−1 /P +�/P β, α2 +�� +. +(3.47b) +The difference between Eqs. (3.47a) and (3.47b) comes from two sources: +U † +α1 /Rα1Uα1 = /P β + (/Rα1 − /P β) + i +�/P β, α1 +� +, +(3.48a) +U † +α1 +�/Rα1, α2 +� +Uα1 = +�/P β, α2 +� ++ +�/Rα1 − /P β, α2 +� +− i +� +α1, +�/P β, α2 +�� +. +(3.48b) +One could go ahead with the calculation keeping track of all these terms for general +β values, but the result is not very illuminating. Instead, let us examine the special +– 20 – + +case β = 0 here. In this case, the right-handed component does not transform: +/Rα1 = /P β=0 = i/∂ . +(3.49) +The middle term of each equation in Eqs. (3.48) is therefore absent, and we have +δα1AΛ +β=0[α2] − δα2AΛ +β=0[α1] ⊃ Tr +� � +�P 2 +β[α1] +�k +γ5α2 +� +− Tr +� � +�P 2 +β +�k +γ5α2 +� +− (α1 ↔ α2) += Tr +� +1+γ5 +2 +� �/PU † +α1 /Rα1Uα1 +�k−1 − +�/P /P β +�k−1 � +/P +�/P β, α2 +� ++ 1+γ5 +2 +�/P /P β +�k−1 /P +� +−iα1, +�/P β, α2 +��� +− (α1 ↔ α2) += Tr +� +1+γ5 +2 +�/P /P β +�k−1 /P +� +/P β, −i [α1, α2] +�� += Tr +� � +�P 2 +β +�k +γ5 [−iα1, α2] +� +. +(3.50) +In the step leading to the second to last line, the first term in the curly brackets gets +canceled upon adding the expression with α1 ↔ α2, while the second term combines +with the latter and we have used the Jacobi identity. +Clearly, summing over all +the kth power relations like in Eq. (3.50) will give us the Wess-Zumino consistency +condition in Eq. (3.37): +δα1AΛ +β=0[α2] − δα2AΛ +β=0[α1] = AΛ +β=0 +� +−i[α1, α2] +� +. +(3.51) +Therefore, we see that in our regularization prescription, β = 0 (meaning βa = 0, +∀a) is always one possible choice to ensure the Wess-Zumino consistency condition +for any symmetry group. However, from the present analysis it is difficult to tell +whether there are other Wess-Zumino consistent choices. We will revisit this issue +in Sec. 5 using the BRST version of the Wess-Zumino condition once we have the +evaluation result for AΛ +β[α]. +4 +Master Formula for the Anomaly From CDE +Now we proceed with the evaluation of the regularized anomaly, starting with Eq. (3.27): +AΛ +β[α] = Tr +� +f +� +− +�P 2 +β +Λ2 +� +α γ5 +� += +� +d4x +� +d4q +(2π)4 tr +� +f +� +− +� +/q + �Pβ +�2 +Λ2 +� +α γ5 +� += +� +d4x +� +d4k +(2π)4 tr +� +Λ4f +� +− +� +/k + +�Pβ +Λ +�2� +α γ5 +� +. +(4.1) +– 21 – + +Here we have rescaled the integration variable kµ ≡ qµ/Λ, so that it is easier to keep +track of the 1/Λ powers. Eventually, we are interested in the Λ → ∞ limit, so in +what follows we will be dropping the O(1/Λ) terms that vanish in this limit. This +can be achieved by applying the simplified CDE, while expanding and truncating the +integrand accordingly: +Λ4f +� +�− +� +/k + +�Pβ +Λ +�2� +� = Λ4f +� +−k2 − 1 +Λ +� +/k �Pβ + �Pβ/k +� +− 1 +Λ2 �P 2 +β +� += Λ4 +� +fu + f ′ +u z + 1 +2f ′′ +u z2 + 1 +6f ′′′ +u z3 + 1 +24f ′′′′ +u z4 +� +, +(4.2) +where we have introduced the following notation for convenience +u ≡ −k2 , +and +z ≡ − 1 +Λ +� +/k �Pβ + �Pβ/k +� +− 1 +Λ2 �P 2 +β . +(4.3) +Plugging this back into Eq. (4.1) and simplifying the expression, we get +AΛ +β[α] = +� +d4x +� +d4k +(2π)4 tr +�� +− Λ2f ′ +u �P 2 +β + 1 +2f ′′ +u �P 4 +β ++ u +24f ′′′ +u +� +�P 2 +βγµ �Pβγµ �Pβ + �Pβγµ �Pβγµ �P 2 +β ++ �Pβγµ �P 2 +βγµ �Pβ + 4 �P 4 +β +�� +α γ5 +� +. +(4.4) +Note that the terms proportional to f ′′′′ +u +can be grouped in pairs that take the form +tr +� +γµ(· · · )γ5 + (· · · )γµγ5� += 0, so they all cancel out; the same is true for a subset of +the f ′′ +u and f ′′′ +u terms, which significantly reduces the number of terms in the result. +Performing the loop momentum integral (after a Wick rotation as usual), we obtain +AΛ +β[α] = +� +d4x +i +16π2 +� +− Λ2 +� +(ufu) +��∞ +0 − +� ∞ +0 +dufu +� +tr0 ++ 1 +2 +� +(uf ′ +u − fu) +��∞ +0 +� +tr1 ++ 1 +12 +� � +u2f ′′ +u − 2uf ′ +u + 2fu +���∞ +0 +� +(2 tr1 − tr2 − tr3) +� +. +(4.5) +– 22 – + +We see that for a general damping function f(u) that satisfies the conditions in +Eqs. (3.22), the calculation yields the result: +AΛ +β[α] = +� +d4x +i +16π2 +� � +Λ2 +� ∞ +0 +duf(u) +� +tr0 +1 +6 +� +tr1 + tr2 + tr3 +� +� +, +(4.6) +where +tr0 ≡ tr +� +�P 2 +βγ5α +� +, +(4.7a) +tr1 ≡ tr +� +�P 4 +βγ5α +� +, +(4.7b) +tr2 ≡ −1 +2 tr +�� �P 2 +βγµ �Pβγµ �Pβ + �Pβγµ �Pβγµ �P 2 +β +� +γ5α +� +, +(4.7c) +tr3 ≡ −1 +2 tr +� +�Pβγµ �P 2 +βγµ �Pβγ5α +� +. +(4.7d) +Eq. (4.6) is our master formula for the regularized anomaly before evaluation of the +Dirac traces. +4.1 +Evaluating the Dirac Traces +In order to evaluate the Dirac traces in Eq. (4.7), it is convenient to use the chirality +decomposition of �Pβ in Eq. (3.43): +�Pβ = /P 1 − γ5 +2 ++ /P β +1 + γ5 +2 +, +(4.8) +where +/P ≡ i/∂ + /G = i/∂ + +� +a +/G +ata , +(4.9a) +/P β ≡ i/∂ + β /G = i/∂ + +� +a +βa /G +ata . +(4.9b) +We also introduce the notation +Gµ +− ≡ P µ − P µ +β = (1 − β) Gµ = +� +a +(1 − βa) Gaµ ta . +(4.10) +The evaluation of tr0 is straightforward: +tr0 = 2 tr +�� +Pµ, P µ +β +� +α +� += −2i(1 − β) tr +� +(∂µGµ) α +� +IBP += 2i(1 − β) tr +� +Gµ(∂µα) +� += 2i +� +a +tr +� +tatb� +(1 − βa) Ga +µ (∂µαb) . +(4.11) +– 23 – + +Turning to tr1, tr2, tr3, we first note that they can be written in the following form:9 +tr1 = 1 +2 tr +� +/P /P β /P +�/P β, α +� +(1 + γ5) +� +, +(4.12a) +tr2 = 1 +2 tr +��/P /P +2 +β + /P +2 +β /P + /P +3��/P β, α +� +(1 + γ5) + /P β /P /P β +�/P β, α +� +(1 − γ5) +� +, (4.12b) +tr3 = −4 tr +�� +PνPµP µ +β + P µ +β PµPν +�� +P ν +β , α +�� +, +(4.12c) +where we have used γµγνγµ = −2γν, γµγνγργµ = 4ηνρ to simplify the products of +gamma matrices. Upon evaluating the Dirac traces we can combine terms in the +sum of all three traces such that all P µ +β factors appear in commutators: +3 +� +i=1 +tri = tr +� +−2 +�� +3 +� +P µ +β , P ν +β +� ++ 2 +� +P µ +β , Gν +− +� +− +� +P ν +β , Gµ +− +� ++ +� +Gµ +−, Gν +− +� +, G−µ +� +− +� +Pβ,µ , +� +P µ +β , Gν +− +� +− +� +Gµ +−, Gν +− +�� +− Gµ +−Gν +−G−µ +� +[Pβ,ν, α] +− iεµνρσ +�� +2Gµ +−Gν +−Gρ +− + +� +3 +� +P µ +β , P ν +β +� ++ 2 +� +P µ +β , Gν +− +� +, Gρ +− +��� +P σ +β , α +� ++ 3 +� +P µ +β , P ν +β +�� +P ρ +β, P σ +β +� +α +�� +. +(4.13) +Having P µ +β in commutators is important because it contains the derivative ∂µ, which +as a functional operator is understood to act on everything to its right. But when it +appears in a commutator, its action is local (or ‘closed’) on the object appearing in +the commutator; for example:10 +� +∂µ, Gµ(x) +� +φ(x) = ∂µGµ(x) φ(x) − Gµ(x) ∂µφ(x) = +� +∂µGµ(x) +� +φ(x) . +(4.14) +4.2 +The Evaluated Master Formula +Gathering the results in Eqs. (4.11) and (4.13) and substituting in Eqs. (4.9b) +and (4.10) for P µ +β and Gµ +−, we obtain our evaluated master formula for the regu- +9To arrive at these expressions, we have used cyclic permutation to move P µ +β to the right in half +of the terms. Generally this is illegal since ‘tr’ is only over the internal space while P µ +β contains ∂µ +which is a spacetime operator. However, such cyclic permutations are innocuous in CDE calculations +of functional traces that arise from evaluating the path integral at one loop. In fact, they have been +used in many previous functional matching calculations. We clarify this subtle point in App. A. +10The local nature of all derivative operators in the CDE is also the reason why the otherwise +illegal cyclic permutation in the internal trace ‘tr’ in intermediate steps actually leads to the correct +result; see App. A for a detailed discussion. +– 24 – + +larized anomaly expressed in the matrix notation: +AΛ +β[α] = +� +d4x +1 +16π2 tr +� +− 2(1 − β) +� +Λ2 +� ∞ +0 +duf(u) +� +Gµ (∂µα) ++ 1 +3(1 − β) +� +i +� +(1 + 4β) (∂µGν) − (1 + 2β) (∂νGµ) − i(1 + 3β2) +� +Gµ, Gν +� +, Gµ� ++ +� +∂2Gν +� ++ i(1 − 2β) +� +(∂µGµ) , Gν +� +− Gµ(1 − β)Gν(1 − β)Gµ +� � +Dν +βα +� +− 1 +2 εµνρσ +�1 +3 +� +(1 − β)Gρ , 2(1 + 2β) (∂µGν) − i(1 + 2β + 3β2)GµGν� � +Dσ +βα +� ++ 4 +� +β (∂µGν) − iβ2GµGν�� +β (∂ρGσ) − iβ2GρGσ� +α +�� +, +(4.15) +where +� +Dµ +βα +� +≡ (∂µα) − iβ[Gµ, α] . +(4.16) +In Eq. (4.15) we have carefully kept the β factors in appropriate places such that +each of them is associated with the gauge field that immediately follows it. +Depending on the application, it is sometimes more convenient to write out the +adjoint components of the master formula in Eq. (4.15), which gives +AΛ +β[α] = +� +d4x +1 +16π2 +� +− +� +a,b +tr +� +tatb� +(1 − βa) +� +2 +� +Λ2 +� ∞ +0 +duf(u) +� +Ga +µ +� +∂µαb� ++ 1 +3 +� +f aef� +(1 + 4βa) (∂µGe +ν) − (1 + 2βa) +� +∂νGe +µ +� ++ +� +1 + 3β2 +a +� +f eghGg +µGh +ν +� +Gfµ +− +� +∂2Ga +ν +� ++ (1 − 2βa)f aef � +∂µGe +µ +� +Gf +ν +�� +∂ναb + βbf bcdGc +ναd�� +− +� +a,b,c,d +tr +� +tatbtctd� 1 +3 (1 − βa)(1 − βb)(1 − βc) Ga +µGb +νGcµ� +∂ναd + βdf defGe +ναf� +− +� +a,b,c +tr +� +{ta, tb} tc� 1 +4 εµνρσ +� +βaβb +� +F aµν +lin ++ βaf adeGdµGeν�� +F bρσ +lin + βbf bfgGfρGgσ� +αc ++ 1 +3 (1 − βb) +� +2(1 + 2βa)F aµν +lin ++ (1 + 2βa + 3β2 +a)f adeGdµGeν� +Gbρ +× +� +∂σαc + βcf cfgGfσαg��� +, +(4.17) +where F µν +lin ≡ (∂µGν) − (∂νGµ) is the part of F µν linear in the gauge fields, and we +have used the fact that β takes the same value within a simple group (only for which +– 25 – + +f abc may be nonzero). +In the next section, we will apply the evaluated master formula, written in matrix +and component forms in Eqs. (4.15) and (4.17), respectively, to obtain explicit results +for various gauge group sectors. Before delving into the details, let us first quickly +note two special β choices which directly relate to the discussion in Sec. 3.4. +• If βa = 1 (∀a), all but the last line in Eq. (4.15) vanishes, and the result takes +a gauge-covariant form: +AΛ +β=1[α] = +� +d4x +� +− +1 +32π2 +� +εµνρσ tr (F µνF ρσα) += +� +d4x +� +− +1 +64π2 +� +tr +� +{ta, tb} tc� +εµνρσ F aµνF bρσαc . +(4.18) +As discussed in Sec. 3.4, the covariant anomaly generically would not satisfy +the Wess-Zumino consistency condition. +However, we also mentioned some +exceptions to this, such as when the anomaly itself is zero. From the equation +above, we see that this can be achieved by the standard anomaly cancellation +condition tr +� +{ta, tb} tc� += 0, where we recall that the internal trace ‘tr’ also +sums over the fermion species. +• If βa = 0 (∀a), we learned from Sec. 3.4 that the Wess-Zumino consistency +condition should be satisfied. In this case, Eq. (4.15) indeed reproduces the +familiar result for the consistent anomaly: +AΛ +β=0[α] = +� +d4x +1 +16π2 tr +� +− 2 +� +Λ2 +� ∞ +0 +duf(u) +� +Gµ (∂µα) ++ 1 +3 +� � +∂2Gν +� ++ i[(∂µGµ) , Gν] + i[Fµν, Gµ] − GµGνGµ� +(∂να) +− 1 +6 εµνρσ +� +Gρ , 2 (∂µGν) − iGµGν� +(∂σα) +� += +� +d4x +� +1 +48π2 εµνρσ tr +� +(∂µα) (GνFρσ + iGνGρGσ) +� +− δαLΛ +ct,0 +� += +� +d4x +� +1 +48π2 tr +�� +ta, tb� +, tc� +εµνρσ (∂µαa) +�� +∂νGb +ρ +� ++ 1 +4 f bdeGd +νGe +ρ +� +Gc +σ +− δαLΛ +ct,0 +� +, +(4.19) +up to an irrelevant anomaly given by the gauge variation of the following local +– 26 – + +counterterm: +LΛ +ct,0 = +1 +16π2 +� +Λ2 +� ∞ +0 +duf(u) +� +tr +� +GµGµ +� ++ +1 +96π2 tr +�� +∂µGµ +�2 − 2iF µνGµGν + 1 +2 GµGνGµGν +� +. +(4.20) +The relevant anomaly in Eq. (4.19) is proportional to tr +� +{ta, tb} tc� +, which +depends on the fermion content of the theory. The symmetries under consid- +eration can be gauged when there is no relevant anomaly, that is, when the +standard anomaly cancellation condition tr +� +{ta, tb} tc� += 0 is satisfied. +5 +Implications of the Master Formula +In this section, we apply Eqs. (4.15) and (4.17) derived in the previous section, which +are evaluation results of our master formula Eq. (4.6), to obtain explicit results for the +anomaly in all possible combinations of the continuous group sectors. We consider +in turn a simple non-Abelian group, semi-simple product of non-Abelian sectors, +product of Abelian sectors, and finally the general case of product of non-Abelian +and Abelian sectors. In each case, we aim to answer the following questions: +• What values of the regularization parameters β are consistent with the Wess- +Zumino condition? +• For these Wess-Zumino consistent β choices, what is the relevant anomaly, and +what are the counterterms associated with the irrelevant anomaly? +• What are the conditions for the relevant anomaly to vanish (in which case the +symmetries under consideration can be gauged in the quantum theory)? +To investigate the first question, we use the BRST form of the Wess-Zumino +consistency condition, which states that (recall the discussion around Eq. (2.12)) +when the gauge variation parameter α is replaced by the ghost field ω, the anomaly +is BRST-closed: +δBRSTAβ[ω] = 0 , +(5.1) +where Aβ[ω] is understood as the renormalized anomaly defined in Eq. (3.24). Since +the gauge variation of local counterterms is always BRST-closed due to the nil- +potency of the BRST transformation, this requires the regularized anomaly is also +BRST-closed: +δBRSTAΛ +β[ω] = 0 . +(5.2) +– 27 – + +We will check this condition up to O(1/Λ) terms. To do so, it is useful to recall that +under the BRST transformation: +δBRSTGµ = Dµω = ∂µω − i +� +Gµ, ω +� +, +(5.3a) +δBRSTFµν = −i +� +Fµν, ω +� +, +(5.3b) +δBRSTω = iω2 , +(5.3c) +δBRST +� +∂µω − iβ +� +Gµ, ω +�� += i(1 − β) +� +ω, ∂µω +� +. +(5.3d) +In answering the second and third questions, we will see how the well-known +results for anomalies are recovered in our formalism with specific (Wess-Zumino con- +sistent) β choices. We will also see that for all the Wess-Zumino consistent anomalies, +the standard anomaly cancellation condition tr +� +{ta, tb} tc� += 0 will guarantee that +the relevant anomaly vanishes (which means the symmetries can be gauged). +5.1 +Simple Non-Abelian Group +For a simple non-Abelian group, all the β factors are degenerate, so we omit their +adjoint indices and simply write all of them as β. We can first verify that the O(Λ2) +term in Eq. (4.15) is BRST-closed: +δBRSTAΛ +β[ω] +�� +O(Λ2) = − 1 +8π2 +� +d4x +� +Λ2 +� ∞ +0 +duf(u) +� +(1 − β) +× tr +� +(∂µω)(∂µω) + i +� +ω , Gµ(∂µω) +�� += 0 . +(5.4) +Note that cyclic permutation of a Grassmann odd matrix in the trace is accompanied +by a minus sign if it passes through an odd number of Grassmann odd matrices, e.g. +tr +� +ω Gµ(∂µω) +� += − tr +� +Gµ(∂µω) ω +� +. +To derive constraints on β from the Wess-Zumino consistency condition, we +need to consider the O(Λ0) terms. The BRST transformation of these terms is quite +tedious. However, as we will show, it turns out sufficient to work out just a subset +of terms. Let us first note that AΛ +β[ω]|O(Λ0) contains terms of the form: +ωG∂3 , +ωG2∂2 , +ωG3∂ , +ωG4 , +(5.5) +whose BRST transformation contains terms of the form:11 +ω2G∂3 , +ω2G2∂2 , +ω2G3∂ , +ω2G4 . +(5.6) +We will see that the ω2G4 and ω2G∂3 terms are sufficient to constrain β. +The ω2G4 terms can only come from BRST transforming the ωG4 terms in +11Note that the ω2∂4 term from BRST transforming the sole ωG∂3 term in Eq. (4.15) vanishes. +– 28 – + +AΛ +β[ω]. Those ωG4 terms that do not involve εµνρσ are easily seen to vanish upon +cyclic permutation, and we are left with +AΛ +β[ω] +�� +G4ω = +� +d4x +1 +24π2 β (1 + β + β2) εµνρσ tr +� +GµGνGρGσω +� += +� +d4x +� +− +1 +192π2 +� +β (1 + β + β2) tr +� +{ta, tb} tc� +× εµνρσf adef bfgGdµGeνGfρGgσωc . +(5.7) +Since δBRSTAΛ +β[ω] +�� +G4ω2 = 0 requires AΛ +β[ω] +�� +G4ω = 0, while (1 + β + β2) is positive- +definite, we see that +δBRSTAΛ +β[ω] +�� +G4ω2 = 0 +=⇒ +β = 0 +or +tr +� +{ta, tb} tc� += 0 . +(5.8) +As discussed around Eq. (4.19), β = 0 reproduces the standard consistent anomaly, +plus an irrelevant piece that is obviously BRST-closed. +The other option is the +standard anomaly cancellation condition tr +� +{ta, tb} tc� += 0; when this is true, the +terms in AΛ +β that are proportional to εµνρσ all vanish. In this case, it remains to +check whether there are additional constraints on the value of β from the terms not +involving εµνρσ. To do so, we focus on the ω2G∂3 terms in δBRSTAΛ +β[ω], for which we +find, after some simplification using cyclic permutation and integration by parts: +δBRSTAΛ +β[ω] +�� +ω2G∂3 = +� +d4x +i +48π2 β (1 − β) tr +�� +∂2� +Gν, ω +� +− +� +(∂2Gν), ω +�� +(∂νω) +� += +� +d4x +1 +48π2 β (1 − β) tr +� +tatb� +× f acd�� +∂2Gcν� +ωd − ∂2� +Gcνωd�� +(∂νωb) . +(5.9) +Here the group theory factor tr +� +tatb� +∝ δab is always non-vanishing, so we see the +only other option (besides β = 0) that makes δBRSTAΛ +β[ω] +�� +ω2G∂3 vanish is β = 1, in +which case the εµνρσ-independent part of AΛ +β simply vanishes. +In summary, we conclude that consistency with the Wess-Zumino condition re- +quires either of the following to be true: +• β = 0, in which case the result is given by Eq. (4.19). This reproduces the +standard consistent anomaly plus an irrelevant piece that is equal to the gauge +variation of the local counterterm given by Eq. (4.20). +As discussed below +Eq. (4.19), for the relevant anomaly to vanish in this case, one needs the stan- +dard anomaly cancellation condition tr +� +{ta, tb} tc� += 0. +• tr +� +{ta, tb} tc� += 0 and β = 1, in which case anomaly cancellation happens and +the regularized anomaly vanishes altogether, AΛ +β[α] = 0. +– 29 – + +5.2 +Product of Non-Abelian Sectors +For a semi-simple product of non-Abelian sectors, the only additional term in AΛ +β[α] +to consider is the tr +� +tatbtctd� +term in Eq. (4.17). Both tr +� +tatb� +and tr +� +{ta, tb} tc� +vanish when the generators belonging to more than one simple sectors are involved, +while tr +� +tatbtctd� +can be nonzero when two of the four generators belong to one simple +sector and the other two belong to another simple sector. +Upon imposing the conditions derived in the previous subsection on each sim- +ple non-Abelian sector, we see that there are only two scenarios. If β = 1 (and +tr +� +{ta, tb} tc� += 0) for either sector, the aforementioned cross term in AΛ +β[α] vanishes +because of the (1 − βa)(1 − βb)(1 − βc) factor. If β = 0 for both sectors, the cross +term is contained in the general result Eq. (4.19), specifically the gauge variation +of the O(G4) counterterm in Eq. (4.20). Therefore, no additional constraints arise +from the Wess-Zumino consistency condition beyond those already derived for each +simple sector. The same is true for the relevant anomaly cancellation condition. +5.3 +Product of Abelian Sectors +For an Abelian gauge group, we can set f abc = 0 and F µν +lin = F µν in Eq. (4.17) to +obtain +AΛ +β[α] = +� +d4x +1 +16π2 +� +− +� +a,b +tr(QaQb) · (1 − βa) +� +2 +� +Λ2 +� ∞ +0 +duf(u) +� +Ga +µ − 1 +3 +� +∂2Ga +µ +��� +∂µαb� +− +� +a,b,c,d +tr(QaQbQcQd) · 1 +3 (1 − βa)(1 − βb)(1 − βc) GaµGbνGc +µ +� +∂ναd� +− +� +a,b,c +tr(QaQbQc) · 1 +8 +� +(1 + βa)(1 + βb) + 1 +3(1 − βa)(1 − βb) +� +εµνρσF a +µνF b +ρσαc +� +, +(5.10) +where we have written the group generators ta as Qa since they are just charges under +the U(1)’s, and ‘tr’ means summing over all chiral fermions. In the tr(QaQbQc) term, +we have integrated by parts and symmetrized the coefficient between a and b: +βaβb + 1 +3(1 + 2βa)(1 − βb) → 1 +4 +� +(1 + βa)(1 + βb) + 1 +3(1 − βa)(1 − βb) +� +. +(5.11) +Under the BRST transformation, only the gauge fields Ga +µ transform nontrivially +– 30 – + +while F a +µν and ωa stay invariant, and we obtain +δBRSTAΛ +β[ω] = +� +d4x +1 +16π2 +� � +a,b +tr(QaQb) · (βa − βb) +× +�� +Λ2 +� ∞ +0 +duf(u) +� +(∂µωa) − 1 +6 +� +∂2∂µωa�� +(∂µωb) ++ +� +a,b,c,d +tr(QaQbQcQd) · 1 +6 (1 − βa)(1 − βb)(βc − βd) +× +� +GaµGb +µ +� +∂νωc�� +∂νωd� ++ 2Ga +µGb +ν +� +∂µωc�� +∂νωd��� +, +(5.12) +where we have used the (anti-)symmetry between the adjoint indices to simplify the +expression. +From Eq. (5.12) we see that the Wess-Zumino consistency condition +δBRSTAΛ +β[ω] = 0 requires the following: +• βa = βb for any two Abelian sectors a, b for which tr(QaQb) ̸= 0. +• Either βa = βb = βc = βd or at least two of them are equal to 1 for any group +of Abelian sectors for which tr(QaQbQcQd) ̸= 0.12 +When these conditions are satisfied, symmetrizing the indices allows one to show +that the tr(QaQb) and tr(QaQbQcQd) terms in Eq. (5.10), if nonzero, are equal to +the gauge variation of local counterterms, and we have: +AΛ +β[α] = +� +d4x +� +−δαL(β) +ct − +1 +128π2 +� +a,b,c +tr(QaQbQc) +× +� +(1 + βa)(1 + βb) + 1 +3(1 − βa)(1 − βb) +� +εµνρσF a +µνF b +ρσαc +� +, (5.13) +where +L(β) +ct = +1 +16π2 +� � +a,b +tr(QaQb)(1 − βa) +�� +Λ2 +� ∞ +0 +duf(u) +� +Ga +µGbµ − 1 +6 +� +∂2Ga +µ +� +Gbµ +� ++ +� +a,b,c,d +tr(QaQbQcQd) · 1 +12 (1 − βa)(1 − βb)(1 − βc) GaµGbνGc +µGd +ν +� +. +(5.14) +Therefore, as in the non-Abelian case, a relevant anomaly may only come from terms +with three gauge group generators. But unlike the non-Abelian case, β values other +12This applies to groups of two, three and four Abelian sectors since a, b, c, d do not have to be +distinct. +– 31 – + +than 0 and 1 are allowed. Again, we see that the standard anomaly cancellation con- +dition tr +� +{ta, tb} tc� += 0 (i.e. tr(QaQbQc) = 0 in the Abelian case) would guarantee +that the relevant anomaly vanishes.13 +U(1)V × U(1)A Example +Let us apply the results above to the classic example of two Abelian sectors U(1)V × +U(1)A. The matter content is assumed to consist of pairs of Weyl fermions with +opposite (identical) charges under U(1)V (U(1)A); the minimal case is that of two +Weyl fermions with (QV , QA) = (1, 1) and (−1, 1), respectively. So the potentially +nonzero traces are: +tr +� +Q2 +V +� +, +tr +� +Q2 +A +� +, +(5.15a) +tr +� +Q2 +V QA +� +, +tr +� +Q3 +A +� +, +(5.15b) +tr +� +Q4 +V +� +, +tr +� +Q2 +V Q2 +A +� +, +tr +� +Q4 +A +� +. +(5.15c) +The fact that tr(Q2 +V Q2 +A) ̸= 0 implies that to satisfy the Wess-Zumino consistency +condition we must choose +βV = βA +or +βV = 1 +or +βA = 1 . +(5.16) +Assuming one of these is true, we can readily obtain the anomaly result from Eq. (5.13): +AΛ +β[α] = +� +d4x +� +−δαL +(βV ,βA) +ct +− +1 +64π2 tr +� +Q2 +V QA +�� +(1 + βV )(1 + βA) + 1 +3(1 − βV )(1 − βA) +� +εµνρσF µν +V F ρσ +A αV +− +1 +128π2 tr +� +Q2 +V QA +�� +(1 + βV )2 + 1 +3(1 − βV )2 +� +εµνρσF µν +V F ρσ +V αA +− +1 +128π2 tr +� +Q3 +A +�� +(1 + βA)2 + 1 +3(1 − βA)2 +� +εµνρσF µν +A F ρσ +A αA +� +. +(5.17) +As discussed in footnote 13, there is in fact an additional possible counterterm, +εµνρσF µν +V V ρAσ (where V and A denote gauge fields), whose gauge variation pro- +duces a linear combination of εµνρσF µν +V F ρσ +A αV and εµνρσF µν +V F ρσ +V αA upon integration +by parts. We will come back to this point shortly. +13One may further ask whether the tr(QaQbQc) terms in Eq. (5.13) may also be irrelevant. Indeed, +there are local counterterms of the form εµνρσF a +µνGb +ρGc +σ one can write down. However, there may +not be enough such counterterms to absorb all the anomalies; in particular, if tr +� +Q3 +a +� +̸= 0 for some +Abelian sector a there must be a relevant anomaly, since the counterterm above vanishes when +a = b = c. +– 32 – + +Eq. (5.17) reproduces the standard result if we further demand that U(1)V is not +anomalous and is preserved by renormalization. This means that we should pick the +βV = 1 option in Eq. (5.16) so that L +(βV ,βA) +ct +does not involve U(1)V -breaking opera- +tors (see Eq. (5.14)). This also rules out the additional counterterm εµνρσF µν +V V ρAσ +discussed above. For U(1)V to be non-anomalous, the coefficient of the αV term in +Eq. (5.17) must vanish, which requires βA = −1 for βV = 1. We conclude that the +standard result corresponds to the specific scheme choice in our formalism: +(βV , βA) = (1 , −1) , +(5.18) +in which case Eq. (5.17) becomes: +AΛ +(1,−1)[α] = +� +d4x +� +−δαL(1,−1) +ct +− +1 +32π2 εµνρσ +� +tr +� +Q2 +V QA +� +F µν +V F ρσ +V + tr +� +Q3 +A +� +· 1 +3 F µν +A F ρσ +A +� +αA +� +, +(5.19) +with the following U(1)V -preserving counterterm: +L(1,−1) +ct += +1 +16π2 +� +tr +� +Q2 +A +�� +2 +� +Λ2 +� ∞ +0 +duf(u) +� +AµAµ − 1 +3 +� +∂2Aµ� +Aµ +� ++ tr +� +Q4 +A +� +· 2 +3 (AµAµ)2 +� +. +(5.20) +It is interesting to note that if we instead choose +(βV , βA) = (0 , 0) , +(5.21) +which is also Wess-Zumino consistent but does not manifestly preserve U(1)V , we +would obtain: +AΛ +(0,0)[α] = +� +d4x +� +−δαL(0,0) +ct +− +1 +48π2 εµνρσ tr +� +Q2 +V QA +� +F µν +V F ρσ +A αV +− +1 +96π2 εµνρσ +� +tr +� +Q2 +V QA +� +F µν +V F ρσ +V + tr +� +Q3 +A +� +F µν +A F ρσ +A +� +αA +� +. +(5.22) +This is in fact related to the standard result Eq. (5.19) by a counterterm: +AΛ +(0,0)[α] = AΛ +(1,−1)[α] + δα +� +d4x +� +L(1,1) +ct +− L(0,0) +ct ++ ∆Lct +� +, +(5.23) +– 33 – + +where +∆Lct = +1 +24π2 εµνρσ tr +� +Q2 +V QA +� +F µν +V V ρAσ . +(5.24) +Therefore, (βV , βA) = (0, 0) actually gives the same relevant anomaly as the standard +result, although at the cost of U(1)V -breaking counterterms. Note that it is impos- +sible to remove both εµνρσF µν +V F ρσ +A αV and εµνρσF µν +V F ρσ +V αA using the counterterm, in +agreement with the familiar result that U(1)V and U(1)A cannot be simultaneously +conserved in the V V A triangle diagram. Also, as discussed in footnote 13, there is +always a relevant U(1)3 +A anomaly which cannot be removed by counterterms. +5.4 +Product of Abelian and Non-Abelian Sectors +Finally, we consider the cross terms in AΛ +β[α] between Abelian and non-Abelian +sectors. These include the tr +� +{ta, tb} tc� +terms in Eq. (4.17) with two of the adjoint +indices in the same non-Abelian sector and the third index in an U(1) sector, and +the tr +� +tatbtctd� +terms with two of the adjoint indices in the same non-Abelian sector +and the other two in either one or two U(1) sectors. So in what follows we focus on +a theory with one simple non-Abelian sector and up to two U(1) sectors, which we +call U(1)A and U(1)B. To ease the presentation we reserve the notation Gµ, F µν, α, +ta that we have been using in the general calculation for the non-Abelian sector here, +while denoting the corresponding objects in the U(1) sectors by Aµ, F µν +A , αA, QA +and Bµ, F µν +B , αB, QB. We use βNA to represent the common β parameter associated +with all the non-Abelian generators, and use βA, βB for the β parameters of the U(1) +sectors. +From the discussion in Sec. 5.1 we know that the only values of βNA consistent +with the Wess-Zumino condition in the non-Abelian sector are 1 and 0. Let us first +consider the simpler βNA = 1 case. Here the tr +� +tatbQAQB +� +terms are all multiplied +by (1−βNA) and vanish, while for the tr +� +tatbQA +� +terms we have (switching to matrix +notation and following Eq. (4.15)): +AΛ +β[α] ⊃ +� +d4x +� +− +1 +32π2 +� +εµνρσ tr +� +F µνF ρσαA ++ 2βAF µνF ρσ +A α + 2(1 − βA)F µνAρ(Dσα) +� += +� +d4x +� +− +1 +32π2 +� +εµνρσ tr +� +F µνF ρσαA + (1 + βA)F µνF ρσ +A α +� +. +(5.25) +To arrive at the last equation we have integrated by parts and used the Bianchi +identity εµνρσ(DσFµν) = 0. Performing the BRST transformation, we find +δBRSTAΛ +β[ω] ⊃ +� +d4x +i +32π2 (1 + βA) εµνρσ tr +� +F µνF ρσ +A ω2� +. +(5.26) +So for these cross terms in the anomaly to be consistent with the Wess-Zumino +– 34 – + +condition, we must have +βA = −1 +or +tr +� +tatbQA +� += 0 +(βNA = 1 case) . +(5.27) +As a result, Eq. (5.25) either vanishes due to tr +� +tatbQA +� += 0, in which case there is no +crossed anomaly, or only the F µνF ρσαA term survives; the latter cannot be obtained +as a local counterterm variation and is therefore a relevant anomaly. In fact, we +have just recovered the non-Abelian generalization of the U(1)V × U(1)A example +in the previous subsection (cf. Eq. (5.19)): swapping U(1)V for a non-anomalous +non-Abelian sector (recall that βNA = 1 requires tr +� +{ta, tb} tc� += 0) leads to the same +crossed anomaly with a chiral U(1). +Next we consider the other option, βNA = 0, for the non-Abelian sector. +In +this case, both the tr +� +tatbQA +� +and tr +� +tatbQAQB +� +terms can be nonzero. After some +algebra we can organize the tr +� +tatbQA +� +terms into the following form: +AΛ +β[α] ⊃ +� +d4x +� +− +1 +96π2 +� +εµνρσ tr +� +F µνF ρσαA + iGµGνF ρσαA + 2GµGνGρGσαA ++ 3 +2(1 + βA)F µν +lin F ρσ +A α − (1 − βA)F µνAρ(∂σα) +� +. +(5.28) +Among the five terms, three (first, third and fourth) are actually BRST-invariant. +Overall, we find Eq. (5.28) has the following BRST transformation: +δBRSTAΛ +β[ω] ⊃ +� +d4x +� +− +1 +96π2 +� +βA εµνρσ tr +� +F µν(∂ρω)(∂σωA) +� +. +(5.29) +For this to vanish, we need +βA = 0 +or +tr +� +tatbQA +� += 0 +(βNA = 0 case) . +(5.30) +So the crossed anomaly in Eq. (5.28) either vanishes due to tr +� +tatbQA +� += 0 or is +contained in the general β = 0 formula Eq. (4.19) as a relevant anomaly. +Meanwhile, for the tr +� +tatbQAQB +� +terms, we find: +AΛ +β[α] ⊃ +� +d4x +� +− +1 +48π2 +� +tr +� +(1 − βA) +� +{Gµ, Gν} Aµ + GµGµAν� +(∂ναB) ++ (1 − βB) +� +{Gµ, Gν} Bµ + GµGµBν� +(∂ναA) ++ 2(1 − βA)(1 − βB) +� +(AµBν + AνBµ) Gµ + AµBµGν� +(∂να) +� +, (5.31) +– 35 – + +which transforms under BRST as: +δBRSTAΛ +β[ω] ⊃ +� +d4x +1 +48π2 tr +� +(βA − βB) +� +2GµGν(∂µωA) + GµGµ(∂νωA) +� +(∂νωB) +− 2(1 − βA)βB +� +Gµ(∂νω)Aµ + Gν(∂µω)Aµ + Gµ(∂µω)Aν� +(∂νωB) +− 2(1 − βB)βA +� +Gµ(∂νω)Bµ + Gν(∂µω)Bµ + Gµ(∂µω)Bν� +(∂νωA) +� +. +(5.32) +For this to vanish, we need +βA = βB = (0 or 1) +or +tr +� +tatbQAQB +� += 0 +(βNA = 0 case) . +(5.33) +So the crossed anomaly in Eq. (5.31) either vanishes due to tr +� +tatbQAQB +� += 0 or +βA = βB = 1, or is contained in the general β = 0 formula Eq. (4.19) as an irrelevant +anomaly. +For both cases discussed above, βNA = 1 and βNA = 0, the relevant part of the +crossed anomaly is proportional to tr +� +tatbQA +� +, so the anomaly cancellation condition +is contained in the standard one, tr +� +{ta, tb} tc� += 0. +6 +Discussion and Future Directions +In this paper, we introduced a novel regularization prescription to calculate anoma- +lies for global and gauge symmetries using CDE. The calculation was performed in +d = 4 spacetime dimensions, thereby avoiding any of the subtleties that arise when +computing anomalies using dimensional regularization. The master formula obtained +in this framework integrates various known results regarding anomalies. +In a companion paper [36], we will extend the formalism developed here to +incorporate the effects of higher dimensional operators into the anomaly calculation. +This has an immediate application to the Standard Model Effective Field Theory +(SMEFT). Recently, arguments that the SMEFT is not anomalous were provided in +Refs. [37, 38]. In Ref. [36], we will give an explicit proof using CDE that SMEFT +is non-anomalous when including operators with general scalar, vector, and tensor +couplings to fermion bilinears. +In future work, we would like to apply this formalism to compute the EFTs that +emerge when integrating out fermions with chiral couplings (for example, integrat- +ing out the top quark in the Standard Model). This is well-known to produce an +EFT with a Wess-Zumino-Witten term [35, 39, 40]. It should be possible to extend +the calculations presented here to reproduce this result in a new way. +This will +require understanding the interplay of the method presented here and the results +for other functional traces that are evaluated using dimensional regularization, since +the functional EFT matching framework relies on the method of regions, which is +– 36 – + +implemented in dimensional regularization. At least for one loop calculations, the +use of different regulators may not cause any particular difficulties. Once this is +understood, functional methods for one-loop matching will be a complete framework +for integrating out any heavy particles with spins 0, 1/2, and 1. +Acknowledgments +We thank Quentin Bonnefoy, Nathaniel Craig, Sungwoo Hong, Markus Luty and +Aneesh Manohar for useful discussions. T.C. is supported by the U.S. Department +of Energy under grant number DE-SC0011640. X.L. is supported by the U.S. De- +partment of Energy under grant numbers DE-SC0009919 and DE-SC0011640. Z.Z. +is supported by the U.S. Department of Energy under grant number DE-SC0011702. +This work was performed in part at Aspen Center for Physics, which is supported +by National Science Foundation grant PHY-1607611. +Appendix +A +Comments on Cyclic Permutation +In this appendix, we clarify a subtle point in performing CDE calculations, i.e., +when (and why) we are allowed to perform cyclic permutations on the argument of +a functional trace ‘ Tr (· · · ) ’, and a lowercase trace ‘ tr (· · · ) ’ which is only over the +internal indices. +We begin by recalling that a functional operator O is a matrix that acts on both +the functional vector space |x⟩ and some internal vector space. The latter is typically +finite dimensional, which we can label by a discrete index i. We can then write out +the concrete relation between the functional trace ‘ Tr ’ and the internal trace ‘ tr ’: +Tr (O) = +� +d4x ⟨x| tr (O) |x⟩ = +� +d4x ⟨x| Oii |x⟩ . +(A.1) +Clearly, the functional trace ‘Tr’ sums over all the indices of the matrix O, and +therefore it is always safe to perform a cyclic permutation: +Tr +� +OAOB� += +� +d4x d4y ⟨x| OA +ij |y⟩ ⟨y| OB +ji |x⟩ += +� +d4y d4x ⟨y| OB +ji |x⟩ ⟨x| OA +ij |y⟩ = Tr +� +OBOA� +. +(A.2) +On the other hand, the internal trace ‘tr’ only sums over a subset of indices for the +matrix O, and therefore it is generically illegal to make cyclic permutations inside +‘tr’ alone: +tr +� +OAOB� +̸= tr +� +OBOA� +⇐⇒ +⟨x| tr +� +OAOB� +|y⟩ ̸= ⟨x| tr +� +OBOA� +|y⟩ . +(A.3) +– 37 – + +Note that after taking the internal trace, the object tr +� +OAOB� +is still a matrix +acting on the functional space spanned by |x⟩. So when we check whether the two +objects tr +� +OAOB� +and tr +� +OBOA� +are equal, it is a comparison of two matrices where +one needs to compare entry by entry, as indicated by the right-hand expression of +Eq. (A.3). Generically, they are not equal and making cyclic permutations inside ‘tr’ +alone is not allowed. +However, in many practical calculations of functional traces, the evaluation re- +sults (after carrying out the loop integrals) are local action-like expressions that +generically have the form (see e.g. Eqs. (4.6) and (4.7)) +Tr (· · · ) = +� +d4x trx +� +OAOBOC · · · +� +, +(A.4) +where the reason for using a slightly different notation ‘ trx (· · · ) ’ will become clear +shortly. When handling expressions like Eq. (A.4), we do sometimes make cyclic +permutations to simplify the calculation: +Sometimes we take : +� +d4x trx +� +OAOB� += +� +d4x trx +� +OBOA� +. +(A.5) +This has been used extensively in Sec. 4, as well as for many functional matching +calculations with CDE in the literature. The purpose of this appendix is to clarify +when and why Eq. (A.5) could hold. The explanation has two important aspects: +• There is a slight abuse of notation ‘ tr ’ in expressions like Eqs. (4.6) and (4.7). +As emphasized by using a different notation ‘ trx ’ above, the traces in Eqs. (A.4) +and (A.5) are not precisely the same objects as the traces in Eq. (A.3) – the +latter are matrices acting on the functional space |x⟩, while the former are +actually elements of those matrices. +• Eq. (A.5) does not hold for generic operators OA, OB. However, if both OAOB +and OBOA are diagonal functional operators in the position basis |x⟩, namely +if they satisfy +tr +� +OAOB� +|x⟩ = tAB(x) |x⟩ , +(A.6a) +tr +� +OBOA� +|x⟩ = tBA(x) |x⟩ . +(A.6b) +for some ordinary functions tAB(x) and tBA(x), then Eq. (A.5) holds. +In what follows, we elaborate on these two aspects in turn. +A.1 +Internal Trace Notation +First, it is clear from Eq. (A.4) that trx (O) must be an ordinary function of the +variable x (similar to a Lagrangian), such that the integral in Eq. (A.4) would yield +– 38 – + +a local action-like result. So trx (O) cannot be a matrix on the functional space |x⟩. +Instead, it should be interpreted as an element of that matrix. +Second, we emphasize that trx (O) is not the following matrix element that one +might naively expect: +trx (O) ̸= ⟨x| tr (O) |x⟩ . +(A.7) +If the above were true, then performing the integral in Eq. (A.5) would give us the +functional trace +� +d4x ⟨x| tr +� +OAOB� +|x⟩ = Tr +� +OAOB� +, +(A.8) +in which cyclic permutation would not be a problem at all, as explained around +Eq. (A.2). But it is clear that Eq. (A.5) is not supposed to yield Tr +� +OAOB� +. The +correct matrix element is +trx (O) = +� +d4y ⟨x| tr (O) |y⟩ . +(A.9) +To understand this subtle point, we need to remind ourselves how we usually obtain +expressions like Eq. (A.4) and hence terms like trx (O) from the CDE evaluation. +Usually, we start with a functional trace like Eq. (4.1) and calculate it using momen- +tum eigenstates: +Tr +� +f +� +iˆ∂µ, U(ˆx) +�� += +� +d4q +(2π)4 +� +q +��� tr +� +f +� +iˆ∂µ, U(ˆx) +�����q +� +. +(A.10) +Using the fact +|q⟩ = +� +d4x |x⟩ ⟨x|q⟩ = +� +d4x e−iqx |x⟩ = +� +d4x e−iqˆx |x⟩ , +(A.11) +we can rewrite Eq. (A.10) as +Tr +� +f +� +iˆ∂µ, U(ˆx) +�� += +� +d4x d4y +� +d4q +(2π)4 +� +x +���eiqˆx tr +� +f +� +iˆ∂µ, U(ˆx) +�� +e−iqˆx���y +� += +� +d4x d4y +� +d4q +(2π)4 +� +x +��� tr +� +f +� +qµ + iˆ∂µ, U(ˆx) +�����y +� += +� +d4x +�� +d4y +� +x +���� +� +d4q +(2π)4 tr +� +f +� +qµ + iˆ∂µ, U(ˆx) +������y +�� += +� +d4x +� � +d4y ⟨x| tr (Of)|y⟩ +� +, +(A.12) +where Of is defined implicitly by the last equation. As indicated in the last line, one +way of understanding the ‘simplified CDE’ is that one Taylor expands the function +– 39 – + +‘ f ’ above and performs the momentum loop integral over qµ to obtain a set of +functional operators of the form tr (Of). This is precisely what we did in deriving +Eqs. (4.6) and (4.7) from Eq. (4.1). Now comparing Eq. (A.12) with Eq. (A.4), we +see that the notation ‘ trx ’ is actually denoting the quantity inside the curly brackets +in Eq. (A.12). Therefore, we have carefully derived the relation in Eq. (A.9). +Let us recall that the definition of the ‘functional vector space’ is the collection +of all the functions φ(x) (usually satisfying certain constraints, such as ∥φ∥2 < ∞ +(under box normalization)), where each function corresponds to a vector |φ⟩: +φ(x) = ⟨x|φ⟩ . +(A.13) +It thus provides us with a linear algebra language for the differential operations. +Specifically, the process of a differential operator ˆf acting on a function φ(x) to yield +a new function +� ˆfφ +� +(x) can be written as the action of a matrix in this linear space: +� ˆfφ +� +(x) = +� +x +�� ˆfφ +� += +� +x +�� ˆf +��φ +� += +� +d4y +� +x +�� ˆf +��y +� +⟨y|φ⟩ . +(A.14) +The key to this dictionary are the matrix elements +� +x +�� ˆf +��y +� +for various differential +operators. When ˆf is an ordinary function such as ˆf = Gµ(x), its matrix is diagonal +in the |x⟩ basis: +⟨x|Gµ|y⟩ = Gµ(x) δ4(x − y) , +(A.15a) +Gµ(x)φ(x) = +� +d4y ⟨x|Gµ|y⟩ ⟨y|φ⟩ = +� +d4y +� +Gµ(x) δ4(x − y) +� +φ(y) . +(A.15b) +When ˆf = ∂µ is a derivative, we have +⟨x|∂µ|y⟩ = +∂ +∂xµ δ4(x − y) , +(A.16a) +∂µφ(x) = +� +d4y ⟨x|∂µ|y⟩ ⟨y|φ⟩ = +∂ +∂xµ +� +d4y δ4(x − y) φ(y) . +(A.16b) +General differential operators, like ˆf +� +iˆ∂µ, U(ˆx) +� +in Eq. (A.12), are built from the two +kinds of operators discussed above. +We note in particular that the constant unity function ‘1’ corresponds to a vector +|1⟩ that satisfies +|1⟩ = +� +d4y |y⟩ ⟨y|1⟩ = +� +d4y |y⟩ . +(A.17) +Therefore, the relation in Eq. (A.9) can be rewritten as +trx (O) = ⟨x| tr (O) |1⟩ = +� +tr (O) 1 +� +(x) , +(A.18) +– 40 – + +where the last expression follows from the differential operation language in Eq. (A.14) +– we are simply taking the differential operator tr (O), acting it on the constant unity +function 1, and then evaluating the resulting function at point x.14 When the func- +tion being acted on is the constant unity function 1, we often suppress it. We also +often suppress the explicit ‘(x)’ when talking about a function. Doing both for the +last expression in Eq. (A.18) leads to our abuse of the notation ‘tr’ in the main text. +From Eq. (A.18), it is immediately clear that +trx (AB · · · C∂µ) = 0 , +(A.19a) +trx (∂µAB · · · C) is a total derivative. +(A.19b) +With these, we can see a quick counterexample to Eq. (A.5): +trx (A∂µBµ) = trx +� +A (∂µBµ) + ABµ∂µ +� += trx +� +A (∂µBµ) +� +, +(A.20a) +trx (BµA∂µ) = 0 . +(A.20b) +Clearly, the two lines are related by a cyclic permutation of Bµ, but they are gener- +ically not equal. +A.2 +Conditions for Cyclic Permutations in Internal Traces +After clarifying the meaning of ‘ trx (· · · ) ’, namely the relation in Eq. (A.9), we see +that Eq. (A.5) does not always hold. However, if both expressions inside the trace +before and after the cyclic permutation are diagonal operators in the position basis +|x⟩, i.e., if Eq. (A.6) is true, then Eq. (A.5) would hold. +To see this, we first note that if +tr +� +OAOB� +|x⟩ = tAB(x) |x⟩ , +(A.21) +then we simply have +trx +� +OAOB� += +� +d4y +� +x +�� tr +� +OAOB���y +� += +� +d4y tAB(y) δ4(x−y) = tAB(x) . (A.22) +Therefore, it is linked with the functional trace as +Tr +� +OAOB� += +� +d4x +� +d4q +(2π)4 ⟨x|q⟩ +� +q +�� tr +� +OAOB���x +� += +� +d4x tAB(x) +� +d4q +(2π)4 ⟨x|q⟩ ⟨q|x⟩ = +� +d4x trx +� +OAOB� � +d4q +(2π)4 . +(A.23) +14See e.g. Sec. 2.2 of Ref. [41] and App. B.2.2 of Ref. [19] for clarifications of this point. +– 41 – + +where +� +d4q +(2π)4 = ⟨x|x⟩ is just a normalization factor. Making use of this relation +between trx(· · · ) and Tr(· · ·), one could take advantage of Eq. (A.2) to perform a +cyclic permutation: +� +d4x trx +� +OAOB� � +d4q +(2π)4 = Tr +� +OAOB� += Tr +� +OBOA� += +� +d4x trx +� +OBOA� � +d4q +(2π)4 . (A.24) +Canceling the normalization factor gives us Eq. (A.5). +Note that one can generalize Eq. (A.5) to the sum of multiple terms: +OAOB +−→ +OA +1 OB +1 + · · · + OA +n OB +n . +(A.25) +In this case, for the steps in Eq. (A.24) to be valid, one only needs the sum to be +diagonal in the position basis |x⟩, namely we have +� +d4x trx +� +OA +1 OB +1 + · · · + OA +n OB +n +� += +� +d4x trx +� +OB +1 OA +1 + · · · + OB +n OA +n +� +, +(A.26) +provided that +tr +� +OA +1 OB +1 + · · · + OA +n OB +n +� +|x⟩ = tAB(x) |x⟩ , +(A.27a) +tr +� +OB +1 OA +1 + · · · + OB +n OA +n +� +|x⟩ = tBA(x) |x⟩ . +(A.27b) +An operator O being diagonal in the position basis |x⟩ is equivalent to the state- +ment that all the derivatives in O are closed. For example, consider the following +differential operator: +O = A∂µBC = A +� +∂µB +� +C + AB∂µC += A +� +∂µB +� +C + AB +� +∂µC +� ++ ABC∂µ , +(A.28) +where A, B, C are diagonal in the |x⟩ basis. The decomposition in the first line follows +from the product rule of the derivative, where the parentheses in the first term has +the usual interpretation – it indicates that ∂µ only acts on B but not anything to +the right of B. (In fact, this notation was already used in Eq. (A.20a).) In this case, +we say that the derivative is closed on B. In contrast, the second term in the first +line has an open derivative that acts on everything to its right. One can further use +the product rule to obtain the decomposition in the second line, where a term with +the derivative closed on C appears, and there is an additional term with an open +derivative. Clearly, terms with closed derivatives, such as A +� +∂µB +� +C and AB +� +∂µC +� +are diagonal operators in the |x⟩ basis, while terms with open derivatives such as +– 42 – + +ABC∂µ are not; see e.g. Eq. (A.16a). +When evaluating a functional trace with simplified CDE, the initial set of op- +erators in the trace trx(· · · ) emerge from evaluating an expression of the form (see +Eq. (A.12)): +� +d4q +(2π)4 tr +� +f +� +qµ + iˆ∂µ, U(ˆx) +�� += tr (Of) . +(A.29) +The operator tr (Of) derived from such an expression, i.e., upon expanding ‘ f ’ and +carrying out the loop momentum integral, is guaranteed to be diagonal in the position +basis |x⟩, because it is known that one could use the trick of ‘original CDE’ to close +all of the derivatives in it (see e.g. App. B.2.3 of Ref. [19]). However, since Of is a +sum of terms, if we perform an arbitrary cyclic permutation on each term: +tr (Of) = tr +� +OA +1 OB +1 + · · · + OA +n OB +n +� +−→ +tr +� +OB +1 OA +1 + · · · + OB +n OA +n +� +, +(A.30) +it is not guaranteed that the operator is still diagonal in the |x⟩ basis, thus invalidat- +ing the operation. Only a subset of cyclic permutations that satisfy the condition in +Eq. (A.27) are ‘legal.’ +Nonetheless, in practical calculations, a very efficient prescription to ensure that +we are only performing legal cyclic permutations is to stipulate that terms with open +derivatives should not be evaluated – one must keep track of all such terms, and +make sure that they get canceled upon summing the terms obtained after the cyclic +permutations. +If they do not get fully canceled, then it is a sign that an illegal +cyclic permutation had been carried out. In this case, one needs to make further +cyclic permutations until the derivatives are all closed. In summary, insisting that +all derivatives must be closed in the end is an efficient way to make sure that we are +carrying out legal cyclic permutations. The calculations in Sec. 4 of the main text (as +well as in many other functional matching calculations with CDE in the literature) +are done in such a manner. +Let us look at a quick example of this: +trx +� +(∂µA) Bµ� += trx +� +∂µABµ − A∂µBµ� +−→ trx +� +∂µABµ − BµA∂µ +� += trx +� +(∂µABµ) + ABµ∂µ − BµA∂µ +� +−→ trx +� +Bµ∂µA − BµA∂µ +� += trx +� +Bµ (∂µA) +� +. +(A.31) +In the first line, we started with an operator (∂µA) Bµ in which the derivative is +closed. We made a cyclic permutation of the second term to arrive at the second +line, where the derivatives are not fully closed, because the last two terms in the +second expression both have open derivatives and they do not cancel each other. If +we were to stop here and evaluate the second line, then following Eq. (A.19) these +– 43 – + +terms are zero and total derivatives that would not feed into the final result: +� +d4x trx +� +(∂µABµ) + ABµ∂µ − BµA∂µ +� += 0 . +(A.32) +This clearly would not agree with the evaluation of the left-hand side of the first line. +The reason is that the second line was obtained by an illegal cyclic permutation. +Now, if we insist that the second line of Eq. (A.31) should not be evaluated since it +has open derivatives, then we are forced to make further cyclic permutations such +that all the derivatives can be closed upon summing the terms. The third line is +an example of such a further cyclic permutation. As soon as the derivatives are all +closed, we can carry out the evaluation. This prescription guarantees that only legal +cyclic permutations would be performed, and we can see that the result obtained in +the third line does agree with the expression we started with in the first line. +References +[1] R. A. 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Murayama, “How to use the Standard Model effective +field theory,” JHEP 01 (2016) 023, arXiv:1412.1837 [hep-ph]. +– 46 – + diff --git a/K9AyT4oBgHgl3EQf6foW/content/tmp_files/load_file.txt b/K9AyT4oBgHgl3EQf6foW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c48726d540728c951f18840f317316105cde44f --- /dev/null +++ b/K9AyT4oBgHgl3EQf6foW/content/tmp_files/load_file.txt @@ -0,0 +1,1515 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf,len=1514 +page_content='CERN-TH-2023-001 Anomalies From the Covariant Derivative Expansion Timothy Cohen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Xiaochuan Lu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 and Zhengkang Zhang5 1 Theoretical Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' CERN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 1211 Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Switzerland 2 Theoretical Particle Physics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' EPFL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 1015 Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Switzerland 3 Institute for Fundamental Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' University of Oregon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eugene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' OR 97403,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' USA 4 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' San Diego,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' La Jolla,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' CA 92093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' USA 5 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' CA 91106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' USA E-mail: tim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='cohen@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='ch, xil224@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='edu, zkzhang@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='edu Abstract: We revisit the calculation of anomalies for global and gauge symmetries in the framework of the Covariant Derivative Expansion (CDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Due to the presence of UV divergences, the result is an ambiguous quantity that depends on the regular- ization procedure and the renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We introduce a class of regulators that facilitate a straightforward evaluation of the anomaly exclusively in d = 4 space- time dimensions using the CDE methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We derive a master formula for the anomaly that integrates various known results into a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='00821v1 [hep-ph] 2 Jan 2023 Contents 1 Introduction 3 2 Anomalies in the Functional Formalism 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Defining the Anomaly 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Connection to the Path Integral Measure 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Connection to Ward Identities 8 3 Regularizing the Anomaly 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 What is Regularization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Anomaly as a Regulated Functional Trace 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Connection to Other Regularization Prescriptions 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 Consistency With the Wess-Zumino Condition 18 4 Master Formula for the Anomaly From CDE 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Evaluating the Dirac Traces 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 The Evaluated Master Formula 24 5 Implications of the Master Formula 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Simple Non-Abelian Group 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Product of Non-Abelian Sectors 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Product of Abelian Sectors 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 Product of Abelian and Non-Abelian Sectors 34 6 Discussion and Future Directions 36 Acknowledgments 37 Appendix 37 A Comments on Cyclic Permutation 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Internal Trace Notation 38 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Conditions for Cyclic Permutations in Internal Traces 41 References 44 – 2 – 1 Introduction It is well known that symmetries of the classical action can be broken by quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This so-called anomaly has far-reaching consequences, from explaining the neutral pion decay to providing critical consistency checks on gauge theories with chi- ral fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Well-established techniques exist for computing anomalies both using Feynman diagrams, and also directly from the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' They can be com- puted for global and gauged symmetries, Abelian and non-Abelian groups, and take ‘consistent’ and/or ‘covariant’ forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The results generally depend on the choice of regulator, and consist of a relevant piece that reflects the IR properties of the theory, and an irrelevant piece that can be absorbed by varying the renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 In this paper, we revisit the calculation of anomalies from the path integral us- ing an approach known as the Covariant Derivative Expansion (CDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This allows us to derive a unified framework that incorporates various types of anomalies into one master formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The CDE was originally invented in the mid-1980s [15–17] to facilitate one-loop calculations of correlation functions purely in terms of functional traces, avoiding the introduction of Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In recent years, the method has been applied in a variety of new settings, which has led to significant theoretical developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' These include the discovery of a variation on the framework, ‘sim- plified CDE’ [18, 19], the incorporation of the method of regions [20], organizing schemes using diagrammatic frameworks [21, 22], as well as techniques that yield effective actions that include all orders in the fields [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' With these developments, the power and efficiency of CDE has been demonstrated for connecting the UV with the IR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', computing low-energy Effective Field Theories (EFTs) from integrating out heavy states in a perturbative UV model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [24] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We now know how to use CDE to perform matching calculations across a mass thresh- old, as well as to extract the renormalization group evolution equations for the EFT couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The CDE has become such a well-developed tool that there now exist packages which automate these calculations [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The practical success of CDE in connecting UV and IR descriptions of quantum field theories motivates applying it to compute the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The approach taken here will be to work exclusively in d = 4 spacetime dimensions, which allows us to avoid any of the complications that arise when attempting to define Weyl fermions in dimensional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 In this paper, we generalize the classic Fujikawa ap- 1There is of course a vast literature on the anomaly, including the excellent reviews Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The story began with its discovery in 1969 by Adler [3] and by Bell and Jackiw [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' It was soon after understood to be one-loop exact [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The connection between the anomaly and the topological winding number of the gauge field was discovered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Of great importance to the approach taken here is Fujikawa’s derivation of the anomaly from the non-invariance of the path integral measure [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2It is well known that handling the γ5 matrix in d ̸= 4 spacetime dimensions is a nontrivial task [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' See Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [32, 33] for recent CDE calculations of anomalies with dimensional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 3 – proach by expressing the anomaly as a functional trace, which must be regularized to be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We introduce a novel regularization prescription, with a set of reg- ulators parameterized by a set of numbers collectively denoted by β, which one can choose based on which symmetries one wishes to preserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We emphasize that our regularization yields unambiguous evaluation results once the values of β are speci- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We derive a master formula for the anomaly using CDE, whose explicit forms are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This master formula encodes a variety of known results for anomaly calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In particular, we examine all possible combinations of continuous symmetry groups, and show in each case how our master formula re- produces the known (relevant) anomaly results, as well as the anomaly cancellation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This establishes that the CDE can accommodate this important effect in perturbative quantum field theory, and sets the stage for its applications to EFT matching across anomalous thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We first review the functional formalism in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2, with an emphasis on the definition of the anomaly and its con- nections to the fermionic path integral measure and the anomalous Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3, we isolate the functional trace that encodes the anomalies and introduce our novel regularization prescription to make it well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We discuss the relation between our regulator and some similar approaches in the literature, and also a suf- ficient condition for it to be consistent with the Wess-Zumino condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4, we carry out the CDE evaluation to obtain our master formula for the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We then demonstrate in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5 that this master formula reproduces various known results regarding anomalies by examining all possible combinations of continuous symme- try groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Some future directions are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A technical clarification regarding CDE manipulations is provided in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2 Anomalies in the Functional Formalism In this section, we briefly review the well-known functional formalism for anomalies, which also serves the purpose of introducing our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Much of this section is drawn from the review article by Bilal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Our discussion here crucially relies on the famous connection between anomalies and the path integral measure first discovered by Fujikawa [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Defining the Anomaly We begin with the definition of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Consider a general gauge theory coupled to a set of left-handed Weyl fermions collectively denoted by χ: L = − 1 4g2F a µνF aµν + χ†¯σµPµχ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) – 4 – where we have defined the Hermitian covariant derivative Pµ ≡ iDµ = i∂µ + Gµ = i∂µ + Ga µta , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) where ta are the (Hermitian) gauge group generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The gauge field strength is given by Fµν = F a µν ta = −i [Pµ, Pν] = (∂µGν) − (∂νGµ) − i [Gµ, Gν] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3) The kinetic term for the gauge fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) should be read as a sum over terms normalized with different gauge couplings in the case of a product gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A gauge transformation can be parameterized by the matrix Uα = eiα = eiαata , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) where the transformation parameters αa = αa(x) are functions of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4), the building blocks of our theory transform as χ → χα = Uαχ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5a) χ† → χ† α = χ†U † α , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5b) P µ → P µ α = UαP µU † α , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5c) Gµ → Gµ α = UαGµU † α + Uα � i∂µU † α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5d) We will use δα to denote the first-order (in α) gauge variation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' for example, δαGµ ≡ (Gµ α − Gµ) �� O(α) = Dµα = ∂µα − i � Gµ, α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) The Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) defines an action that is gauge invariant at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, quantum effects can spoil gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If this happens, we say that the theory has an anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To define the anomaly, we consider the bosonic effective action W[G], computed from the path integral by integrating out the fermions, while treating the gauge field as a classical background: eiW[G] ≡ � DχDχ† eiSf[χ,χ†,G] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) where Sf ≡ � d4x χ†¯σµPµχ is the fermion bilinear part of the classical action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If we would also like to treat the gauge field Gµ as a dynamical quantum field by – 5 – performing its path integral, � DGDχDχ† ei � d4x L = � DG e i � − 1 4g2 � d4x F a µνF aµν+W[G] � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) we need W[G] to be gauge invariant (upon regularization and renormalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gauge invariance of the classical action Sf does not guarantee that of W[G], since quantum effects (due to the fermionic path integral measure) can break gauge in- variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The anomaly functional A[α], which we also simply refer to as the anomaly, can be defined by taking the gauge variation of the bosonic effective action W[G]:3 A[α] ≡ � d4x αa(x)Aa(x) ≡ δαW[G] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) If A[α] = 0, the theory is anomaly-free and the path integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) yields a well- behaved quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If A[α] ̸= 0 but is equal to the gauge variation of a local action, A[α] = δα(− � d4x Lct), it is called an irrelevant anomaly and can be removed by renormalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', by adding local counterterms Lct to the Lagrangian (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [34] for a systematic study of such counterterms);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' in this case the (renormalized) quantum theory is also well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' On the other hand, a nonzero A[α] that cannot be written as the gauge variation of a local action, called a relevant anomaly, implies that the gauge theory is not well-defined at the quantum level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' in this case, the anomaly is an IR effect and cannot be removed by renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The definition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) we adopt here is known as the consistent anomaly, in the sense that it should – if properly regularized – satisfy the Wess-Zumino consistency condition [35]: δα1A[α2] − δα2A[α1] = A � −i[α1, α2] � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) which is a direct consequence of the Lie algebra: (δα1δα2 − δα2δα1)W[G] = δ−i[α1,α2]W[G] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11) The Wess-Zumino consistency condition is also equivalent to the statement that the anomaly is BRST-closed when α is replaced by the ghost field ω = ωata: A[ω] = δBRSTW[G] =⇒ δBRSTA[ω] = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) which follows from the nil-potency of the BRST transformation, δ2 BRST = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, since the bosonic effective action W[G] is not a local functional of the gauge field Gµ, the fact that A[ω] = δBRSTW[G] does not mean that the anomaly is BRST-exact 3Note that in such variations, we restrict to the set of α(x) that fall off fast enough at infinity such that one can always use integration by parts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In particular, a constant α(x) does not belong to this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 6 – on the space of local functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Anomalies that are BRST-exact on this space can be absorbed by local counterterms, A[ω] = δBRST(− � d4x Lct), and are the irrelevant anomalies, while the relevant anomalies are given by nontrivial BRST cohomology classes (closed but not exact) on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Finally, we note that while we have focused on gauge symmetries in the discussion above, anomalies of global symmetries can be treated in the same framework by artificially gauging all the (classical) global symmetries of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Concretely, we introduce auxiliary gauge fields for all the global symmetries as part of Gµ, and take Uα to also include local transformations associated with the global symmetry generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Then A[α] as defined above will also contain anomalies of the global symmetries, and a nonzero value of A[α] implies that the classical global symmetry cannot be gauged in the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In what follows, we will assume this artificial gauging has been done for all the classical global symmetries of interest, and will not distinguish between global and gauge symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Connection to the Path Integral Measure As explained above, the classical action Sf in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) is gauge invariant, so the only possible source of the anomaly is the path integral measure over the fermionic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Specifically, performing the transformation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) changes the measure by a Jacobian factor: DχαDχ† α = J −1 α DχDχ† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) Therefore, we have eiW[Gα] = � DχDχ† eiSf[χ,χ†,Gα] = � DχαDχ† α eiSf[χα,χ† α,Gα] = � J −1 α DχDχ† eiSf[χ,χ†,G] = eiW[G] � J −1 α DχDχ† eiSf[χ,χ†,G] � DχDχ† eiSf[χ,χ†,G] = eiW[G] � J −1 α � G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) In the first line, we just relabeled the dummy integration variables, χ → χα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' in the second line, we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) and the gauge invariance of the classical ac- tion Sf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' in the last line, we multiplied and divided the expression by eiW[G] = � DχDχ† eiSf[χ,χ†,G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Taking the logarithm of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14), we arrive at a relation between the Jacobian factor4 and the anomaly: − i log � J −1 α � G = W[Gα] − W[G] = A[α] + O(α2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) 4We note that sometimes in the literature, � J −1 α � G is simply written as J −1 α (G) or just J −1 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This might give an impression that it does not depend on the details of the action Sf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Throughout this paper, we manifestly write it as an expectation value � J −1 α � G to emphasize that it is a quantum expectation value and a priori may depend on what interactions are included in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 7 – We see that when the quantum expectation value of the Jacobian factor is trivial, there is no anomaly � J −1 α � G = 1 =⇒ A[α] = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16) while anomalies are associated with the quantum breaking of classical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Connection to Ward Identities The connection between the anomaly and the Ward identities can be made by noting δαW[G] = � d4x � δαGa µ(x) � δW δGa µ(x) = � d4x (Dµα)a δW δGa µ(x) = − � d4x αa(x) � Dµ δW δGµ(x) �a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) Comparing this to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9), we get � Dµ δW δGµ(x) �a = −Aa(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) Meanwhile, since Ga µ acts as a source for the fermion current Jaµ = χ†¯σµtaχ, we have δW δGa µ(x) = ⟨Jaµ⟩G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) Together, these imply (Dµ ⟨Jµ⟩G)a = −Aa(x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' the fermion current is covariant up to the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The BRST symmetry that is critical to the quantization of gauge theory requires (Dµ ⟨Jµ⟩G)a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This makes the connection between anomaly cancellation and consistency of gauge theory precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20), or equivalently Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18), as a generating functional for the Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' First, let us explicitly write out the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18): ∂µ � δW δGa µ � + f abc Gb µ δW δGc µ = −Aa(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) Now taking the kth functional derivative, we get ∂µ � δk+1W δGa µδGb1 µ1 · · · δGbk µk ������ G=0 + k � i=1 f abic δkW δGb1 µ1 · · · δGc µi · · · δGbk µk ����� G=0 = − δkAa(x) δGb1 µ1 · · · δGbk µk ����� G=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22) – 8 – These are the anomalous Ward identities, and are often written in terms of the connected correlation functions of the fermion currents: ∂µ⟨Jµ,aJµ1,b1 · · · Jµk,bk⟩conn + k � i=1 f abic⟨Jµ1,b1 · · · Jµi,c · · · Jµkbk⟩conn = − δkAa(x) δGb1 µ1 · · · δGbk µk ����� G=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) We see that a Gk term in Aa(x) corresponds to a mismatch between the (k+1)-point and k-point correlation functions of the fermion currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) is sometimes taken as a definition of the anomaly in renormalized perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In the case of an irrelevant anomaly, one can add local counterterms which give additional contributions to the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22) and correspond to choosing a different renormalization scheme for the current correlators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A relevant anomaly, on the other hand, constitutes a genuine violation of the classical Ward identities that cannot be remedied by renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' It is also worth noting that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) can be used to prove that Aa(x) truncates at a finite power of the gauge field Gµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3 Regularizing the Anomaly The definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) does not fully specify the value of the anomaly, because (the gauge variation of) the bosonic effective action W[G] is not well-defined in the absence of a regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this section, we introduce our regularization prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Then the CDE evaluation of the regularized anomaly will be presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Before discussing the case of anomalies, we first review the basic idea of reg- ularization and illustrate the role of regularization prescriptions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 using some simple toy series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (Experts can safely skip this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=') Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2, we introduce our regularization prescription for the anomaly, motivated by its conve- nience for evaluating the functional trace using CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Specifically, we will be working in strictly d = 4 spacetime dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', we will not be using dimensional reg- ularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Instead, we will insert a damping factor into the functional trace, in a similar spirit to heat kernel regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In fact, we will introduce a class of such damping factors parameterized by a set of numbers β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' different choices of these β parameters correspond to different regularization schemes and will lead to different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3, we comment on the connection between our regularization pre- scription and some familiar approaches in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In particular, we will see that both the heat kernel and Pauli-Villars regulators can be viewed as specific in- carnations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Finally, we check our regularization prescription against the Wess-Zumino consistency condition in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4, and show how it may be satisfied or violated depending on the choice of β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 9 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 What is Regularization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this subsection, we illustrate the role of regularization with some simple toy series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In particular, we demonstrate how different regularization prescriptions correspond to different definitions for a non-converging series and hence generically lead to different results upon evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We will also clarify the allowed manipulations for a non- converging series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' When we encounter a series that is not convergent, its sum does not have a well-defined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, it is often useful to promote such a series into a ‘func- tion series,’ where the summands are functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' these functions must reproduce the original series term by term when their argument takes a particular value (or limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Then we can define the sum through analytic continuation: we first sum the func- tion series inside its convergence region to obtain an analytic function, and then take the limit corresponding to the original series to define the value of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This regularization procedure leads to a regulated (finite) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us explain how this works using a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Consider the series s1 = ∞ � k=0 2k = 1 + 2 + 4 + 8 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) Clearly, this is a non-converging series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, we could associate it with the function series s1 ⇐⇒ � ∞ � k=0 xk ������ x=2 regularization −−−−−−−−→ 1 1 − x ���� x=2 = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) This function series converges to f1(x) = 1 1−x within the disk |x| < 1, but not at x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' But we can take f1(x = 2) as the definition for the sum s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This is what we mean by a regulated series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Another example is the famous zeta function regularization originally used by Euler: the diverging series s2 = ∞ � k=1 k = 1 + 2 + 3 + 4 + · · · (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3) can be regularized as s2 ⇐⇒ � ∞ � k=1 1 ks ������ s=−1 regularization −−−−−−−−→ ζ(s) �� s=−1 = − 1 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) As mentioned above, when we promote a non-converging series into a function series, we require that the function series reproduces the original series term by term when evaluated at a certain point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Clearly, this does not uniquely specify the choice: – 10 – given a non-converging series, one can usually promote it into many different function series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' These correspond to different regularization schemes and serve as different definitions of the sum of the original series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To see this concretely, let us consider the following toy series s0 = ∞ � k=0 (−1)k = 1 − 1 + 1 − 1 + 1 − 1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) To regularize this series, we could choose to promote it to any of the following set of function series parameterized by a number β: fβ(τ) = τ 0 − τ 1+β + τ 2 − τ 3+β + τ 4 − τ 5+β + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) Then we have s0 ⇐⇒ fβ(τ) �� τ→1 regularization −−−−−−−−→ 1 − τ 1+β 1 − τ 2 ����� τ→1 = 1 + β 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) We see that with different values for β, the original non-converging series s0 can be defined/regularized to take different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If we are going to regularize a non-converging series with a function series that is absolutely convergent (in its convergence region), then one can shuffle and/or group terms in the latter without changing its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Alternatively, one could shuffle and/or group terms first in the original non-converging series, and then regularize the new expression with an absolutely convergent function series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This second way will lead to the same result upon evaluation, and it is sometimes more convenient because the series is easier to massage before promoting it into a function series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, when we shuffle and/or group terms in the original non-converging series to go from one expression to another, we have to remember that none of these expressions is well-defined yet, so it is not appropriate to say that they are equal (‘=’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Instead, they are just ‘equivalent’ in the sense that they would be equal if one were to regularize them with the same absolutely convergent function series (with the same shuffling and/or grouping of terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this paper, we use the symbol ‘≃’ to denote this equivalence relation between non-converging series (see equations below starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us take the same toy series example s0 to illustrate this point, as well as the use of the ‘≃’ notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Since the function series Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) is absolutely convergent within the disk |τ| < 1, we can group its terms to get another series: fβ(τ) group terms −−−−−−−→ �fβ(τ) ≡ ∞ � k=0 � τ 2k − τ 2k+1+β� analytic continuation −−−−−−−−−−−−→ 1 − τ 1+β 1 − τ 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) – 11 – which has the same analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Alternatively, one could first group terms in the original non-converging series: s0 ≃ �s0 ≡ ∞ � k=0 (1 − 1) = 0 + 0 + 0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) Note that we have used the ‘≃’ sign here between s0 and �s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The new series �s0 is a converging series and does have a default definition, so a regularization for �s0 is not mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, one could still use the function series �fβ(τ) to regularize it, because � τ 2k − τ 2k+1+β���� τ=1 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) would also reproduce the series �s0 term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' With this regularization, one would then get the same evaluation result 1+β 2 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Our use of the ‘≃’ sign here is emphasizing this: s0 and �s0 are equal only when we use the same regularization prescription for them (although one of them has a different default definition in the absence of regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We note in particular that performing cyclic permutations inside a trace is a typical type of shuffling and/or grouping of terms: tr (AB) = � i �� a AiaBai � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11a) tr (BA) = � a �� i BaiAia � = � a �� i AiaBai � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11b) The two traces are related by a change of summation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' When the matrices A and B are infinite dimensional, such as in the case of functional traces, each trace is a sum over a (double) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If the series is not convergent and needs regularization to be well-defined, then it is not appropriate to claim that the two traces are equal, as we have just explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Instead, we should use the ‘≃’ sign: tr (AB) ≃ tr (BA) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) to emphasize that they would be equal when we use the same absolutely convergent function series to regulate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Anomaly as a Regulated Functional Trace Let us now turn to the case of interest in this paper, the anomaly functional A[α] defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' First, we would like to isolate the functional trace that encodes the anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We start with the definition of W[Gα], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) with Gµ replaced by – 12 – Gµ α according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' It can be formally written as a functional determinant: eiW[Gα] = � DχDχ† eiSf[χ,χ†,Gα] = det � Uα ¯σµPµ U † α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) Taking the logarithm and expanding in α, we get W[Gα] = −i log det � Uα ¯σµPµ U † α � ≃ −i log det (¯σµPµ + iα ¯σµPµ − ¯σµPµ iα) + O(α2) ≃ −i log det (¯σµPµ) − i log det � 1 + 1 ¯σνPν (iα ¯σµPµ − ¯σµPµ iα) � + O(α2) ≃ W[G] − i Tr log � 1 + 1 ¯σνPν (iα ¯σµPµ − ¯σµPµ iα) � + O(α2) ≃ W[G] + Tr � 1 ¯σνPν (α ¯σµPµ − ¯σµPµ α) � + O(α2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) According to the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9), the leading order contribution to the differ- ence W[Gα] − W[G] gives the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Therefore, we obtain A[α] ≃ Tr � 1 ¯σνPν � α ¯σµPµ − ¯σµPµ α �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) At this point, we have formally written the anomaly as a functional trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' How- ever, we emphasize that the functional trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) is the sum of a series that is not convergent, so it does not have a definite value and requires regularization to become well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As elaborated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1, different regularization prescriptions can yield different results upon evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In fact, the same is true for the expression in each line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Therefore, we have used the notation ‘≃’ to emphasize that these expressions are not exactly equal ‘=’ unless they are regularized in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' One may also attempt to perform a cyclic permutation within the functional trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15), so the two terms appear to cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1, such a cyclic permutation amounts to shuffling and/or grouping terms in the original non-converging series to obtain a new series: A[α] ≃ Tr[0] = 0 + 0 + 0 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16) Although this new series is zero term by term, which is convergent and hence has a default definition without a regulator, this does not contradict the statement that regularization prescriptions exist that yield a nonzero value for this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As em- phasized by the ‘≃’ sign, the two expressions above would only be equal under the – 13 – same regularization prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The default definition of the right-hand side (which gives zero) corresponds to one particular choice of regularization (a trivial one), so its evaluation result would not be equal to that of the left-hand side if a different regularization prescription is chosen for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To motivate our regulator, let us first check what would happen if we go ahead and evaluate the functional trace in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) with CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Focusing on the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='¯σνPν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='α ¯σµPµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='(2π)4 tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='¯σν(qν + Pν) α ¯σµ(qµ + Pµ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='(2π)4 tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='k=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='−σλqλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='¯στPτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='�k σνqν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='α ¯σµ(qµ + Pµ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='(2π)4 tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='k=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='− /q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 /P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='�k /q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 α (/q + /P) 1 − γ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='(2π)4 tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='k=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='− /q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 �Pβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='�k /q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='q2 α (/q + �Pβ) 1 − γ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='(2π)4 tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='/q + �Pβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='α (/q + �Pβ) 1 − γ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='≃ Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='�Pβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='α �Pβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 − γ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) The first line above simply follows from the definition of the functional trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5 To obtain the second line, we have performed a Taylor expansion in terms of the Hermi- tian covariant derivative Pµ, an operation called the Covariant Derivative Expansion (CDE) in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6 To get the third line, we used the following identity between Pauli matrices and the Dirac gamma matrices: tr � (σµ1¯σν1) · · · (σµk¯σνk) � = tr � (γµ1γν1) · · · (γµkγνk) 1 − γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) 5See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) for a more detailed explanation of the shift Pµ → qµ + Pµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We note that the internal traces ‘tr’ from here on in the main text are actually what we denote by ‘trx’ in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' See App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1, especially the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) for a careful clarification on this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 6More precisely, the operation here is called ‘simplified CDE’ [18], in which one makes a Taylor expansion directly in terms of the ‘open’ covariant derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This is different from the ‘original CDE’ [15–17] where one inserts additional factors to ‘close’ the covariant derivatives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' put them into commutators) before performing the Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' See the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) for an elaboration on open vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' closed derivatives in functional operators, and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [19] for a detailed discussion on simplified vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' original CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 14 – Starting from the fourth line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17), we have introduced the β-modified covariant derivative:7 �Pβ ≡ i/∂ + /G �1 − γ5 2 + β 1 + γ5 2 � ≡ i/∂ + � a /G ata �1 − γ5 2 + βa 1 + γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) Finally, in the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17), we rewrote the result as a functional trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Note that we take the βa parameters to be degenerate within each simple gauge group sector so that βata (no sum over a) satisfy the same Lie algebra as ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Here and in what follows, we explicitly write out the summation over adjoint indices when the presence of βa results in more than two identical adjoint indices in an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) has allowed us to convert the two-component ex- pression (left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17)) into a four-component expression (last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17)) with an insertion of the projector operator 1−γ5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The same procedure goes through when both terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) are present, so we have A[α] ≃ Tr � 1 �Pβ � α �Pβ − �Pβ α � 1 − γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) At this stage, it seems that the β parameters could take arbitrary values without affecting the value of the expression, because it comes with the factor 1+γ5 2 which will get annihilated by the projector 1−γ5 2 at the end of the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, we stress that this β-parameterized functional trace is still the sum of a non-converging series, so we need to introduce a regulator to make it well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As we will see below, once we regulate this expression, different β’s will define different values for the functional trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Motivated by the form of the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20), we choose to insert a damping factor to define the regularized anomaly: AΛ β[α] ≡ Tr � f � − �P 2 β Λ2 � 1 �Pβ � α �Pβ − �Pβ α � 1 − γ5 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) where the function f(u) satisfies the following conditions: f(0) = 1 , f(+∞) = 0 , � ∞ 0 duf(u) well-defined , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22a) 7Note that when β ̸= 1, the operator �Pβ is not gauge covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This is the reason why we will not always get a covariant anomaly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' see discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 15 – undnf dun ���� u=0 = undnf dun ���� u→+∞ = 0 for n ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22b) Typical examples of such functions are f(u) = e−u , and f(u) = 2 (1 + u)(2 + u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) The renormalized anomaly is then given by Aβ[α] ≡ lim Λ→∞ � AΛ β[α] + δα � d4x LΛ ct � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='24) where LΛ ct is the local counterterm Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Note in particular that the regularized anomaly AΛ β[α] generically contains an O(Λ2) piece that is irrelevant for β values satisfying the Wess-Zumino consistency condition, in which case we should include operators with appropriate O(Λ2) coefficients in LΛ ct to obtain a finite result for the renormalized anomaly Aβ[α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' LΛ ct can also contain O(Λ0) counterterms, and their coefficients specify the renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Having included the damping factor f � − �P 2 β/Λ2� , the functional trace AΛ β[α] is now the sum of an absolutely convergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' So at this point one is free to manipulate this expression, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' perform cyclic permutations while maintaining a genuine ‘=’ sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Our regularization prescription Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) is designed to facilitate the evaluation with CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In particular, the damping factor inserted commutes with the β-modified covariant derivative: � f � − �P 2 β Λ2 � , �Pβ � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='25) Also note from the definition of �Pβ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) that it anticommutes with γ5: �Pβγ5 = −γ5 �Pβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='26) Making use of these identities, we can simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) to AΛ β[α] = Tr � f � − �P 2 β Λ2 � α γ5 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27) from which it is clear that the evaluation result will depend on the parameters β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' One interpretation of this regulator is that the β parameters determine the com- bination of background fields that are turned on when computing the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This effectively forces the path integral measure DχDχ† to be organized according to the – 16 – eigenmodes of the operator �Pβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8 We will proceed with the evaluation of AΛ β[α] in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4, after the next two sub- sections which discuss how our prescription connects to other familiar regularization approaches, and how the Wess-Zumino consistency condition is satisfied or violated by different choices of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Connection to Other Regularization Prescriptions Let us make a few comments on the connection between our regularization prescrip- tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) and some approaches that often appear in the literature, in particular, heat kernel regularization and Pauli-Villars regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Regularizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) with a heat kernel regulator, one obtains AHK β ≡ Tr � e �P 2 β/Λ2 1 �Pβ � α �Pβ − �Pβ α � 1 − γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21), we see that the heat kernel regularization amounts to choosing the damping function to be Heat kernel : f(u) = e−u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='29) Alternatively, regularizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) with one Pauli-Villars field, one obtains APV,1 β ≡ Tr �� 1 �Pβ − 1 �Pβ − Λ � � α �Pβ − �Pβ α � 1 − γ5 2 � = Tr � −Λ �Pβ( �Pβ − Λ) � α �Pβ − �Pβ α � 1 − γ5 2 � = Tr � −Λ �Pβ − Λ α γ5 � = Tr � −Λ2 �P 2 β − Λ2 α γ5 � = Tr � −Λ2 �P 2 β − Λ2 1 �Pβ � α �Pβ − �Pβ α � 1 − γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='30) Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21), we see that this amounts to choosing the damping func- tion to be Pauli-Villars with one regulator field : f(u) = 1 1 + u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='31) Note that this damping factor does not satisfy all the conditions listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22), and hence would not regulate all the divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This motivates considering Pauli- Villars regularization with three regulator fields, for which one obtains the regularized 8We leave implicit possible analytic continuations needed to make �Pβ a Hermitian operator that has a well-defined eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 17 – anomaly as APV,3 β ≡ Tr �� 1 �Pβ − 1 �Pβ − M1 + 1 �Pβ − M2 − 1 �Pβ − M3 � � α �Pβ − �Pβ α � 1 − γ5 2 � = Tr � −(M1 − M2 + M3) �P 2 β + 2M1M3 �Pβ − M1M2M3 ( �Pβ − M1)( �Pβ − M2)( �Pβ − M3) α γ5 � = Tr � M 2 1M 2 3 (2 �P 2 β − M 2 2) ( �P 2 β − M 2 1)( �P 2 β − M 2 2)( �P 2 β − M 2 3) α γ5 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='32) where we have assumed the relation M 2 1 − M 2 2 + M 2 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If we now take M 2 1 = M 2 3 = Λ2 , and M 2 2 = 2Λ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='33) this simplifies to APV,3 β = Tr � 2Λ4 ( �P 2 β − Λ2)( �P 2 β − 2Λ2) 1 �Pβ � α �Pβ − �Pβ α � 1 − γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='34) Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21), we see that this amounts to choosing the damping func- tion to be Pauli-Villars with three regulator fields : f(u) = 2 (1 + u)(2 + u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='35) This damping function does satisfy all the conditions listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22), and will successfully regularize all the divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 Consistency With the Wess-Zumino Condition Since we have adopted the definition of anomaly as the gauge variation of the bosonic effective action: A[α] ≡ δαW[G] = (W[Gα] − W[G])|O(α) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='36) we expect it to satisfy the Wess-Zumino consistency condition, as reviewed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, an implicit assumption behind this expectation is that there is a well- defined W[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Importantly, our regularization prescription presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 is directly applied to δαW[G], instead of W[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, the Wess-Zumino consis- tency condition may not be satisfied, because generic β values may not correspond to applying the same (or ‘consistent’) regularization prescription to W[Gα] and W[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this subsection, we check the Wess-Zumino consistency condition for the regular- ized anomaly AΛ β[α] at the level of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27), and give a partial but general answer – 18 – to the question of what β values lead to a Wess-Zumino consistent anomaly: δα1AΛ β[α2] − δα2AΛ β[α1] ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='= AΛ β � −i[α1, α2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='37) We will revisit this question in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5 after evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Using the expression of AΛ β[α] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27), we can write the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='37) as (it is understood that we will be dropping terms of order O(α2 1, α2 2) throughout this subsection): δα1AΛ β[α2] = Tr � f � − �P 2 β[α1] Λ2 � γ5α2 � − Tr � f � − �P 2 β Λ2 � γ5α2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='38) where �Pβ[α1] denotes the gauge transformation of �Pβ: �Pβ[α1] ≡ i/∂ + /Gα1 �1 − γ5 2 + β 1 + γ5 2 � = i/∂ + � Uα1 /GU † α1 + Uα1 � i/∂U † α1 � � �1 − γ5 2 + β 1 + γ5 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='39) We note that when β = 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='38) is quite easy to calculate because �Pβ=1 = /P transforms covariantly and so does the damping factor: �Pβ=1[α1] = Uα1 �Pβ=1U † α1 =⇒ f � − �P 2 β=1[α1] Λ2 � = Uα1f � − �P 2 β=1 Λ2 � U † α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='40) This leads us to the so-called covariant anomaly that satisfies δα1AΛ β=1[α2] = Tr � f � − �P 2 β=1 Λ2 � γ5 � U † α1 α2 Uα1 − α2 � � = AΛ β=1 � −i[α1, α2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='41) We see that this covariant anomaly generically would not satisfy the Wess-Zumino consistency condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' it is off by a factor of two compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='37): δα1AΛ β=1[α2] − δα2AΛ β=1[α1] = 2 AΛ β=1 � −i[α1, α2] � ̸= AΛ β=1 � −i[α1, α2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='42) The only exceptions are when the anomaly itself vanishes AΛ β=1[α] = 0 (once summed over fermion species) or when the two gauge transformations under consideration commute, [α1, α2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In these cases, the Wess-Zumino consistency condition itself is trivial, and the covariant anomaly is also a consistent anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' For general β values, �Pβ does not transform covariantly, and calculating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='38) – 19 – is more tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' It is useful to write �Pβ in terms of its chirality components: �Pβ = /P 1 − γ5 2 + /P β 1 + γ5 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='43) with /P = i/∂ + /G = i/∂ + � a /G ata , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='44a) /P β ≡ i/∂ + β /G = i/∂ + � a βa /G ata .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='44b) The left-handed component is gauge covariant, but the right-handed component transforms in a complicated manner for general β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To proceed, let us rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='39) also in terms of its chirality components: �Pβ −→ �Pβ[α1] = /Lα1 1 − γ5 2 + /Rα1 1 + γ5 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='45) with /P −→ /Lα1 ≡ Uα1 /PU † α1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='46a) /P β −→ /Rα1 ≡ i/∂ + β � Uα1 /GU † α1 + Uα1 � i/∂U † α1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='46b) To check the Wess-Zumino consistency condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='37), we can Taylor expand the damping factors in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='38) and examine a general kth power term therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We have Tr � � �P 2 β[α1] �k γ5α2 � = Tr �� �/Rα1 /Lα1 �k 1−γ5 2 + �/Lα1 /Rα1 �k 1+γ5 2 � γ5α2 � = Tr � 1+γ5 2 �/PU † α1 /Rα1Uα1 �k−1 /P U † α1 �/Rα1, α2 � Uα1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='47a) Tr � � �P 2 β �k γ5α2 � = Tr � 1+γ5 2 �/P /P β �k−1 /P �/P β, α2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='47b) The difference between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='47a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='47b) comes from two sources: U † α1 /Rα1Uα1 = /P β + (/Rα1 − /P β) + i �/P β, α1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='48a) U † α1 �/Rα1, α2 � Uα1 = �/P β, α2 � + �/Rα1 − /P β, α2 � − i � α1, �/P β, α2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='48b) One could go ahead with the calculation keeping track of all these terms for general β values, but the result is not very illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Instead, let us examine the special – 20 – case β = 0 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, the right-handed component does not transform: /Rα1 = /P β=0 = i/∂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='49) The middle term of each equation in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='48) is therefore absent, and we have δα1AΛ β=0[α2] − δα2AΛ β=0[α1] ⊃ Tr � � �P 2 β[α1] �k γ5α2 � − Tr � � �P 2 β �k γ5α2 � − (α1 ↔ α2) = Tr � 1+γ5 2 � �/PU † α1 /Rα1Uα1 �k−1 − �/P /P β �k−1 � /P �/P β, α2 � + 1+γ5 2 �/P /P β �k−1 /P � −iα1, �/P β, α2 ��� − (α1 ↔ α2) = Tr � 1+γ5 2 �/P /P β �k−1 /P � /P β, −i [α1, α2] �� = Tr � � �P 2 β �k γ5 [−iα1, α2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='50) In the step leading to the second to last line, the first term in the curly brackets gets canceled upon adding the expression with α1 ↔ α2, while the second term combines with the latter and we have used the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Clearly, summing over all the kth power relations like in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='50) will give us the Wess-Zumino consistency condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='37): δα1AΛ β=0[α2] − δα2AΛ β=0[α1] = AΛ β=0 � −i[α1, α2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='51) Therefore, we see that in our regularization prescription, β = 0 (meaning βa = 0, ∀a) is always one possible choice to ensure the Wess-Zumino consistency condition for any symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, from the present analysis it is difficult to tell whether there are other Wess-Zumino consistent choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We will revisit this issue in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5 using the BRST version of the Wess-Zumino condition once we have the evaluation result for AΛ β[α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4 Master Formula for the Anomaly From CDE Now we proceed with the evaluation of the regularized anomaly, starting with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27): AΛ β[α] = Tr � f � − �P 2 β Λ2 � α γ5 � = � d4x � d4q (2π)4 tr � f � − � /q + �Pβ �2 Λ2 � α γ5 � = � d4x � d4k (2π)4 tr � Λ4f � − � /k + �Pβ Λ �2� α γ5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) – 21 – Here we have rescaled the integration variable kµ ≡ qµ/Λ, so that it is easier to keep track of the 1/Λ powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eventually, we are interested in the Λ → ∞ limit, so in what follows we will be dropping the O(1/Λ) terms that vanish in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This can be achieved by applying the simplified CDE, while expanding and truncating the integrand accordingly: Λ4f � �− � /k + �Pβ Λ �2� � = Λ4f � −k2 − 1 Λ � /k �Pβ + �Pβ/k � − 1 Λ2 �P 2 β � = Λ4 � fu + f ′ u z + 1 2f ′′ u z2 + 1 6f ′′′ u z3 + 1 24f ′′′′ u z4 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) where we have introduced the following notation for convenience u ≡ −k2 , and z ≡ − 1 Λ � /k �Pβ + �Pβ/k � − 1 Λ2 �P 2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3) Plugging this back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) and simplifying the expression, we get AΛ β[α] = � d4x � d4k (2π)4 tr �� − Λ2f ′ u �P 2 β + 1 2f ′′ u �P 4 β + u 24f ′′′ u � �P 2 βγµ �Pβγµ �Pβ + �Pβγµ �Pβγµ �P 2 β + �Pβγµ �P 2 βγµ �Pβ + 4 �P 4 β �� α γ5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) Note that the terms proportional to f ′′′′ u can be grouped in pairs that take the form tr � γµ(· · · )γ5 + (· · · )γµγ5� = 0, so they all cancel out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' the same is true for a subset of the f ′′ u and f ′′′ u terms, which significantly reduces the number of terms in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Performing the loop momentum integral (after a Wick rotation as usual), we obtain AΛ β[α] = � d4x i 16π2 � − Λ2 � (ufu) ��∞ 0 − � ∞ 0 dufu � tr0 + 1 2 � (uf ′ u − fu) ��∞ 0 � tr1 + 1 12 � � u2f ′′ u − 2uf ′ u + 2fu ���∞ 0 � (2 tr1 − tr2 − tr3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) – 22 – We see that for a general damping function f(u) that satisfies the conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22), the calculation yields the result: AΛ β[α] = � d4x i 16π2 � � Λ2 � ∞ 0 duf(u) � tr0 +1 6 � tr1 + tr2 + tr3 � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) where tr0 ≡ tr � �P 2 βγ5α � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7a) tr1 ≡ tr � �P 4 βγ5α � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7b) tr2 ≡ −1 2 tr �� �P 2 βγµ �Pβγµ �Pβ + �Pβγµ �Pβγµ �P 2 β � γ5α � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7c) tr3 ≡ −1 2 tr � �Pβγµ �P 2 βγµ �Pβγ5α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7d) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) is our master formula for the regularized anomaly before evaluation of the Dirac traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Evaluating the Dirac Traces In order to evaluate the Dirac traces in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7), it is convenient to use the chirality decomposition of �Pβ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='43): �Pβ = /P 1 − γ5 2 + /P β 1 + γ5 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) where /P ≡ i/∂ + /G = i/∂ + � a /G ata , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9a) /P β ≡ i/∂ + β /G = i/∂ + � a βa /G ata .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9b) We also introduce the notation Gµ − ≡ P µ − P µ β = (1 − β) Gµ = � a (1 − βa) Gaµ ta .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) The evaluation of tr0 is straightforward: tr0 = 2 tr �� Pµ, P µ β � α � = −2i(1 − β) tr � (∂µGµ) α � IBP = 2i(1 − β) tr � Gµ(∂µα) � = 2i � a tr � tatb� (1 − βa) Ga µ (∂µαb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11) – 23 – Turning to tr1, tr2, tr3, we first note that they can be written in the following form:9 tr1 = 1 2 tr � /P /P β /P �/P β, α � (1 + γ5) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12a) tr2 = 1 2 tr ��/P /P 2 β + /P 2 β /P + /P 3��/P β, α � (1 + γ5) + /P β /P /P β �/P β, α � (1 − γ5) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12b) tr3 = −4 tr �� PνPµP µ β + P µ β PµPν �� P ν β , α �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12c) where we have used γµγνγµ = −2γν, γµγνγργµ = 4ηνρ to simplify the products of gamma matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Upon evaluating the Dirac traces we can combine terms in the sum of all three traces such that all P µ β factors appear in commutators: 3 � i=1 tri = tr � −2 �� 3 � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' P ν β � + 2 � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gν − � − � P ν β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gµ − � + � Gµ −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gν − � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' G−µ � − � Pβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gν − � − � Gµ −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gν − �� − Gµ −Gν −G−µ � [Pβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' α] − iεµνρσ �� 2Gµ −Gν −Gρ − + � 3 � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' P ν β � + 2 � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gν − � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Gρ − ��� P σ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' α � + 3 � P µ β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' P ν β �� P ρ β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' P σ β � α �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) Having P µ β in commutators is important because it contains the derivative ∂µ, which as a functional operator is understood to act on everything to its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' But when it appears in a commutator, its action is local (or ‘closed’) on the object appearing in the commutator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' for example:10 � ∂µ, Gµ(x) � φ(x) = ∂µGµ(x) φ(x) − Gµ(x) ∂µφ(x) = � ∂µGµ(x) � φ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 The Evaluated Master Formula Gathering the results in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) and substituting in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9b) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) for P µ β and Gµ −, we obtain our evaluated master formula for the regu- 9To arrive at these expressions, we have used cyclic permutation to move P µ β to the right in half of the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Generally this is illegal since ‘tr’ is only over the internal space while P µ β contains ∂µ which is a spacetime operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, such cyclic permutations are innocuous in CDE calculations of functional traces that arise from evaluating the path integral at one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In fact, they have been used in many previous functional matching calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We clarify this subtle point in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 10The local nature of all derivative operators in the CDE is also the reason why the otherwise illegal cyclic permutation in the internal trace ‘tr’ in intermediate steps actually leads to the correct result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 24 – larized anomaly expressed in the matrix notation: AΛ β[α] = � d4x 1 16π2 tr � − 2(1 − β) � Λ2 � ∞ 0 duf(u) � Gµ (∂µα) + 1 3(1 − β) � i � (1 + 4β) (∂µGν) − (1 + 2β) (∂νGµ) − i(1 + 3β2) � Gµ, Gν � , Gµ� + � ∂2Gν � + i(1 − 2β) � (∂µGµ) , Gν � − Gµ(1 − β)Gν(1 − β)Gµ � � Dν βα � − 1 2 εµνρσ �1 3 � (1 − β)Gρ , 2(1 + 2β) (∂µGν) − i(1 + 2β + 3β2)GµGν� � Dσ βα � + 4 � β (∂µGν) − iβ2GµGν�� β (∂ρGσ) − iβ2GρGσ� α �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) where � Dµ βα � ≡ (∂µα) − iβ[Gµ, α] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) we have carefully kept the β factors in appropriate places such that each of them is associated with the gauge field that immediately follows it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Depending on the application, it is sometimes more convenient to write out the adjoint components of the master formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' which gives AΛ β[α] = � d4x 1 16π2 � − � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='b tr � tatb� (1 − βa) � 2 � Λ2 � ∞ 0 duf(u) � Ga µ � ∂µαb� + 1 3 � f aef� (1 + 4βa) (∂µGe ν) − (1 + 2βa) � ∂νGe µ � + � 1 + 3β2 a � f eghGg µGh ν � Gfµ − � ∂2Ga ν � + (1 − 2βa)f aef � ∂µGe µ � Gf ν �� ∂ναb + βbf bcdGc ναd�� − � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='d tr � tatbtctd� 1 3 (1 − βa)(1 − βb)(1 − βc) Ga µGb νGcµ� ∂ναd + βdf defGe ναf� − � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='c tr � {ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' tb} tc� 1 4 εµνρσ � βaβb � F aµν lin + βaf adeGdµGeν�� F bρσ lin + βbf bfgGfρGgσ� αc + 1 3 (1 − βb) � 2(1 + 2βa)F aµν lin + (1 + 2βa + 3β2 a)f adeGdµGeν� Gbρ × � ∂σαc + βcf cfgGfσαg��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) where F µν lin ≡ (∂µGν) − (∂νGµ) is the part of F µν linear in the gauge fields, and we have used the fact that β takes the same value within a simple group (only for which – 25 – f abc may be nonzero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In the next section, we will apply the evaluated master formula, written in matrix and component forms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17), respectively, to obtain explicit results for various gauge group sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Before delving into the details, let us first quickly note two special β choices which directly relate to the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If βa = 1 (∀a), all but the last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) vanishes, and the result takes a gauge-covariant form: AΛ β=1[α] = � d4x � − 1 32π2 � εµνρσ tr (F µνF ρσα) = � d4x � − 1 64π2 � tr � {ta, tb} tc� εµνρσ F aµνF bρσαc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4, the covariant anomaly generically would not satisfy the Wess-Zumino consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, we also mentioned some exceptions to this, such as when the anomaly itself is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' From the equation above, we see that this can be achieved by the standard anomaly cancellation condition tr � {ta, tb} tc� = 0, where we recall that the internal trace ‘tr’ also sums over the fermion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If βa = 0 (∀a), we learned from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 that the Wess-Zumino consistency condition should be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) indeed reproduces the familiar result for the consistent anomaly: AΛ β=0[α] = � d4x 1 16π2 tr � − 2 � Λ2 � ∞ 0 duf(u) � Gµ (∂µα) + 1 3 � � ∂2Gν � + i[(∂µGµ) , Gν] + i[Fµν, Gµ] − GµGνGµ� (∂να) − 1 6 εµνρσ � Gρ , 2 (∂µGν) − iGµGν� (∂σα) � = � d4x � 1 48π2 εµνρσ tr � (∂µα) (GνFρσ + iGνGρGσ) � − δαLΛ ct,0 � = � d4x � 1 48π2 tr �� ta, tb� , tc� εµνρσ (∂µαa) �� ∂νGb ρ � + 1 4 f bdeGd νGe ρ � Gc σ − δαLΛ ct,0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) up to an irrelevant anomaly given by the gauge variation of the following local – 26 – counterterm: LΛ ct,0 = 1 16π2 � Λ2 � ∞ 0 duf(u) � tr � GµGµ � + 1 96π2 tr �� ∂µGµ �2 − 2iF µνGµGν + 1 2 GµGνGµGν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) The relevant anomaly in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) is proportional to tr � {ta, tb} tc� , which depends on the fermion content of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The symmetries under consid- eration can be gauged when there is no relevant anomaly, that is, when the standard anomaly cancellation condition tr � {ta, tb} tc� = 0 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5 Implications of the Master Formula In this section, we apply Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) derived in the previous section, which are evaluation results of our master formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6), to obtain explicit results for the anomaly in all possible combinations of the continuous group sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We consider in turn a simple non-Abelian group, semi-simple product of non-Abelian sectors, product of Abelian sectors, and finally the general case of product of non-Abelian and Abelian sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In each case, we aim to answer the following questions: What values of the regularization parameters β are consistent with the Wess- Zumino condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' For these Wess-Zumino consistent β choices, what is the relevant anomaly, and what are the counterterms associated with the irrelevant anomaly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' What are the conditions for the relevant anomaly to vanish (in which case the symmetries under consideration can be gauged in the quantum theory)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To investigate the first question, we use the BRST form of the Wess-Zumino consistency condition, which states that (recall the discussion around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12)) when the gauge variation parameter α is replaced by the ghost field ω, the anomaly is BRST-closed: δBRSTAβ[ω] = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) where Aβ[ω] is understood as the renormalized anomaly defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Since the gauge variation of local counterterms is always BRST-closed due to the nil- potency of the BRST transformation, this requires the regularized anomaly is also BRST-closed: δBRSTAΛ β[ω] = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) – 27 – We will check this condition up to O(1/Λ) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To do so, it is useful to recall that under the BRST transformation: δBRSTGµ = Dµω = ∂µω − i � Gµ, ω � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3a) δBRSTFµν = −i � Fµν, ω � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3b) δBRSTω = iω2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3c) δBRST � ∂µω − iβ � Gµ, ω �� = i(1 − β) � ω, ∂µω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3d) In answering the second and third questions, we will see how the well-known results for anomalies are recovered in our formalism with specific (Wess-Zumino con- sistent) β choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We will also see that for all the Wess-Zumino consistent anomalies, the standard anomaly cancellation condition tr � {ta, tb} tc� = 0 will guarantee that the relevant anomaly vanishes (which means the symmetries can be gauged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Simple Non-Abelian Group For a simple non-Abelian group, all the β factors are degenerate, so we omit their adjoint indices and simply write all of them as β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We can first verify that the O(Λ2) term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) is BRST-closed: δBRSTAΛ β[ω] �� O(Λ2) = − 1 8π2 � d4x � Λ2 � ∞ 0 duf(u) � (1 − β) × tr � (∂µω)(∂µω) + i � ω , Gµ(∂µω) �� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) Note that cyclic permutation of a Grassmann odd matrix in the trace is accompanied by a minus sign if it passes through an odd number of Grassmann odd matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' tr � ω Gµ(∂µω) � = − tr � Gµ(∂µω) ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To derive constraints on β from the Wess-Zumino consistency condition, we need to consider the O(Λ0) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The BRST transformation of these terms is quite tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, as we will show, it turns out sufficient to work out just a subset of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us first note that AΛ β[ω]|O(Λ0) contains terms of the form: ωG∂3 , ωG2∂2 , ωG3∂ , ωG4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) whose BRST transformation contains terms of the form:11 ω2G∂3 , ω2G2∂2 , ω2G3∂ , ω2G4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) We will see that the ω2G4 and ω2G∂3 terms are sufficient to constrain β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The ω2G4 terms can only come from BRST transforming the ωG4 terms in 11Note that the ω2∂4 term from BRST transforming the sole ωG∂3 term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 28 – AΛ β[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Those ωG4 terms that do not involve εµνρσ are easily seen to vanish upon cyclic permutation, and we are left with AΛ β[ω] �� G4ω = � d4x 1 24π2 β (1 + β + β2) εµνρσ tr � GµGνGρGσω � = � d4x � − 1 192π2 � β (1 + β + β2) tr � {ta, tb} tc� × εµνρσf adef bfgGdµGeνGfρGgσωc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) Since δBRSTAΛ β[ω] �� G4ω2 = 0 requires AΛ β[ω] �� G4ω = 0, while (1 + β + β2) is positive- definite, we see that δBRSTAΛ β[ω] �� G4ω2 = 0 =⇒ β = 0 or tr � {ta, tb} tc� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) As discussed around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19), β = 0 reproduces the standard consistent anomaly, plus an irrelevant piece that is obviously BRST-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The other option is the standard anomaly cancellation condition tr � {ta, tb} tc� = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' when this is true, the terms in AΛ β that are proportional to εµνρσ all vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, it remains to check whether there are additional constraints on the value of β from the terms not involving εµνρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To do so, we focus on the ω2G∂3 terms in δBRSTAΛ β[ω], for which we find, after some simplification using cyclic permutation and integration by parts: δBRSTAΛ β[ω] �� ω2G∂3 = � d4x i 48π2 β (1 − β) tr �� ∂2� Gν, ω � − � (∂2Gν), ω �� (∂νω) � = � d4x 1 48π2 β (1 − β) tr � tatb� × f acd�� ∂2Gcν� ωd − ∂2� Gcνωd�� (∂νωb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) Here the group theory factor tr � tatb� ∝ δab is always non-vanishing, so we see the only other option (besides β = 0) that makes δBRSTAΛ β[ω] �� ω2G∂3 vanish is β = 1, in which case the εµνρσ-independent part of AΛ β simply vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In summary, we conclude that consistency with the Wess-Zumino condition re- quires either of the following to be true: β = 0, in which case the result is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This reproduces the standard consistent anomaly plus an irrelevant piece that is equal to the gauge variation of the local counterterm given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As discussed below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19), for the relevant anomaly to vanish in this case, one needs the stan- dard anomaly cancellation condition tr � {ta, tb} tc� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' tr � {ta, tb} tc� = 0 and β = 1, in which case anomaly cancellation happens and the regularized anomaly vanishes altogether, AΛ β[α] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 29 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Product of Non-Abelian Sectors For a semi-simple product of non-Abelian sectors, the only additional term in AΛ β[α] to consider is the tr � tatbtctd� term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Both tr � tatb� and tr � {ta, tb} tc� vanish when the generators belonging to more than one simple sectors are involved, while tr � tatbtctd� can be nonzero when two of the four generators belong to one simple sector and the other two belong to another simple sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Upon imposing the conditions derived in the previous subsection on each sim- ple non-Abelian sector, we see that there are only two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If β = 1 (and tr � {ta, tb} tc� = 0) for either sector, the aforementioned cross term in AΛ β[α] vanishes because of the (1 − βa)(1 − βb)(1 − βc) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If β = 0 for both sectors, the cross term is contained in the general result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19), specifically the gauge variation of the O(G4) counterterm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Therefore, no additional constraints arise from the Wess-Zumino consistency condition beyond those already derived for each simple sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The same is true for the relevant anomaly cancellation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 Product of Abelian Sectors For an Abelian gauge group, we can set f abc = 0 and F µν lin = F µν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) to obtain AΛ β[α] = � d4x 1 16π2 � − � a,b tr(QaQb) · (1 − βa) � 2 � Λ2 � ∞ 0 duf(u) � Ga µ − 1 3 � ∂2Ga µ ��� ∂µαb� − � a,b,c,d tr(QaQbQcQd) · 1 3 (1 − βa)(1 − βb)(1 − βc) GaµGbνGc µ � ∂ναd� − � a,b,c tr(QaQbQc) · 1 8 � (1 + βa)(1 + βb) + 1 3(1 − βa)(1 − βb) � εµνρσF a µνF b ρσαc � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) where we have written the group generators ta as Qa since they are just charges under the U(1)’s, and ‘tr’ means summing over all chiral fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In the tr(QaQbQc) term, we have integrated by parts and symmetrized the coefficient between a and b: βaβb + 1 3(1 + 2βa)(1 − βb) → 1 4 � (1 + βa)(1 + βb) + 1 3(1 − βa)(1 − βb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11) Under the BRST transformation, only the gauge fields Ga µ transform nontrivially – 30 – while F a µν and ωa stay invariant, and we obtain δBRSTAΛ β[ω] = � d4x 1 16π2 � � a,b tr(QaQb) · (βa − βb) × �� Λ2 � ∞ 0 duf(u) � (∂µωa) − 1 6 � ∂2∂µωa�� (∂µωb) + � a,b,c,d tr(QaQbQcQd) · 1 6 (1 − βa)(1 − βb)(βc − βd) × � GaµGb µ � ∂νωc�� ∂νωd� + 2Ga µGb ν � ∂µωc�� ∂νωd��� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) where we have used the (anti-)symmetry between the adjoint indices to simplify the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) we see that the Wess-Zumino consistency condition δBRSTAΛ β[ω] = 0 requires the following: βa = βb for any two Abelian sectors a, b for which tr(QaQb) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Either βa = βb = βc = βd or at least two of them are equal to 1 for any group of Abelian sectors for which tr(QaQbQcQd) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12 When these conditions are satisfied, symmetrizing the indices allows one to show that the tr(QaQb) and tr(QaQbQcQd) terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10), if nonzero, are equal to the gauge variation of local counterterms, and we have: AΛ β[α] = � d4x � −δαL(β) ct − 1 128π2 � a,b,c tr(QaQbQc) × � (1 + βa)(1 + βb) + 1 3(1 − βa)(1 − βb) � εµνρσF a µνF b ρσαc � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) where L(β) ct = 1 16π2 � � a,b tr(QaQb)(1 − βa) �� Λ2 � ∞ 0 duf(u) � Ga µGbµ − 1 6 � ∂2Ga µ � Gbµ � + � a,b,c,d tr(QaQbQcQd) · 1 12 (1 − βa)(1 − βb)(1 − βc) GaµGbνGc µGd ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) Therefore, as in the non-Abelian case, a relevant anomaly may only come from terms with three gauge group generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' But unlike the non-Abelian case, β values other 12This applies to groups of two, three and four Abelian sectors since a, b, c, d do not have to be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 31 – than 0 and 1 are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Again, we see that the standard anomaly cancellation con- dition tr � {ta, tb} tc� = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' tr(QaQbQc) = 0 in the Abelian case) would guarantee that the relevant anomaly vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13 U(1)V × U(1)A Example Let us apply the results above to the classic example of two Abelian sectors U(1)V × U(1)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The matter content is assumed to consist of pairs of Weyl fermions with opposite (identical) charges under U(1)V (U(1)A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' the minimal case is that of two Weyl fermions with (QV , QA) = (1, 1) and (−1, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' So the potentially nonzero traces are: tr � Q2 V � , tr � Q2 A � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15a) tr � Q2 V QA � , tr � Q3 A � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15b) tr � Q4 V � , tr � Q2 V Q2 A � , tr � Q4 A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15c) The fact that tr(Q2 V Q2 A) ̸= 0 implies that to satisfy the Wess-Zumino consistency condition we must choose βV = βA or βV = 1 or βA = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16) Assuming one of these is true, we can readily obtain the anomaly result from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13): AΛ β[α] = � d4x � −δαL (βV ,βA) ct − 1 64π2 tr � Q2 V QA �� (1 + βV )(1 + βA) + 1 3(1 − βV )(1 − βA) � εµνρσF µν V F ρσ A αV − 1 128π2 tr � Q2 V QA �� (1 + βV )2 + 1 3(1 − βV )2 � εµνρσF µν V F ρσ V αA − 1 128π2 tr � Q3 A �� (1 + βA)2 + 1 3(1 − βA)2 � εµνρσF µν A F ρσ A αA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) As discussed in footnote 13, there is in fact an additional possible counterterm, εµνρσF µν V V ρAσ (where V and A denote gauge fields), whose gauge variation pro- duces a linear combination of εµνρσF µν V F ρσ A αV and εµνρσF µν V F ρσ V αA upon integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We will come back to this point shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 13One may further ask whether the tr(QaQbQc) terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) may also be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Indeed, there are local counterterms of the form εµνρσF a µνGb ρGc σ one can write down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, there may not be enough such counterterms to absorb all the anomalies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' in particular, if tr � Q3 a � ̸= 0 for some Abelian sector a there must be a relevant anomaly, since the counterterm above vanishes when a = b = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 32 – Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) reproduces the standard result if we further demand that U(1)V is not anomalous and is preserved by renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This means that we should pick the βV = 1 option in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16) so that L (βV ,βA) ct does not involve U(1)V -breaking opera- tors (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This also rules out the additional counterterm εµνρσF µν V V ρAσ discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' For U(1)V to be non-anomalous, the coefficient of the αV term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) must vanish, which requires βA = −1 for βV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We conclude that the standard result corresponds to the specific scheme choice in our formalism: (βV , βA) = (1 , −1) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) in which case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) becomes: AΛ (1,−1)[α] = � d4x � −δαL(1,−1) ct − 1 32π2 εµνρσ � tr � Q2 V QA � F µν V F ρσ V + tr � Q3 A � 1 3 F µν A F ρσ A � αA � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) with the following U(1)V -preserving counterterm: L(1,−1) ct = 1 16π2 � tr � Q2 A �� 2 � Λ2 � ∞ 0 duf(u) � AµAµ − 1 3 � ∂2Aµ� Aµ � + tr � Q4 A � 2 3 (AµAµ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20) It is interesting to note that if we instead choose (βV , βA) = (0 , 0) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) which is also Wess-Zumino consistent but does not manifestly preserve U(1)V , we would obtain: AΛ (0,0)[α] = � d4x � −δαL(0,0) ct − 1 48π2 εµνρσ tr � Q2 V QA � F µν V F ρσ A αV − 1 96π2 εµνρσ � tr � Q2 V QA � F µν V F ρσ V + tr � Q3 A � F µν A F ρσ A � αA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22) This is in fact related to the standard result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) by a counterterm: AΛ (0,0)[α] = AΛ (1,−1)[α] + δα � d4x � L(1,1) ct − L(0,0) ct + ∆Lct � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) – 33 – where ∆Lct = 1 24π2 εµνρσ tr � Q2 V QA � F µν V V ρAσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='24) Therefore, (βV , βA) = (0, 0) actually gives the same relevant anomaly as the standard result, although at the cost of U(1)V -breaking counterterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Note that it is impos- sible to remove both εµνρσF µν V F ρσ A αV and εµνρσF µν V F ρσ V αA using the counterterm, in agreement with the familiar result that U(1)V and U(1)A cannot be simultaneously conserved in the V V A triangle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Also, as discussed in footnote 13, there is always a relevant U(1)3 A anomaly which cannot be removed by counterterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4 Product of Abelian and Non-Abelian Sectors Finally, we consider the cross terms in AΛ β[α] between Abelian and non-Abelian sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' These include the tr � {ta, tb} tc� terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) with two of the adjoint indices in the same non-Abelian sector and the third index in an U(1) sector, and the tr � tatbtctd� terms with two of the adjoint indices in the same non-Abelian sector and the other two in either one or two U(1) sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' So in what follows we focus on a theory with one simple non-Abelian sector and up to two U(1) sectors, which we call U(1)A and U(1)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To ease the presentation we reserve the notation Gµ, F µν, α, ta that we have been using in the general calculation for the non-Abelian sector here, while denoting the corresponding objects in the U(1) sectors by Aµ, F µν A , αA, QA and Bµ, F µν B , αB, QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We use βNA to represent the common β parameter associated with all the non-Abelian generators, and use βA, βB for the β parameters of the U(1) sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' From the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 we know that the only values of βNA consistent with the Wess-Zumino condition in the non-Abelian sector are 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us first consider the simpler βNA = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Here the tr � tatbQAQB � terms are all multiplied by (1−βNA) and vanish, while for the tr � tatbQA � terms we have (switching to matrix notation and following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15)): AΛ β[α] ⊃ � d4x � − 1 32π2 � εµνρσ tr � F µνF ρσαA + 2βAF µνF ρσ A α + 2(1 − βA)F µνAρ(Dσα) � = � d4x � − 1 32π2 � εµνρσ tr � F µνF ρσαA + (1 + βA)F µνF ρσ A α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='25) To arrive at the last equation we have integrated by parts and used the Bianchi identity εµνρσ(DσFµν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Performing the BRST transformation, we find δBRSTAΛ β[ω] ⊃ � d4x i 32π2 (1 + βA) εµνρσ tr � F µνF ρσ A ω2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='26) So for these cross terms in the anomaly to be consistent with the Wess-Zumino – 34 – condition, we must have βA = −1 or tr � tatbQA � = 0 (βNA = 1 case) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27) As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='25) either vanishes due to tr � tatbQA � = 0, in which case there is no crossed anomaly, or only the F µνF ρσαA term survives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' the latter cannot be obtained as a local counterterm variation and is therefore a relevant anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In fact, we have just recovered the non-Abelian generalization of the U(1)V × U(1)A example in the previous subsection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19)): swapping U(1)V for a non-anomalous non-Abelian sector (recall that βNA = 1 requires tr � {ta, tb} tc� = 0) leads to the same crossed anomaly with a chiral U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Next we consider the other option, βNA = 0, for the non-Abelian sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, both the tr � tatbQA � and tr � tatbQAQB � terms can be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' After some algebra we can organize the tr � tatbQA � terms into the following form: AΛ β[α] ⊃ � d4x � − 1 96π2 � εµνρσ tr � F µνF ρσαA + iGµGνF ρσαA + 2GµGνGρGσαA + 3 2(1 + βA)F µν lin F ρσ A α − (1 − βA)F µνAρ(∂σα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) Among the five terms, three (first, third and fourth) are actually BRST-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Overall, we find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) has the following BRST transformation: δBRSTAΛ β[ω] ⊃ � d4x � − 1 96π2 � βA εµνρσ tr � F µν(∂ρω)(∂σωA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='29) For this to vanish, we need βA = 0 or tr � tatbQA � = 0 (βNA = 0 case) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='30) So the crossed anomaly in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) either vanishes due to tr � tatbQA � = 0 or is contained in the general β = 0 formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) as a relevant anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Meanwhile, for the tr � tatbQAQB � terms, we find: AΛ β[α] ⊃ � d4x � − 1 48π2 � tr � (1 − βA) � {Gµ, Gν} Aµ + GµGµAν� (∂ναB) + (1 − βB) � {Gµ, Gν} Bµ + GµGµBν� (∂ναA) + 2(1 − βA)(1 − βB) � (AµBν + AνBµ) Gµ + AµBµGν� (∂να) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='31) – 35 – which transforms under BRST as: δBRSTAΛ β[ω] ⊃ � d4x 1 48π2 tr � (βA − βB) � 2GµGν(∂µωA) + GµGµ(∂νωA) � (∂νωB) − 2(1 − βA)βB � Gµ(∂νω)Aµ + Gν(∂µω)Aµ + Gµ(∂µω)Aν� (∂νωB) − 2(1 − βB)βA � Gµ(∂νω)Bµ + Gν(∂µω)Bµ + Gµ(∂µω)Bν� (∂νωA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='32) For this to vanish, we need βA = βB = (0 or 1) or tr � tatbQAQB � = 0 (βNA = 0 case) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='33) So the crossed anomaly in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='31) either vanishes due to tr � tatbQAQB � = 0 or βA = βB = 1, or is contained in the general β = 0 formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) as an irrelevant anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' For both cases discussed above, βNA = 1 and βNA = 0, the relevant part of the crossed anomaly is proportional to tr � tatbQA � , so the anomaly cancellation condition is contained in the standard one, tr � {ta, tb} tc� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 6 Discussion and Future Directions In this paper, we introduced a novel regularization prescription to calculate anoma- lies for global and gauge symmetries using CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The calculation was performed in d = 4 spacetime dimensions, thereby avoiding any of the subtleties that arise when computing anomalies using dimensional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The master formula obtained in this framework integrates various known results regarding anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In a companion paper [36], we will extend the formalism developed here to incorporate the effects of higher dimensional operators into the anomaly calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This has an immediate application to the Standard Model Effective Field Theory (SMEFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Recently, arguments that the SMEFT is not anomalous were provided in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [36], we will give an explicit proof using CDE that SMEFT is non-anomalous when including operators with general scalar, vector, and tensor couplings to fermion bilinears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In future work, we would like to apply this formalism to compute the EFTs that emerge when integrating out fermions with chiral couplings (for example, integrat- ing out the top quark in the Standard Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This is well-known to produce an EFT with a Wess-Zumino-Witten term [35, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' It should be possible to extend the calculations presented here to reproduce this result in a new way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This will require understanding the interplay of the method presented here and the results for other functional traces that are evaluated using dimensional regularization, since the functional EFT matching framework relies on the method of regions, which is – 36 – implemented in dimensional regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' At least for one loop calculations, the use of different regulators may not cause any particular difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Once this is understood, functional methods for one-loop matching will be a complete framework for integrating out any heavy particles with spins 0, 1/2, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Acknowledgments We thank Quentin Bonnefoy, Nathaniel Craig, Sungwoo Hong, Markus Luty and Aneesh Manohar for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Department of Energy under grant number DE-SC0011640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' De- partment of Energy under grant numbers DE-SC0009919 and DE-SC0011640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Department of Energy under grant number DE-SC0011702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Appendix A Comments on Cyclic Permutation In this appendix, we clarify a subtle point in performing CDE calculations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', when (and why) we are allowed to perform cyclic permutations on the argument of a functional trace ‘ Tr (· · · ) ’, and a lowercase trace ‘ tr (· · · ) ’ which is only over the internal indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We begin by recalling that a functional operator O is a matrix that acts on both the functional vector space |x⟩ and some internal vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The latter is typically finite dimensional, which we can label by a discrete index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We can then write out the concrete relation between the functional trace ‘ Tr ’ and the internal trace ‘ tr ’: Tr (O) = � d4x ⟨x| tr (O) |x⟩ = � d4x ⟨x| Oii |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) Clearly, the functional trace ‘Tr’ sums over all the indices of the matrix O, and therefore it is always safe to perform a cyclic permutation: Tr � OAOB� = � d4x d4y ⟨x| OA ij |y⟩ ⟨y| OB ji |x⟩ = � d4y d4x ⟨y| OB ji |x⟩ ⟨x| OA ij |y⟩ = Tr � OBOA� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) On the other hand, the internal trace ‘tr’ only sums over a subset of indices for the matrix O, and therefore it is generically illegal to make cyclic permutations inside ‘tr’ alone: tr � OAOB� ̸= tr � OBOA� ⇐⇒ ⟨x| tr � OAOB� |y⟩ ̸= ⟨x| tr � OBOA� |y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3) – 37 – Note that after taking the internal trace, the object tr � OAOB� is still a matrix acting on the functional space spanned by |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' So when we check whether the two objects tr � OAOB� and tr � OBOA� are equal, it is a comparison of two matrices where one needs to compare entry by entry, as indicated by the right-hand expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Generically, they are not equal and making cyclic permutations inside ‘tr’ alone is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, in many practical calculations of functional traces, the evaluation re- sults (after carrying out the loop integrals) are local action-like expressions that generically have the form (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7)) Tr (· · · ) = � d4x trx � OAOBOC · · · � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) where the reason for using a slightly different notation ‘ trx (· · · ) ’ will become clear shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' When handling expressions like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4), we do sometimes make cyclic permutations to simplify the calculation: Sometimes we take : � d4x trx � OAOB� = � d4x trx � OBOA� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) This has been used extensively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4, as well as for many functional matching calculations with CDE in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The purpose of this appendix is to clarify when and why Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) could hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The explanation has two important aspects: There is a slight abuse of notation ‘ tr ’ in expressions like Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As emphasized by using a different notation ‘ trx ’ above, the traces in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) are not precisely the same objects as the traces in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3) – the latter are matrices acting on the functional space |x⟩, while the former are actually elements of those matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) does not hold for generic operators OA, OB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, if both OAOB and OBOA are diagonal functional operators in the position basis |x⟩, namely if they satisfy tr � OAOB� |x⟩ = tAB(x) |x⟩ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6a) tr � OBOA� |x⟩ = tBA(x) |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6b) for some ordinary functions tAB(x) and tBA(x), then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In what follows, we elaborate on these two aspects in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1 Internal Trace Notation First, it is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) that trx (O) must be an ordinary function of the variable x (similar to a Lagrangian), such that the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) would yield – 38 – a local action-like result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' So trx (O) cannot be a matrix on the functional space |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Instead, it should be interpreted as an element of that matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Second, we emphasize that trx (O) is not the following matrix element that one might naively expect: trx (O) ̸= ⟨x| tr (O) |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) If the above were true, then performing the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) would give us the functional trace � d4x ⟨x| tr � OAOB� |x⟩ = Tr � OAOB� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='8) in which cyclic permutation would not be a problem at all, as explained around Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' But it is clear that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) is not supposed to yield Tr � OAOB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The correct matrix element is trx (O) = � d4y ⟨x| tr (O) |y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) To understand this subtle point, we need to remind ourselves how we usually obtain expressions like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4) and hence terms like trx (O) from the CDE evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Usually, we start with a functional trace like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1) and calculate it using momen- tum eigenstates: Tr � f � iˆ∂µ, U(ˆx) �� = � d4q (2π)4 � q ��� tr � f � iˆ∂µ, U(ˆx) �����q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) Using the fact |q⟩ = � d4x |x⟩ ⟨x|q⟩ = � d4x e−iqx |x⟩ = � d4x e−iqˆx |x⟩ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='11) we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='10) as Tr � f � iˆ∂µ, U(ˆx) �� = � d4x d4y � d4q (2π)4 � x ���eiqˆx tr � f � iˆ∂µ, U(ˆx) �� e−iqˆx���y � = � d4x d4y � d4q (2π)4 � x ��� tr � f � qµ + iˆ∂µ, U(ˆx) �����y � = � d4x �� d4y � x ���� � d4q (2π)4 tr � f � qµ + iˆ∂µ, U(ˆx) ������y �� = � d4x � � d4y ⟨x| tr (Of)|y⟩ � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) where Of is defined implicitly by the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As indicated in the last line, one way of understanding the ‘simplified CDE’ is that one Taylor expands the function – 39 – ‘ f ’ above and performs the momentum loop integral over qµ to obtain a set of functional operators of the form tr (Of).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This is precisely what we did in deriving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='7) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Now comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='4), we see that the notation ‘ trx ’ is actually denoting the quantity inside the curly brackets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Therefore, we have carefully derived the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us recall that the definition of the ‘functional vector space’ is the collection of all the functions φ(x) (usually satisfying certain constraints, such as ∥φ∥2 < ∞ (under box normalization)), where each function corresponds to a vector |φ⟩: φ(x) = ⟨x|φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='13) It thus provides us with a linear algebra language for the differential operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Specifically, the process of a differential operator ˆf acting on a function φ(x) to yield a new function � ˆfφ � (x) can be written as the action of a matrix in this linear space: � ˆfφ � (x) = � x �� ˆfφ � = � x �� ˆf ��φ � = � d4y � x �� ˆf ��y � ⟨y|φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) The key to this dictionary are the matrix elements � x �� ˆf ��y � for various differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' When ˆf is an ordinary function such as ˆf = Gµ(x), its matrix is diagonal in the |x⟩ basis: ⟨x|Gµ|y⟩ = Gµ(x) δ4(x − y) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15a) Gµ(x)φ(x) = � d4y ⟨x|Gµ|y⟩ ⟨y|φ⟩ = � d4y � Gµ(x) δ4(x − y) � φ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='15b) When ˆf = ∂µ is a derivative, we have ⟨x|∂µ|y⟩ = ∂ ∂xµ δ4(x − y) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16a) ∂µφ(x) = � d4y ⟨x|∂µ|y⟩ ⟨y|φ⟩ = ∂ ∂xµ � d4y δ4(x − y) φ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16b) General differential operators, like ˆf � iˆ∂µ, U(ˆx) � in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12), are built from the two kinds of operators discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We note in particular that the constant unity function ‘1’ corresponds to a vector |1⟩ that satisfies |1⟩ = � d4y |y⟩ ⟨y|1⟩ = � d4y |y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='17) Therefore, the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9) can be rewritten as trx (O) = ⟨x| tr (O) |1⟩ = � tr (O) 1 � (x) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) – 40 – where the last expression follows from the differential operation language in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14) – we are simply taking the differential operator tr (O), acting it on the constant unity function 1, and then evaluating the resulting function at point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='14 When the func- tion being acted on is the constant unity function 1, we often suppress it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We also often suppress the explicit ‘(x)’ when talking about a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Doing both for the last expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18) leads to our abuse of the notation ‘tr’ in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='18), it is immediately clear that trx (AB · · · C∂µ) = 0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19a) trx (∂µAB · · · C) is a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19b) With these, we can see a quick counterexample to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5): trx (A∂µBµ) = trx � A (∂µBµ) + ABµ∂µ � = trx � A (∂µBµ) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20a) trx (BµA∂µ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20b) Clearly, the two lines are related by a cyclic permutation of Bµ, but they are gener- ically not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 Conditions for Cyclic Permutations in Internal Traces After clarifying the meaning of ‘ trx (· · · ) ’, namely the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='9), we see that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) does not always hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, if both expressions inside the trace before and after the cyclic permutation are diagonal operators in the position basis |x⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='6) is true, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) would hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' To see this, we first note that if tr � OAOB� |x⟩ = tAB(x) |x⟩ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='21) then we simply have trx � OAOB� = � d4y � x �� tr � OAOB���y � = � d4y tAB(y) δ4(x−y) = tAB(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='22) Therefore, it is linked with the functional trace as Tr � OAOB� = � d4x � d4q (2π)4 ⟨x|q⟩ � q �� tr � OAOB���x � = � d4x tAB(x) � d4q (2π)4 ⟨x|q⟩ ⟨q|x⟩ = � d4x trx � OAOB� � d4q (2π)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='23) 14See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [41] and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [19] for clarifications of this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' – 41 – where � d4q (2π)4 = ⟨x|x⟩ is just a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Making use of this relation between trx(· · · ) and Tr(· · ·), one could take advantage of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2) to perform a cyclic permutation: � d4x trx � OAOB� � d4q (2π)4 = Tr � OAOB� = Tr � OBOA� = � d4x trx � OBOA� � d4q (2π)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='24) Canceling the normalization factor gives us Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Note that one can generalize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='5) to the sum of multiple terms: OAOB −→ OA 1 OB 1 + · · · + OA n OB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='25) In this case, for the steps in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='24) to be valid, one only needs the sum to be diagonal in the position basis |x⟩, namely we have � d4x trx � OA 1 OB 1 + · · · + OA n OB n � = � d4x trx � OB 1 OA 1 + · · · + OB n OA n � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='26) provided that tr � OA 1 OB 1 + · · · + OA n OB n � |x⟩ = tAB(x) |x⟩ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27a) tr � OB 1 OA 1 + · · · + OB n OA n � |x⟩ = tBA(x) |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27b) An operator O being diagonal in the position basis |x⟩ is equivalent to the state- ment that all the derivatives in O are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' For example, consider the following differential operator: O = A∂µBC = A � ∂µB � C + AB∂µC = A � ∂µB � C + AB � ∂µC � + ABC∂µ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='28) where A, B, C are diagonal in the |x⟩ basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The decomposition in the first line follows from the product rule of the derivative, where the parentheses in the first term has the usual interpretation – it indicates that ∂µ only acts on B but not anything to the right of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (In fact, this notation was already used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='20a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=') In this case, we say that the derivative is closed on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In contrast, the second term in the first line has an open derivative that acts on everything to its right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' One can further use the product rule to obtain the decomposition in the second line, where a term with the derivative closed on C appears, and there is an additional term with an open derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Clearly, terms with closed derivatives, such as A � ∂µB � C and AB � ∂µC � are diagonal operators in the |x⟩ basis, while terms with open derivatives such as – 42 – ABC∂µ are not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' When evaluating a functional trace with simplified CDE, the initial set of op- erators in the trace trx(· · · ) emerge from evaluating an expression of the form (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='12)): � d4q (2π)4 tr � f � qµ + iˆ∂µ, U(ˆx) �� = tr (Of) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='29) The operator tr (Of) derived from such an expression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=', upon expanding ‘ f ’ and carrying out the loop momentum integral, is guaranteed to be diagonal in the position basis |x⟩, because it is known that one could use the trick of ‘original CDE’ to close all of the derivatives in it (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' However, since Of is a sum of terms, if we perform an arbitrary cyclic permutation on each term: tr (Of) = tr � OA 1 OB 1 + · · · + OA n OB n � −→ tr � OB 1 OA 1 + · · · + OB n OA n � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='30) it is not guaranteed that the operator is still diagonal in the |x⟩ basis, thus invalidat- ing the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Only a subset of cyclic permutations that satisfy the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='27) are ‘legal.’ Nonetheless, in practical calculations, a very efficient prescription to ensure that we are only performing legal cyclic permutations is to stipulate that terms with open derivatives should not be evaluated – one must keep track of all such terms, and make sure that they get canceled upon summing the terms obtained after the cyclic permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If they do not get fully canceled, then it is a sign that an illegal cyclic permutation had been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In this case, one needs to make further cyclic permutations until the derivatives are all closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' In summary, insisting that all derivatives must be closed in the end is an efficient way to make sure that we are carrying out legal cyclic permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The calculations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 4 of the main text (as well as in many other functional matching calculations with CDE in the literature) are done in such a manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Let us look at a quick example of this: trx � (∂µA) Bµ� = trx � ∂µABµ − A∂µBµ� −→ trx � ∂µABµ − BµA∂µ � = trx � (∂µABµ) + ABµ∂µ − BµA∂µ � −→ trx � Bµ∂µA − BµA∂µ � = trx � Bµ (∂µA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='31) In the first line, we started with an operator (∂µA) Bµ in which the derivative is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' We made a cyclic permutation of the second term to arrive at the second line, where the derivatives are not fully closed, because the last two terms in the second expression both have open derivatives and they do not cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' If we were to stop here and evaluate the second line, then following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='19) these – 43 – terms are zero and total derivatives that would not feed into the final result: � d4x trx � (∂µABµ) + ABµ∂µ − BµA∂µ � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='32) This clearly would not agree with the evaluation of the left-hand side of the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The reason is that the second line was obtained by an illegal cyclic permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Now, if we insist that the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content='31) should not be evaluated since it has open derivatives, then we are forced to make further cyclic permutations such that all the derivatives can be closed upon summing the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' The third line is an example of such a further cyclic permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' As soon as the derivatives are all closed, we can carry out the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' This prescription guarantees that only legal cyclic permutations would be performed, and we can see that the result obtained in the third line does agree with the expression we started with in the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Bertlmann, Anomalies in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQf6foW/content/2301.00821v1.pdf'} +page_content=' Bilal, “Lectures on Anomalies,” arXiv:0802.' metadata={'source': 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Kim,1 Guoqing Wang,1, 2 and Paola Cappellaro1, 2, 4, ‡ +1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA +2Department of Nuclear Science and Engineering, +Massachusetts Institute of Technology, Cambridge, MA 02139, USA +3Department of Chemistry and Institute for Soldier Nanotechnologies, +Massachusetts Institute of Technology, Cambridge, MA 02139, USA +4Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA +Alkali metal ions such as sodium and potassium cations play fundamental roles in biology. De- +veloping highly sensitive and selective methods to both detect and quantify these ions is of consid- +erable importance for medical diagnostics and bioimaging. Fluorescent nanoparticles have emerged +as powerful tools for nanoscale imaging, but their optical properties need to be supplemented with +specificity to particular chemical and biological signals in order to provide further information about +biological processes. Nitrogen-vacancy (NV) centers in diamond are particularly attractive as fluo- +rescence markers, thanks to their optical stability, biocompatibility and further ability to serve as +highly sensitive quantum sensors of temperature, magnetic and electric fields in ambient conditions. +In this work, by covalently grafting crown ether structures on the surface of nanodiamonds (NDs), +we build sensors that are capable of detecting specific alkali ions such as sodium cations. We will +show that the presence of these metal ions modifies the charge state of NV centers inside the ND, +which can then be read out by measuring their photoluminescence spectrum. Our work paves the +way for designing selective biosensors based on NV centers in diamond. +I. +INTRODUCTION +Alkali ions such as sodium and potassium play an es- +sential role in biological systems and their concentration +is tightly regulated and vary in different bodily fluids. +For example, sodium and potassium ions are responsible +for maintaining fluid and electrolyte balance: the typical +Na+ concentration is 135-150 mM in human blood and +only less than 30 mM in intracellular fluid, while these +proportions are inverted for K+ – below 5 mM in blood +and about 150 mM in intracellular fluid [1, 2]. Fluctua- +tions in their concentrations are usually problematic and +can lead to various physiological disorders and diseases, +including cardiovascular disease and hypertension [3, 4]. +Measurement of their concentrations would thus be of +great interest both in understanding the functions of the +ions for studying cellular physiology and in clinical ex- +aminations. +Clinical laboratories typically use ion-selective elec- +trodes [5] or flame photometry [6] to perform measure- +ments, but these techniques require a sample volume +as large as several mL. To overcome this issue while +achieving a high spatial and temporal resolution, small +molecule-based fluorescence sensors would be favorable. +Developing small molecular or nanoparticle probes for +sodium or potassium ions has seen substantial advances +in recent years [1, 2, 7, 8]. Unlike instrumentation meth- +ods in clinical labs, these probes could not only provide a +∗ The authors contributed equally to this work.; Current address: +Global Technology Applied Research, JPMorgan Chase, New +York, NY 10017 USA +† The authors contributed equally to this work. +‡ pcappell@mit.edu +good spatio-temporal resolution but also potentially cross +the cell membrane and be used for intracellular measure- +ments. +One of the key challenges for designing such ion sen- +sors is to distinguish the target from other common metal +ions. As an example, a sodium sensor should be able to +tell the difference between sodium (Na+) and other high- +concentration ions in biological system, such as potas- +sium (K+), magnesium (Mg2+) and calcium (Ca2+) ions. +In this context, the crown ether family has been widely +used in designing alkali metal sensors thanks to its capa- +bility of binding alkali ions selectively [9]. Crown ethers +are cyclic chemical compounds that consist of a ring con- +taining several ether groups, which can bound specific +cations inside the ring. For example, 18-crown-6 is known +to have a high affinity for potassium cation whereas 15- +crown-5 for sodium cation and 12-crown-4 for lithium +cation. +Using crown ether together with a fluorescent +moiety, highly sensitive metal ion sensor can then be +built [9, 10]. However, the fluorescent dye molecules used +by most of the existing metal ion sensors can suffer from +photobleaching and fast efflux from cells after loading. +Nanoparticles, on the other hand, might provide stable +fluorescent signal and last long in cells [11, 12]. In par- +ticular, nanodiamonds (NDs) are bio-compatible and can +have stable fluorescence due to inner spin defects, the +most-studied one of which is the nitrogen-vacancy (NV) +center. The NV center is a point defect in diamond lat- +tice consisting of a substitutional nitrogen atom with an +adjacent vacancy. +The negatively-charged NV− defect +has a triplet ground state energy with a zero-field split- +ting of (2π)2.87 GHz and can show quantum effects even +at room temperature. In recent years, the NV system +has been widely used in various quantum applications +such as quantum sensors [13–18] and quantum informa- + +2 +10-1 +100 +101 +surface Na+ density [nm-2] +0.4 +0.6 +0.8 +1 +[NV-] +-20 +-15 +-10 +-5 +0 +depth (nm) +-4 +-2 +0 +2 +4 +6 +Energy (eV) +VB +CB +NV-/0 +Fermi level +NV0 +NV- +a +b +c +d +e +12-crown-4 +15-crown-5 +18-crown-6 ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +Air +Diamond +-20 +-15 +-10 +-5 +0 +depth (nm) +-5 +0 +5 +Energy (eV) +VB +CB +NV-/0 +Fermi level ++ ++ ++ ++ ++ +Air +Diamond +NV- +FIG. 1. a. Schematics of the ion sensor. NV center-containing NDs are firstly coated with crown ether. Upon their introduction, +sodium cations will be trapped in the 15-crown-5 compounds. b. Energy band schematic of diamond-air interface. Positive +charges on the diamond surface lead to upward bending of the conduction band (CB) and valence band (VB) as well as the +NV−/0 transition level. The dashed line indicates the Fermi level. NV0 is the dominant NV charge state when the NV−/0 +transition level is above the Fermi level, while NV− dominates in the opposite case. The surface density of Na+ is taken to +be 0.1 nm−2 here. c. Energy band structure when the surface density of Na+ is 1 nm−2 d. Simulated fraction of negatively +charged NV states versus surface Na+ density. e. Generalization into other type of crown ethers, which can selectively bind +certain cations and form complexes. +tion processors [19, 20]. +In particular, the NV defect +is known to be highly sensitive to certain external sig- +nals such as magnetic fields and surrounding charge en- +vironments. Depending on the local charge distributions +and external electrical manipulations, the NV state could +have dynamical transitions between the widely-studied +negative NV− state and a neutral state NV0 [21]. The +charge state of NV centers can then serve as a meter to +yield the information of its environment [18, 22–25]. +In this work, we design and demonstrate an alkali ion +sensor based on NV centers in crown-ether-functionalized +nanodiamonds. As a proof-of-principle example, we use +15-crown-5 to detect sodium ions. The formation of 15- +crown-5-Na+ complexes on nanodiamond surface will suf- +ficiently modify the charge environment of the NV cen- +ters inside, and thus alter the charge states of NV defects. +We thus measure the photoluminescence (PL) emission +spectrum of NV defects in nanodiamonds under continu- +ous green laser illumination, from which we can extract +the fraction of each NV charge state. We find that the +NV’s charge state is influenced both by the surface charge +profile and by the power of illumination laser. To demon- +strate the sensor’s capability in detecting the ions pres- +ence we exploit the emergence of a distinctive laser power +dependence of the NV− fraction’s upon introduction of +sodium cations in the sample, that is clearly different +from the pure ND dependence. We further demonstrate +the specificity of the NV-based sensor by several control +experiments. While we uses 15-crown-5 in our proof-of- +principle experiments, we remark that a different type +of crown ether structure can be adapted to probe other +alkali ions, including lithium, potassium and cesium ions. +II. +RESULTS +A. +Principles of the sensor +We start by introducing the basic detection mechanism +of the sensor. As shown in Fig. 1(a), a fluorescent ND +with initial carboxyl group terminations is firstly coated +with crown ether via covalent bond. This yields a neutral +surface layer in contrast to the negative layer due to the +carboxyl groups. Due to the chelate effect and macro- +cyclic effect, depending on its cavity size, crown ether +exhibits strong affinities for specific alkali ions [9, 10]. +Upon introducing these ions, stable complexes will form, +which in turn lead to positive charges on the ND surface. + +Na+Na15-crown-5 +NVcentersK+Na+Li+3 +550 +600 +650 +700 +750 +800 +wavelength (nm) +0 +0.2 +0.4 +0.6 +0.8 +1 +Intensity (a.u.) +[NV-] = 0.754 +Experiment +NV0 +NV- +total fit +102 +103 +Laser power [ W] +0 +0.2 +0.4 +0.6 +0.8 +1 +[NV-] +Spot1 +Spot2 +Spot3 +Spot4 +a +b +c +FIG. 2. a. Typical PL spectrum for unfunctionalized NDs (carboxyl-terminated) acquired under 532 nm laser illumination. +We perform linear fitting of the spectrum to extract the fraction of NV− charge state. The peak-normalized NV−(0) spectrum +is from reference [26]. The Raman peak at around 547 nm is attributed to the silicon wafer over which we deposit our samples. +Oscillations of PL intensity after 700 nm are due to the etaloning effect on the CCD camera. b. Distribution of NV− fraction +for carboxyl-terminated NDs in the absence and presence of sodium ions. The laser power is fixed to be 0.5 mW. Sodium +cations are added via mixing NaCl solutions with samples of interest hereafter. c. Typical laser power dependence of NV− +fraction of carboxyl-terminated NDs for four different spots. The errorbars are the fitting errors (5 percent). Inset shows the +recombination and ionization processes of the two NV charge states. +For example, the 15-crown-5 structure we studied in this +work can strongly bond Na+ and form complexes, while +its interaction and affinity with K+ are weak. While we +focus on 15-crown-5 and sodium cation in this work, the +scheme is general and can be extended to probe other +alkali or alkaline earth metal ions using different types of +crown ethers. For example, in Fig. 1(e) we show the case +of detecting Li+ and K+. +The accumulated charges on the ND surface can ef- +fectively result in bending of the valence band and con- +duction band in the diamond lattice [18, 25, 27]. Sim- +ilar as hydrogen-terminated diamond [27, 28], as shown +in Fig. 1(b-c), a positive charge layer on the surface can +lead to upward bending of both bands and the bending of +the NV−/0 level, which indicates the energy at which the +NV defect loses or takes up one electron. In other words, +this transition level indeed corresponds to the transition +from the neutrally to the negatively charged NV states. +When its relative position with respect to Fermi level is +lower (higher), the defect tends to absorb (lose) one elec- +tron. Due to the positive charge layer on the surface, +when close to the surface, this transition level is shifted +above the Fermi level (dashed line in Fig. 1(b-c)) and +then the NV defect is ionized from NV− to NV0. This +effect is particularly important for shallow NVs or NVs +in ND, while at larger depths the band bending effect is +weak and the Fermi level is above the NV−/0 level and +the dominant charge state would be NV−. We note that +the NV0 can further lose an electron and becomes a non- +fluorescent state NV+ [21], which is inaccessible here. +To quantitatively characterize the above model, we +perform numerical simulations to study the relative ratio +of different NV states (see Methods for detailed infor- +mation). +The NV−/0 level corresponds to the ground +state of NV−. It’s predicted to be in the band gap and +it’s 2.8 eV above the valence band maximum [21]. +In +Fig. 1(c) we show the simulated band bending effect due +to sodium cations on the surface of ND with a diame- +ter of 40 nm when the surface ion density is 1 nm−2. +At a depth of 12.5 nm, we observe that the dominant +species of NV defects switches from neutrally to nega- +tively charged states. We note that close to the surface, +the valence band is shifted above the Fermi level, indi- +cating p-type surface conductivity due to the formation +of a two-dimensional hole gas. With Fermi-Dirac distri- +bution, one can then extract the fraction of NV− as a +function of surface cation density (here we use sodium +ion as an example). +As expected, in Fig. 1(d) we ob- +serve that a larger density of cations on the ND surface +will serve as electron acceptor, thus leading to a smaller +NV− fraction in the diamond lattice. +B. +Experiments +To demonstrate the proposed mechanism for ion sens- +ing, we coat NDs with 15-crown-5 via EDC coupling +(see Methods for details) and use the sensor to detect +sodium cations. In order to measure the change of NV− +fraction induced upon the surface charge, we measure +the spectrum of nanodiamond ensembles using a Raman +spectroscopy (see Methods for details). NV− and NV0 +have different PL spectra [26, 29], in particular, the zero- +phonon-line (ZPL) of NV− is at 637 nm while it is 575 +nm for NV0. To approximately determine the NV− frac- +tion, we fit the acquired spectrum of our samples I(λ) to +a linear combination of spectrum of NV− and NV0: +I(λ) = a{[NV −]I− + (1 − [NV −])I0}, +(1) +where I−(0) denotes the peak-normalized spectrum for +a NV−(0) and a is a normalization factor. In Fig. 2(a) + +CB +二 +NV- +NVO +NV +NV01 +VB7 +6 +ND +5 +ND + Na+ (34 mM) +occurrence +ND + Na* (68 mM) +4 +3 +2 +1 +0 +0.2 +0.4 +0.6 +0.8 +[NV]4 +102 +103 +104 +105 +Laser power [ W] +0 +0.2 +0.4 +0.6 +[NV-] +0 +0.2 +0.4 +0.6 +0.8 +1 +Max NV- fraction +0 +0.2 +0.4 +0.6 +0.8 +1 +Abs change +ND +NDCE5 +NDCE5+Na+ +NDCE5+K+ +ND+Na+ +ND +NDCE5 +NDCE5+Na+ +5+Na +5+Na +NDCE5+K+ +ND+Na+ +0 +0.2 +0.2 +0.4 +0.4 +0.6 +0.6 +0.8 +0.8 +1 +Max NV +Max NV- fraction + fraction +0 +0.2 +0.2 +0.4 +0.4 +0.6 +0.6 +0.8 +0.8 +1 +Abs change +Abs change +0 +101 +102 +103 +104 +105 +Laser power [ W] +0 +0.2 +0.4 +0.6 +0.8 +1 +[NV-] +ND +NDCE5 +NDCE5+Na+ +NDCE5+K+ +550 +600 +650 +700 +750 +800 +wavelength (nm) +0 +0.2 +0.4 +0.6 +0.8 +1 +Intensity (a.u.) +25 uW +50 +250 +500 +2500 +5000 +a +c +b +d +FIG. 3. a. PL spectrum of NDCE5 under different illumination powers. b. Typical laser power dependence of NV− fraction +for various samples. Measurements are performed in order of increasing power. c. Memory curve of NDCE5 sample. The solid +curve were first measured in power ascending order and then we went back to low powers (dashed line, in power descending +order). d. Comparison of different samples using the coordinates of maximal reachable NV− fraction and absolute change +of NV− fraction when one varies illumination power. The lighter data points show the results for individual spots, while the +opaque data is the average results of different samples. The dashed line is when the change of NV− fraction equals the maximal, +i.e., when starting from [NV−]=0 at low power. +we give an example of a typical spectrum for carboxyl- +terminated nanodiamonds measured under continuous +532 nm laser illumination. The fitting here yields that +the fraction of NV− is [NV−] = 0.754, which is compa- +rable to the value observed in bulk diamond [22, 30]. +As a control test, we first demonstrate that the charge +state of NV centers in the carboxyl-terminated nanodi- +amonds (labelled as ND in the plots) is not sensitive to +sodium cations. First, the ion itself shows no fluorescence +under 532 nm laser. Then, as plotted in Fig. 2(b), the dis- +tribution of [NV−] shows little dependence on the concen- +tration of sodium ions. The laser power here is fixed to be +0.5 mW. We further present typical illumination power +dependence curves of [NV−] for these unfunctionalized +NDs in the absence of external ions (Fig. 2(c)). +The +green excitation dynamically modulates the NV charge +state between neutral and negative. Due to NV’s suc- +cessive absorption of two photons with wavelength less +than 637 nm (corresponds to 1.946 eV), an extra elec- +tron of NV− can be ejected into the conduction band. +The first photon pumps the NV− from ground state to +excited state while the second photon takes the electron +to the conduction band. In the opposite case, a neutrally +charged NV defect can take up one electron from the dia- +mond valence band band via absorption of a photon with +energy larger than 2.156 eV (wavelength 575 nm). Under +532 nm illumination here, both the electron-NV recom- +bination and the NV− ionization processes can happen, +yielding an equilibrium NV− fraction around 0.7. Again, +this is similar as NV centers in bulk diamond [22, 30]. To +further study the power dependence of the recombination +or ionization process separately, another laser with longer +wavelength (for example, 632 nm) is required to induce +one-directional charge conversion process while suppress- +ing the reverse one. +We then switch to the crown-ether-functionalized nan- +odiamond sample and measure its fluorescence spectrum +under the same 532 nm laser illumination. +The mea- +surement is performed from low laser power to high +power chronologically. Fig. 3(a) clearly shows that now +the emission spectrum for 15-crown-5 coated nanodia- +mond (labelled as NDCE5 hereafter) is power-dependent. +When one increases the laser power, the spectrum shifts +rightward, corresponding to an increasing NV− fraction. + +5 +We note that overall the fraction of negative charge state +for NDCE5 is smaller than unfunctionalized ND even +when the laser power is high. This can be attributed to +the fact that NDCE5 is supposed to have neutral charge +on the surface, while the carboxyl-terminated NDs tend +to have negatively charged surface. +For a comparison, we then show the the power- +dependence of [NV−] for different samples in Fig. 3(b). +While for unfunctionalized NDs we observe no depen- +dence on illumination power, for all crown-ether-coated +samples studied in this work, we see that when the power +is low the NV− fraction is approximately zero and it then +keeps increasing as a function of illumination power. A +similar phenomenon has been observed in near-surface +shallow NV centers [22] and this is in contrast to NV +centers in certain bulk samples [29], where the NV−-to- +NV0 ratio decreases as a function of the laser intensity. +The power dependence observed here emerges from the +interplay between the ionization and recombination rates +and the charge transfer rate from NV− to neighboring +traps [22, 31]. +These trap states are likely originated +from neighboring defects such as vacancy complexes. +While the NDCE5 curve typically shows a saturated +fraction of about 0.5, after adding sodium cations, the +NV− fraction is considerably lower than 0.5 (usually be- +low 0.4) even at high laser power. We attribute this to the +positive charge on ND surface yielding the band bend- +ing effect. We further show the data for NDCE5 in the +presence of potassium ions. In contrast to sodium ions, +15-crown-5 has low affinity for potassium cation and the +sample shows a high [NV−] at large laser power. +We note that we observe a memory effect for the +NDCE5 sample studied above. That is, after finishing +the spectrum measurement at high laser power, we set +the power to low values and re-evaluate the properties +of spectrum. Fig. 3(c) shows that the NV− fraction re- +mains at a high value (above 0.5) even when the laser +power is set as low as 50 µW. The memory effect per- +sists for at least hours. While a detailed study on the +charge dynamics and electron transfer process is needed, +we suspect that high laser power preferentially transfers +an excess electron from local traps to the NV defect [22]. +Finally, in Fig. 3(d) we summarize the results for var- +ious samples we evaluated. +Here we plot the absolute +change of [NV−] as a function of the maximal [NV−] the +sample can achieve at high illumination power. We note +that in the presence of sodium ions, the maximal frac- +tion is considerably lower than other samples, including +NDCE5 and unfunctionalized NDs. Again, this is due to +the fact that 15-crown-5 can form complexes with sodium +cations, leading to a positive charge layer on the ND sur- +face. This in turn results in the band bending effect and +thus in a lower NV− fraction even when the laser illu- +mination power is high. Surprisingly, we find that in the +presence of K+, at high power most of the NV defects can +be converted to NV−. Further study is needed to inves- +tigate the interaction between 15-crown-5 and potassium +cations. +III. +DISCUSSIONS +While our results demonstrate that the NV charge ra- +tio can be used to detect the presence of targeted cations, +the large inhomogeneities in the ND properties prevented +us from obtaining quantitative results on the [NV−] den- +sity. As our experimental setup lacked the ability to re- +peatedly addressing the same ND, we had to perform +measurements over many spatial spots to average out +the inhomogeneities among NDs and extract significant +differences as a function of sample properties. The inho- +mogeneities in fluorescence intensity, initial charge ratio +and ionization(deionization) rates are induced by vari- +ous factors, including inefficient surface coating of crown +ether, spatial density profile of NV centers and nitrogen +atoms, as well as distinct local charge environments. A +well-calibrated, single ND sensor would avoid this costly +repeated measurement process to extract the behavior +distribution, and greatly improve the sensitivity of the +sensor. NV centers with pre-characterized local charge +environments and surface charge densities might be used +to reduce the overhead in measuring many spatial spots. +While we used an all-optical approach to read out +the charge state information of NV defects via their PL +spectrum measurement, we remark that one might ex- +tract the same information from the signal contrast of +optical-detected magnetic resonance (ODMR), given the +fact that NV0 and NV− have distinct ground state spin +configurations. A larger NV− fraction will yield a better +signal contrast at the resonance frequency (2.87 GHz in +the absence of external magnetic fields). In this case, a +microwave pulse would be required for the ODMR exper- +iment. With the help of microwave, the surface charge +layer might also be probed by monitoring the transverse +relaxation time of the NV− charge state, as it can induce +fluctuating electrical field that can interact with the NV +centers shortening its relaxation time [32, 33]. +In conclusion, we designed a sodium ion sensor based +on NV centers in crown-ether-functionalized nanodia- +monds. The charge state of NV centers shows a strong +dependence on the surface charge profile and can be de- +tected by measuring the PL spectrum of NV centers. +While we focused on sodium cations here, the sensing +mechanism can be extended to detect other ions such +as K+ by changing the surface crown ether structures. +These NV-based sensors have a stable PL signal and can +provide excellent spatial resolutions, thus opening new +opportunities for monitoring ion concentrations in bio- +logical systems and for cellular physiology. +ACKNOWLEDGEMENT +This work was supported in part by the U.S. Army +Research Office through Grant W911NF-15-1-0548 and +by the National Science Foundation. +D.K. acknowledges the support from the National In- +stitute of General Medical Sciences with award Number + +6 +T32GM007753. The content is solely the responsibility +of the authors and does not necessarily represent the of- +ficial views of the National Institute of General Medical +Sciences or the National Institutes of Health. +COMPETING FINANCIAL INTEREST +The authors declare no competing financial interests. +AUTHOR CONTRIBUTIONS +C.L. and S.-X. L. L. proposed the sensor scheme and +designed the experiment. +P.C. supervised the project. +S.-X. L. L. conducted the sensor synthesis and character- +ization. C.L. performed the PL spectrum measurement +and data analysis, with partial input from G.W. The nu- +merical simulations were performed by C.L. and D.K. +using the nextnano software. C.L. wrote the paper with +the contributions from all authors. All authors discussed +the results. +Appendix A: Experiment details +1. +Nanodiamond sample +The +nanodiamonds +(Adamas +Nanotechnologies, +NDNV40nmHi10ml) in this work had an average size of +35-40 nm and were milled from high pressure high tem- +perature (HPHT) micro-size particles. +Before milling, +these particles were irradiated with 2-3 MeV electrons +followed by annealing at 850◦C for 2 hrs. Substitutional +nitrogen content in the starting micro-size material was +around 100 ppm, while the concentration of NV defect +is found to be 1-2 ppm, corresponding to around 12-14 +color centers per 40 nm nanoparticle [34]. +2. +Chemical synthesis and characterization +Commercial reagents were purchased from Sigma- +Aldrich and TCI and used as received unless otherwise +noted. Following reported procedures [35, 36], nanodi- +amonds were first treated with a mixture of acids to +remove the surface graphite contaminants and metal- +lic impurities, generating carboxyl groups for the sub- +sequent functionalization by EDC coupling. +In par- +ticular, nanodiamonds were dispersed in a 3:1 mix- +ture of concentrated sulfuric acid and nitric acid and +stirred overnight at room temperature. +After neutral- +ization with aqueous NaOH (1 M), the resulting nan- +odiamonds were cleaned using several centrifugation, +washing, and redispersion cycles with Milli-Q water. +EDC coupling with amino crown ether was performed +by adding an excess of 1-(3-dimethylaminopropyl)-3- +ethylcarbodiimide hydrochloride and 2-aminomethyl-15- +crown-5 to the nanodiamonds dispersion and the reac- +tion mixture was stirred overnight at room temperature, +followed by several centrifugation, washing, and redis- +persion cycles with Milli-Q water. The resulting nanodi- +monds were characterized by a Bruker Alpha II FTIR +spectrometer with a Diamond Crystal ATR (attenuated +total reflectance) accessory and a K-alpha+ X-ray Photo- +electron Spectrometer system (Thermo Scientific) using +a Al Kα radiation source (Fig. 4). +3. +Spectrum measurement and fitting +The photoluminescence (PL) spectrum of the NV cen- +ters under 532 nm laser excitation is collected via a confo- +cal Raman microscope (Renishaw inVia microscope with +a 1024 × 256 pixel CCD camera) at room temperature. +We put the samples on a silicon wafer to avoid unwanted +Raman peaks. +To approximately extract the fraction of NV− charge +state we first peak-normalize the measured curves and +then perform linear fitting of the spectrum from 560 nm +to approximately 750 nm based on the single NV− and +NV0 reference spectra [26]. +Appendix B: Simulations +To qualitatively understand the change of NV charge +state in the presence of surface charges, numerical sim- +ulations are performed. We first assume rapid thermal +equilibrium state after the photoexcitation process of the +NV centers [25, 27]. +The charge state of NV center +is then exclusively governed by the relative position of +the NV−/0 charge transition level with respect to the +Fermi level, neglecting the complex ground, excited and +metastable states of all defect states. We first perform +numerical simulations of the band bending effect using +the nextnano software [37], a tool for simulation of elec- +tronic semiconductor nanodevices. With realistic param- +eters, one can use it to calculate the band structure of +multi-dimensional devices. +In this work, we perform three-dimensional simulations +on a nanodiamond of spherical shape with a diameter +d = 40 nm. In the simulation, the Poisson equation is +discretized on a grid with grid size 0.5 nm using the fi- +nite differences method and solved iteratively in a self- +consistent manner. The Poisson equation is coupled with +the single-band effective mass Schr¨odinger equation via +the charge density, as described by the wave functions in +the diamond [25, 27]. Negative charge (electron) donors +in the lattice mainly include the substitutional nitrogen +atoms (ionization energy 1.7 eV) and here we take their +concentration to be 100 ppm (as expected from the ND +used in experiments) and assume uniform distributions +as a function of depth below the diamond surface. We + +7 +a +b +c +FIG. 4. Characterization of NDCE5 sample. a. FT-IR spectrum of NDCE5. b-c. XPS C 1s and N 1s spectra of NDCE5. +take the nitrogen-to-NV conversion ratio to be 1%, cor- +responding to a total NV defect concentration of 1 ppm. +The boundary condition is determined by the surface +charges (depth x = 0) that are contributed by the 15- +crown-5-Na+ complexes or 15-crown-5 structures. 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Yadav, +Diamond and Related Materials 94, 172 (2019). +[37] A. +Trellakis, +T. +Zibold, +T. +Andlauer, +S. +Birner, +R. +K. +Smith, +R. +Morschl, +and +P. +Vogl, +Journal of Computational Electronics 5, 285 (2006). + diff --git a/LNE1T4oBgHgl3EQfYwTJ/content/tmp_files/load_file.txt b/LNE1T4oBgHgl3EQfYwTJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40f2e7b8e1e93aa44114975143d19610dac41944 --- /dev/null +++ b/LNE1T4oBgHgl3EQfYwTJ/content/tmp_files/load_file.txt @@ -0,0 +1,734 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf,len=733 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='03143v1 [quant-ph] 9 Jan 2023 Ion sensors with crown ether-functionalized nanodiamonds Changhao Li,1, 2, ∗ Shao-Xiong Lennon Luo,3, † Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Kim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='1 Guoqing Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2 and Paola Cappellaro1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' ‡ 1Research Laboratory of Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' USA 2Department of Nuclear Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' USA 3Department of Chemistry and Institute for Soldier Nanotechnologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' USA 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' USA Alkali metal ions such as sodium and potassium cations play fundamental roles in biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' De- veloping highly sensitive and selective methods to both detect and quantify these ions is of consid- erable importance for medical diagnostics and bioimaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Fluorescent nanoparticles have emerged as powerful tools for nanoscale imaging, but their optical properties need to be supplemented with specificity to particular chemical and biological signals in order to provide further information about biological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Nitrogen-vacancy (NV) centers in diamond are particularly attractive as fluo- rescence markers, thanks to their optical stability, biocompatibility and further ability to serve as highly sensitive quantum sensors of temperature, magnetic and electric fields in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In this work, by covalently grafting crown ether structures on the surface of nanodiamonds (NDs), we build sensors that are capable of detecting specific alkali ions such as sodium cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We will show that the presence of these metal ions modifies the charge state of NV centers inside the ND, which can then be read out by measuring their photoluminescence spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Our work paves the way for designing selective biosensors based on NV centers in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' INTRODUCTION Alkali ions such as sodium and potassium play an es- sential role in biological systems and their concentration is tightly regulated and vary in different bodily fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' For example, sodium and potassium ions are responsible for maintaining fluid and electrolyte balance: the typical Na+ concentration is 135-150 mM in human blood and only less than 30 mM in intracellular fluid, while these proportions are inverted for K+ – below 5 mM in blood and about 150 mM in intracellular fluid [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Fluctua- tions in their concentrations are usually problematic and can lead to various physiological disorders and diseases, including cardiovascular disease and hypertension [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Measurement of their concentrations would thus be of great interest both in understanding the functions of the ions for studying cellular physiology and in clinical ex- aminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Clinical laboratories typically use ion-selective elec- trodes [5] or flame photometry [6] to perform measure- ments, but these techniques require a sample volume as large as several mL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To overcome this issue while achieving a high spatial and temporal resolution, small molecule-based fluorescence sensors would be favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Developing small molecular or nanoparticle probes for sodium or potassium ions has seen substantial advances in recent years [1, 2, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Unlike instrumentation meth- ods in clinical labs, these probes could not only provide a ∗ The authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Current address: Global Technology Applied Research, JPMorgan Chase, New York, NY 10017 USA † The authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' ‡ pcappell@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='edu good spatio-temporal resolution but also potentially cross the cell membrane and be used for intracellular measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' One of the key challenges for designing such ion sen- sors is to distinguish the target from other common metal ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As an example, a sodium sensor should be able to tell the difference between sodium (Na+) and other high- concentration ions in biological system, such as potas- sium (K+), magnesium (Mg2+) and calcium (Ca2+) ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In this context, the crown ether family has been widely used in designing alkali metal sensors thanks to its capa- bility of binding alkali ions selectively [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Crown ethers are cyclic chemical compounds that consist of a ring con- taining several ether groups, which can bound specific cations inside the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' For example, 18-crown-6 is known to have a high affinity for potassium cation whereas 15- crown-5 for sodium cation and 12-crown-4 for lithium cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Using crown ether together with a fluorescent moiety, highly sensitive metal ion sensor can then be built [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' However, the fluorescent dye molecules used by most of the existing metal ion sensors can suffer from photobleaching and fast efflux from cells after loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Nanoparticles, on the other hand, might provide stable fluorescent signal and last long in cells [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In par- ticular, nanodiamonds (NDs) are bio-compatible and can have stable fluorescence due to inner spin defects, the most-studied one of which is the nitrogen-vacancy (NV) center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The NV center is a point defect in diamond lat- tice consisting of a substitutional nitrogen atom with an adjacent vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The negatively-charged NV− defect has a triplet ground state energy with a zero-field split- ting of (2π)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='87 GHz and can show quantum effects even at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In recent years, the NV system has been widely used in various quantum applications such as quantum sensors [13–18] and quantum informa- 2 10-1 100 101 surface Na+ density [nm-2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 [NV-] 20 15 10 5 0 depth (nm) 4 2 0 2 4 6 Energy (eV) VB CB NV-/0 Fermi level NV0 NV- a b c d e 12-crown-4 15-crown-5 18-crown-6 + + + + + + + + + + + + + + + Air Diamond 20 15 10 5 0 depth (nm) 5 0 5 Energy (eV) VB CB NV-/0 Fermi level + + + + + Air Diamond NV- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Schematics of the ion sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' NV center-containing NDs are firstly coated with crown ether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Upon their introduction, sodium cations will be trapped in the 15-crown-5 compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Energy band schematic of diamond-air interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Positive charges on the diamond surface lead to upward bending of the conduction band (CB) and valence band (VB) as well as the NV−/0 transition level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The dashed line indicates the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' NV0 is the dominant NV charge state when the NV−/0 transition level is above the Fermi level, while NV− dominates in the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The surface density of Na+ is taken to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='1 nm−2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Energy band structure when the surface density of Na+ is 1 nm−2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Simulated fraction of negatively charged NV states versus surface Na+ density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Generalization into other type of crown ethers, which can selectively bind certain cations and form complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' tion processors [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In particular, the NV defect is known to be highly sensitive to certain external sig- nals such as magnetic fields and surrounding charge en- vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Depending on the local charge distributions and external electrical manipulations, the NV state could have dynamical transitions between the widely-studied negative NV− state and a neutral state NV0 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The charge state of NV centers can then serve as a meter to yield the information of its environment [18, 22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In this work, we design and demonstrate an alkali ion sensor based on NV centers in crown-ether-functionalized nanodiamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As a proof-of-principle example, we use 15-crown-5 to detect sodium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The formation of 15- crown-5-Na+ complexes on nanodiamond surface will suf- ficiently modify the charge environment of the NV cen- ters inside, and thus alter the charge states of NV defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We thus measure the photoluminescence (PL) emission spectrum of NV defects in nanodiamonds under continu- ous green laser illumination, from which we can extract the fraction of each NV charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We find that the NV’s charge state is influenced both by the surface charge profile and by the power of illumination laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To demon- strate the sensor’s capability in detecting the ions pres- ence we exploit the emergence of a distinctive laser power dependence of the NV− fraction’s upon introduction of sodium cations in the sample, that is clearly different from the pure ND dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We further demonstrate the specificity of the NV-based sensor by several control experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While we uses 15-crown-5 in our proof-of- principle experiments, we remark that a different type of crown ether structure can be adapted to probe other alkali ions, including lithium, potassium and cesium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Principles of the sensor We start by introducing the basic detection mechanism of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(a), a fluorescent ND with initial carboxyl group terminations is firstly coated with crown ether via covalent bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' This yields a neutral surface layer in contrast to the negative layer due to the carboxyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Due to the chelate effect and macro- cyclic effect, depending on its cavity size, crown ether exhibits strong affinities for specific alkali ions [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Upon introducing these ions, stable complexes will form, which in turn lead to positive charges on the ND surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Na+Na15-crown-5 NVcentersK+Na+Li+3 550 600 650 700 750 800 wavelength (nm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=') [NV-] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='754 Experiment NV0 NV- total fit 102 103 Laser power [ W] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 [NV-] Spot1 Spot2 Spot3 Spot4 a b c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Typical PL spectrum for unfunctionalized NDs (carboxyl-terminated) acquired under 532 nm laser illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We perform linear fitting of the spectrum to extract the fraction of NV− charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The peak-normalized NV−(0) spectrum is from reference [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The Raman peak at around 547 nm is attributed to the silicon wafer over which we deposit our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Oscillations of PL intensity after 700 nm are due to the etaloning effect on the CCD camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Distribution of NV− fraction for carboxyl-terminated NDs in the absence and presence of sodium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The laser power is fixed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Sodium cations are added via mixing NaCl solutions with samples of interest hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Typical laser power dependence of NV− fraction of carboxyl-terminated NDs for four different spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The errorbars are the fitting errors (5 percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Inset shows the recombination and ionization processes of the two NV charge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' For example, the 15-crown-5 structure we studied in this work can strongly bond Na+ and form complexes, while its interaction and affinity with K+ are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While we focus on 15-crown-5 and sodium cation in this work, the scheme is general and can be extended to probe other alkali or alkaline earth metal ions using different types of crown ethers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(e) we show the case of detecting Li+ and K+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The accumulated charges on the ND surface can ef- fectively result in bending of the valence band and con- duction band in the diamond lattice [18, 25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Sim- ilar as hydrogen-terminated diamond [27, 28], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(b-c), a positive charge layer on the surface can lead to upward bending of both bands and the bending of the NV−/0 level, which indicates the energy at which the NV defect loses or takes up one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In other words, this transition level indeed corresponds to the transition from the neutrally to the negatively charged NV states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' When its relative position with respect to Fermi level is lower (higher), the defect tends to absorb (lose) one elec- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Due to the positive charge layer on the surface, when close to the surface, this transition level is shifted above the Fermi level (dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(b-c)) and then the NV defect is ionized from NV− to NV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' This effect is particularly important for shallow NVs or NVs in ND, while at larger depths the band bending effect is weak and the Fermi level is above the NV−/0 level and the dominant charge state would be NV−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We note that the NV0 can further lose an electron and becomes a non- fluorescent state NV+ [21], which is inaccessible here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To quantitatively characterize the above model, we perform numerical simulations to study the relative ratio of different NV states (see Methods for detailed infor- mation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The NV−/0 level corresponds to the ground state of NV−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' It’s predicted to be in the band gap and it’s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 eV above the valence band maximum [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(c) we show the simulated band bending effect due to sodium cations on the surface of ND with a diame- ter of 40 nm when the surface ion density is 1 nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' At a depth of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5 nm, we observe that the dominant species of NV defects switches from neutrally to nega- tively charged states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We note that close to the surface, the valence band is shifted above the Fermi level, indi- cating p-type surface conductivity due to the formation of a two-dimensional hole gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' With Fermi-Dirac distri- bution, one can then extract the fraction of NV− as a function of surface cation density (here we use sodium ion as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As expected, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 1(d) we ob- serve that a larger density of cations on the ND surface will serve as electron acceptor, thus leading to a smaller NV− fraction in the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Experiments To demonstrate the proposed mechanism for ion sens- ing, we coat NDs with 15-crown-5 via EDC coupling (see Methods for details) and use the sensor to detect sodium cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In order to measure the change of NV− fraction induced upon the surface charge, we measure the spectrum of nanodiamond ensembles using a Raman spectroscopy (see Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' NV− and NV0 have different PL spectra [26, 29], in particular, the zero- phonon-line (ZPL) of NV− is at 637 nm while it is 575 nm for NV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To approximately determine the NV− frac- tion, we fit the acquired spectrum of our samples I(λ) to a linear combination of spectrum of NV− and NV0: I(λ) = a{[NV −]I− + (1 − [NV −])I0}, (1) where I−(0) denotes the peak-normalized spectrum for a NV−(0) and a is a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2(a) CB 二 NV- NVO NV NV01 VB7 6 ND 5 ND + Na+ (34 mM) occurrence ND + Na* (68 mM) 4 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 [NV]4 102 103 104 105 Laser power [ W] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 [NV-] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Max NV- fraction 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Abs change ND NDCE5 NDCE5+Na+ NDCE5+K+ ND+Na+ ND NDCE5 NDCE5+Na+ 5+Na 5+Na NDCE5+K+ ND+Na+ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Max NV Max NV- fraction fraction 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Abs change Abs change 0 101 102 103 104 105 Laser power [ W] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 [NV-] ND NDCE5 NDCE5+Na+ NDCE5+K+ 550 600 650 700 750 800 wavelength (nm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='8 1 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=') 25 uW 50 250 500 2500 5000 a c b d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' PL spectrum of NDCE5 under different illumination powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Typical laser power dependence of NV− fraction for various samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Measurements are performed in order of increasing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Memory curve of NDCE5 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The solid curve were first measured in power ascending order and then we went back to low powers (dashed line, in power descending order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Comparison of different samples using the coordinates of maximal reachable NV− fraction and absolute change of NV− fraction when one varies illumination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The lighter data points show the results for individual spots, while the opaque data is the average results of different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The dashed line is when the change of NV− fraction equals the maximal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=', when starting from [NV−]=0 at low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' we give an example of a typical spectrum for carboxyl- terminated nanodiamonds measured under continuous 532 nm laser illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The fitting here yields that the fraction of NV− is [NV−] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='754, which is compa- rable to the value observed in bulk diamond [22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As a control test, we first demonstrate that the charge state of NV centers in the carboxyl-terminated nanodi- amonds (labelled as ND in the plots) is not sensitive to sodium cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' First, the ion itself shows no fluorescence under 532 nm laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Then, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2(b), the dis- tribution of [NV−] shows little dependence on the concen- tration of sodium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The laser power here is fixed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We further present typical illumination power dependence curves of [NV−] for these unfunctionalized NDs in the absence of external ions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The green excitation dynamically modulates the NV charge state between neutral and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Due to NV’s suc- cessive absorption of two photons with wavelength less than 637 nm (corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='946 eV), an extra elec- tron of NV− can be ejected into the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The first photon pumps the NV− from ground state to excited state while the second photon takes the electron to the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In the opposite case, a neutrally charged NV defect can take up one electron from the dia- mond valence band band via absorption of a photon with energy larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='156 eV (wavelength 575 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Under 532 nm illumination here, both the electron-NV recom- bination and the NV− ionization processes can happen, yielding an equilibrium NV− fraction around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Again, this is similar as NV centers in bulk diamond [22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To further study the power dependence of the recombination or ionization process separately, another laser with longer wavelength (for example, 632 nm) is required to induce one-directional charge conversion process while suppress- ing the reverse one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We then switch to the crown-ether-functionalized nan- odiamond sample and measure its fluorescence spectrum under the same 532 nm laser illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The mea- surement is performed from low laser power to high power chronologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3(a) clearly shows that now the emission spectrum for 15-crown-5 coated nanodia- mond (labelled as NDCE5 hereafter) is power-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' When one increases the laser power, the spectrum shifts rightward, corresponding to an increasing NV− fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 5 We note that overall the fraction of negative charge state for NDCE5 is smaller than unfunctionalized ND even when the laser power is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' This can be attributed to the fact that NDCE5 is supposed to have neutral charge on the surface, while the carboxyl-terminated NDs tend to have negatively charged surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' For a comparison, we then show the the power- dependence of [NV−] for different samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While for unfunctionalized NDs we observe no depen- dence on illumination power, for all crown-ether-coated samples studied in this work, we see that when the power is low the NV− fraction is approximately zero and it then keeps increasing as a function of illumination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' A similar phenomenon has been observed in near-surface shallow NV centers [22] and this is in contrast to NV centers in certain bulk samples [29], where the NV−-to- NV0 ratio decreases as a function of the laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The power dependence observed here emerges from the interplay between the ionization and recombination rates and the charge transfer rate from NV− to neighboring traps [22, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' These trap states are likely originated from neighboring defects such as vacancy complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While the NDCE5 curve typically shows a saturated fraction of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5, after adding sodium cations, the NV− fraction is considerably lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5 (usually be- low 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='4) even at high laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We attribute this to the positive charge on ND surface yielding the band bend- ing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We further show the data for NDCE5 in the presence of potassium ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In contrast to sodium ions, 15-crown-5 has low affinity for potassium cation and the sample shows a high [NV−] at large laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We note that we observe a memory effect for the NDCE5 sample studied above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' That is, after finishing the spectrum measurement at high laser power, we set the power to low values and re-evaluate the properties of spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3(c) shows that the NV− fraction re- mains at a high value (above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5) even when the laser power is set as low as 50 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The memory effect per- sists for at least hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While a detailed study on the charge dynamics and electron transfer process is needed, we suspect that high laser power preferentially transfers an excess electron from local traps to the NV defect [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3(d) we summarize the results for var- ious samples we evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Here we plot the absolute change of [NV−] as a function of the maximal [NV−] the sample can achieve at high illumination power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We note that in the presence of sodium ions, the maximal frac- tion is considerably lower than other samples, including NDCE5 and unfunctionalized NDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Again, this is due to the fact that 15-crown-5 can form complexes with sodium cations, leading to a positive charge layer on the ND sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' This in turn results in the band bending effect and thus in a lower NV− fraction even when the laser illu- mination power is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Surprisingly, we find that in the presence of K+, at high power most of the NV defects can be converted to NV−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Further study is needed to inves- tigate the interaction between 15-crown-5 and potassium cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' DISCUSSIONS While our results demonstrate that the NV charge ra- tio can be used to detect the presence of targeted cations, the large inhomogeneities in the ND properties prevented us from obtaining quantitative results on the [NV−] den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' As our experimental setup lacked the ability to re- peatedly addressing the same ND, we had to perform measurements over many spatial spots to average out the inhomogeneities among NDs and extract significant differences as a function of sample properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The inho- mogeneities in fluorescence intensity, initial charge ratio and ionization(deionization) rates are induced by vari- ous factors, including inefficient surface coating of crown ether, spatial density profile of NV centers and nitrogen atoms, as well as distinct local charge environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' A well-calibrated, single ND sensor would avoid this costly repeated measurement process to extract the behavior distribution, and greatly improve the sensitivity of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' NV centers with pre-characterized local charge environments and surface charge densities might be used to reduce the overhead in measuring many spatial spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While we used an all-optical approach to read out the charge state information of NV defects via their PL spectrum measurement, we remark that one might ex- tract the same information from the signal contrast of optical-detected magnetic resonance (ODMR), given the fact that NV0 and NV− have distinct ground state spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' A larger NV− fraction will yield a better signal contrast at the resonance frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='87 GHz in the absence of external magnetic fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In this case, a microwave pulse would be required for the ODMR exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' With the help of microwave, the surface charge layer might also be probed by monitoring the transverse relaxation time of the NV− charge state, as it can induce fluctuating electrical field that can interact with the NV centers shortening its relaxation time [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In conclusion, we designed a sodium ion sensor based on NV centers in crown-ether-functionalized nanodia- monds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The charge state of NV centers shows a strong dependence on the surface charge profile and can be de- tected by measuring the PL spectrum of NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' While we focused on sodium cations here, the sensing mechanism can be extended to detect other ions such as K+ by changing the surface crown ether structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' These NV-based sensors have a stable PL signal and can provide excellent spatial resolutions, thus opening new opportunities for monitoring ion concentrations in bio- logical systems and for cellular physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Army Research Office through Grant W911NF-15-1-0548 and by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' acknowledges the support from the National In- stitute of General Medical Sciences with award Number 6 T32GM007753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The content is solely the responsibility of the authors and does not necessarily represent the of- ficial views of the National Institute of General Medical Sciences or the National Institutes of Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' COMPETING FINANCIAL INTEREST The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' proposed the sensor scheme and designed the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' conducted the sensor synthesis and character- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' performed the PL spectrum measurement and data analysis, with partial input from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The nu- merical simulations were performed by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' using the nextnano software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' wrote the paper with the contributions from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' All authors discussed the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Appendix A: Experiment details 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Nanodiamond sample The nanodiamonds (Adamas Nanotechnologies, NDNV40nmHi10ml) in this work had an average size of 35-40 nm and were milled from high pressure high tem- perature (HPHT) micro-size particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Before milling, these particles were irradiated with 2-3 MeV electrons followed by annealing at 850◦C for 2 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Substitutional nitrogen content in the starting micro-size material was around 100 ppm, while the concentration of NV defect is found to be 1-2 ppm, corresponding to around 12-14 color centers per 40 nm nanoparticle [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Chemical synthesis and characterization Commercial reagents were purchased from Sigma- Aldrich and TCI and used as received unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Following reported procedures [35, 36], nanodi- amonds were first treated with a mixture of acids to remove the surface graphite contaminants and metal- lic impurities, generating carboxyl groups for the sub- sequent functionalization by EDC coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In par- ticular, nanodiamonds were dispersed in a 3:1 mix- ture of concentrated sulfuric acid and nitric acid and stirred overnight at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' After neutral- ization with aqueous NaOH (1 M), the resulting nan- odiamonds were cleaned using several centrifugation, washing, and redispersion cycles with Milli-Q water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' EDC coupling with amino crown ether was performed by adding an excess of 1-(3-dimethylaminopropyl)-3- ethylcarbodiimide hydrochloride and 2-aminomethyl-15- crown-5 to the nanodiamonds dispersion and the reac- tion mixture was stirred overnight at room temperature, followed by several centrifugation, washing, and redis- persion cycles with Milli-Q water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The resulting nanodi- monds were characterized by a Bruker Alpha II FTIR spectrometer with a Diamond Crystal ATR (attenuated total reflectance) accessory and a K-alpha+ X-ray Photo- electron Spectrometer system (Thermo Scientific) using a Al Kα radiation source (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Spectrum measurement and fitting The photoluminescence (PL) spectrum of the NV cen- ters under 532 nm laser excitation is collected via a confo- cal Raman microscope (Renishaw inVia microscope with a 1024 × 256 pixel CCD camera) at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We put the samples on a silicon wafer to avoid unwanted Raman peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' To approximately extract the fraction of NV− charge state we first peak-normalize the measured curves and then perform linear fitting of the spectrum from 560 nm to approximately 750 nm based on the single NV− and NV0 reference spectra [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Appendix B: Simulations To qualitatively understand the change of NV charge state in the presence of surface charges, numerical sim- ulations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We first assume rapid thermal equilibrium state after the photoexcitation process of the NV centers [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The charge state of NV center is then exclusively governed by the relative position of the NV−/0 charge transition level with respect to the Fermi level, neglecting the complex ground, excited and metastable states of all defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We first perform numerical simulations of the band bending effect using the nextnano software [37], a tool for simulation of elec- tronic semiconductor nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' With realistic param- eters, one can use it to calculate the band structure of multi-dimensional devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In this work, we perform three-dimensional simulations on a nanodiamond of spherical shape with a diameter d = 40 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In the simulation, the Poisson equation is discretized on a grid with grid size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='5 nm using the fi- nite differences method and solved iteratively in a self- consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The Poisson equation is coupled with the single-band effective mass Schr¨odinger equation via the charge density, as described by the wave functions in the diamond [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Negative charge (electron) donors in the lattice mainly include the substitutional nitrogen atoms (ionization energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='7 eV) and here we take their concentration to be 100 ppm (as expected from the ND used in experiments) and assume uniform distributions as a function of depth below the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' We 7 a b c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Characterization of NDCE5 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' FT-IR spectrum of NDCE5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' b-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' XPS C 1s and N 1s spectra of NDCE5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' take the nitrogen-to-NV conversion ratio to be 1%, cor- responding to a total NV defect concentration of 1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The boundary condition is determined by the surface charges (depth x = 0) that are contributed by the 15- crown-5-Na+ complexes or 15-crown-5 structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Con- sidering the strong affinity for sodium cation of 15-crown- 5, we assume that the density of surface crown ethers is identical to the density of surface positive ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' After getting the relative position of the charge tran- sition level E−/0 with respect to the Fermi level EF , we use the Fermi-Dirac distribution to extract the average fraction of NV− charge state at depth x: ⟨n−(x)⟩ = 1 1 + e(E−/0−EF )/kBT where kB is the Boltzmann constant and T = 300 K is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' The total fraction of NV− in the whole nanodiamond lattice with radius r can then be calculated by performing the integral [NV −] = � r 0 ⟨n−(x)⟩g(x)dx where g(x) is the normalized NV defect density profile in the diamond lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=', � r 0 g(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' In our simula- tions, we assume uniform distributions of NV defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Liu, D.' metadata={'source': 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D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Daranciang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Dai, Nano letters 8, 586 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Lamy, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=' Sallin, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=') 406 404 402 400 398 396 394 Binding Energy (eV)C 1s Intensity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=') C-O C=0 292 290 288 286 284 282 Binding Energy (eV)NDCE5 Transmittance (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfYwTJ/content/2301.03143v1.pdf'} +page_content=') N-H Aliphatic C-H Amide C=O C-0 3500 3000 2500 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+nonadiabaticity in highly-compressed I-43d-TaH3 superconductor +Evgueni F. Talantsev1,2,*,** +1M.N. Miheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, +18, S. Kovalevskoy St., Ekaterinburg, 620108, Russia +2NANOTECH Centre, Ural Federal University, 19 Mira St., Ekaterinburg, 620002, +Russia +*corresponding author +**corresponding author’s E-mail: evgeny.f.talantsev@gmail.com + +Abstract +Recently He et al (2022 arXiv2212.13739) reported on the discovery of high-temperature +superconductivity in highly-compressed polyhydride of tantalum. At pressure 𝑃 = 197 𝐺𝑃𝑎, +the polyhydride I-43d-phase of TaH3 exhibits zero-resistance transition temperature 𝑇𝑐,𝑧𝑒𝑟𝑜 = +25.6 𝐾. Measurements of the low-temperature magnetoresistance showed that this +superconductor has the ground state upper critical field (defined by the zero resistance +criterion) 𝐵𝑐2(0) = 11 ± 1 𝑇𝑒𝑠𝑙𝑎. Here, we performed detailed analysis of the reported +experimental data by He et al (2022 arXiv2212.13739) and deduced several parameters of the +I-43d-phase of TaH3: (a) the Debye temperature, 𝑇𝜃 = 263 𝐾, (b) the electron-phonon +coupling constant, 𝜆𝑒−𝑝ℎ = 1.53 ± 0.13; (c) the Fermi temperature 𝑇𝐹 = 1324 ± 74 𝐾; (d) +the strength of nonadiabaticity, +𝑇𝜃 +𝑇𝐹 = 0.19 ± 0.01; (e) and the ratio of +𝑇𝑐 +𝑇𝐹 = 0.0185 ± 0.010 +which implies that I-43d-phase of TaH3 falls in unconventional superconductors band in the +Uemura plot. Deduced parameters indicate that the I-43d-phase of TaH3 (P = 197 GPa) can +be classified as typical unconventional high-temperature superconductor. + + + +2 + +Debye temperature, electron-phonon coupling constant, and moderate +nonadiabaticity in highly-compressed I-43d-TaH3 superconductor +I. Introduction +The discovery of near-room temperature superconductivity in highly compressed sulphur +hydride by Drozdov et al [1] manifested a new era in superconductivity. This research field +represents one of the most fascinating exploration in modern physics where advanced first +principles calculations is essential part of the quest [2-42]. +From 2015 till now, more than two dozens of high-temperature and near-room +temperature superconducting polyhydride phases have been discovered [1-20]. Recently, He +et al [40] reported on the discovery of a new highly-compressed polyhydride of tantalum +which exhibits the zero-resistance superconducting transition temperature 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.6 𝐾 at +pressure of 𝑃 = 197 𝐺𝑃𝑎. The superconducting critical temperature is close to the boiling +point of neon, 𝑇𝐵,𝑛𝑒𝑜𝑛 = 27.1 𝐾, and well above the boiling point of hydrogen, 𝑇𝐵,𝐻2 = +20.3 𝐾, and thus, this polyhydride can be classified as high-temperature superconductor. The +analysis of X-ray diffraction (XRD) pattern showed that the superconducting state is +attributed to the I-43d-phase of TaH3 [43]. +Here, we performed detailed analysis of the temperature dependent magnetoresistance +𝑅(𝑇, 𝐵) data reported by He et al [43] and showed that the I-43d-phase of TaH3 exhibits +conventional resistive transition width broadening vs applied magnetic field, 𝐵. Also, from +𝑅(𝑇, 𝐵) data several fundamental parameters of the I-43d-phase of TaH3 (𝑃 = 197 𝐺𝑃𝑎) +have been extracted: +(1) the Debye temperature, 𝑇𝜃 = 263 𝐾, +(2) the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.53 ± 0.13; +(3) the ground state coherence length, 𝜉(0) = 1.53 ± 0.13; +(4) the Fermi temperature 𝑇𝐹 = 1324 ± 74 𝐾; + +3 + +(5) the strength of nonadiabaticity, +𝑇𝜃 +𝑇𝐹 = 0.20 ± 0.01; +(6) +𝑇𝑐 +𝑇𝐹 = 0.019 ± 0.01, which implies that the I-43d-phase of TaH3 falls in +unconventional superconductors band in the Uemura plot. + +II. Results and Discussion +2.1. Debye temperature +Debye temperature, 𝑇𝜃, can be deduced as a free-fitting parameter of the fit of +temperature dependent resistance, 𝑅(𝑇), to the saturated resistance model within +conventional phenomenology of the Bloch-Grüneisen (BG) equation [44-50]: +𝑅(𝑇) = +1 +1 +𝑅𝑠𝑎𝑡+ +1 +𝑅0+𝐴×( 𝑇 +𝑇𝜃 +) +5 +×∫ +𝑥5 +(𝑒𝑥−1)(1−𝑒−𝑥) +𝑇𝜃 +𝑇 +0 +𝑑𝑥 + + + + + + +(1) +where 𝑅𝑠𝑎𝑡, 𝑅0, 𝑇𝜃 and 𝐴 are free-fitting parameters. In Figure 1 we showed the fit of the +𝑅(𝑇) dataset measured by He et al [43] for the tantalum hydride compressed at 𝑃 = +197 𝐺𝑃𝑎. + + +Figure 1. Temperature dependent resistance data, R(T), for compressed tantalum hydride (I-43d- +phase of TaH3 at P = 197 GPa) and data fit to Eq. 1 (raw data reported by He et al [43]). Green balls +indicate the bounds for which R(T) data was used for the fit to Eq. 1. Deduced 𝑇𝜃 = 263.7 ± 0.3 𝐾, +𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.6 𝐾, 𝜆𝑒−𝑝ℎ = 1.53, fit quality is 0.99998. 95% confidence bands are shown. + +0.35 +I-43d-TaH (P = 197 GPa) +0.30 +2e-ph = 1.53 +0.25 +T。= 263.7 ± 0.3 K +(U) +R. +resistance +0.20 +0.15 +0.10 +raw R(T) +fit to BG model +0.05 +fitted R(T) range +T += 25.6 K +c,zero +0.00 +0 +40 +80 +120 +160 +200 +240 +280 +320 +temperature (K)4 + + +2.2. The electron phonon coupling constant +From the deduced 𝑇𝜃 and measured 𝑇𝑐, which we defined by strict resistance criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚 → 0 (in the given case, the ratio was +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚 = 0.02), where 𝑅𝑛𝑜𝑟𝑚 is the sample +resistance at the onset of the superconducting transition, the electron-phonon coupling +constant, 𝜆𝑒−𝑝ℎ, can be calculated as the root of McMillan equation [51-54]: +𝑇𝑐 = ( +1 +1.45) × 𝑇𝜃 × 𝑒 +−( +1.04(1+𝜆𝑒−𝑝ℎ) +𝜆𝑒−𝑝ℎ−𝜇∗(1+0.62𝜆𝑒−𝑝ℎ)) +× 𝑓1 × 𝑓2 +∗ + + +(2) +where +𝑓1 = (1 + ( +𝜆𝑒−𝑝ℎ +2.46(1+3.8𝜇∗)) +3 2 +⁄ +) +1 3 +⁄ + + + + + + +(3) +𝑓2 +∗ = 1 + (0.0241 − 0.0735 × 𝜇∗) × 𝜆𝑒−𝑝ℎ +2 +. + + + + +(4) +where 𝜇∗ is the Coulomb pseudopotential, which it was assumed to be 𝜇∗ = 0.13 (which is +typical value utilized in the first principles calculation for many electron-phonon mediated +superconductors [38,40-42]). We deduced 𝜆𝑒−𝑝ℎ = 1.53 (Fig. 1), which is in a conventional +range for other highly compressed hydride superconductors [21,54,55]. + +2.3. Ground state coherence length +To deduce the ground state coherence length, 𝜉(0), we fitted the upper critical field +datatset, 𝐵𝑐2(𝑇), to analytical approximant of the Werthamer-Helfand-Hohenberg model +[56,57], which was proposed by Baumgartner et al [58]: +𝐵𝑐2(𝑇) = +1 +0.693 × +𝜙0 +2𝜋𝜉2(0) × ((1 − +𝑇 +𝑇𝑐) − 0.153 × (1 − +𝑇 +𝑇𝑐) +2 +− 0.152 × (1 − +𝑇 +𝑇𝑐) +4 +) (11) +where 𝜙0 = +ℎ +2𝑒 is the superconducting flux quantum, ℎ = 6.626 × 10−34 𝐽 ⋅ 𝑠 is Planck +constant, 𝑒 = 1.602 × 10−19 𝐶, and 𝜉(0) and 𝑇𝑐 ≡ 𝑇𝑐(𝐵 = 0) are free fitting parameters. Eq. +11 is designated as B-WHH model in Fig. 2. + +5 + + +Figure 2. The upper critical field data, Bc2(T), for compressed tantalum hydride (I-43d-phase of TaH3 +at P = 197 GPa) and data fit to Eq. 11. Raw R(T,B) dataset reported by He et al [43]. Definition +criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚 = 0.02 was used. Deduced parameters are: 𝜉(0) = 2.33 ± 0.02 𝑛𝑚, 𝑇𝑐 = 21.8 ± +0.2 𝐾. Fit quality is 0.9943. 95% confidence bands are shown by pink shadow areas. + +𝐵𝑐2(𝑇) dataset was deduced from R(T,B) curves reported by He et al [43] in their Figure +2,a [43], for which we utilized the same criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚 = 0.02, which was used to define +𝑇𝑐 above. Obtained 𝐵𝑐2(𝑇) data and data fit are shown in Figure 2. Deduced 𝜉(0) = 5.45 ± +0.10 𝑛𝑚. + +2.4. The Fermi temperature +To calculate the Fermi temperature, we utilized the equation [59,60]: +𝑇𝐹 = +𝜋2𝑚𝑒 +8∙𝑘𝐵 × (1 + 𝜆𝑒−𝑝ℎ) × 𝜉2(0) × ( +𝛼∙𝑘𝐵∙𝑇𝑐 +ℏ +) +2 +, + + + +(12) +where 𝑚𝑒 = 9.109 × 10−31 𝑘𝑔 is bare electron mass, ℏ = 1.055 × 10−34 𝐽 ⋅ 𝑠 is reduced +Planck constant, 𝑘𝐵 = 1.381 × 10−23 𝑚2 ⋅ 𝑘𝑔 ⋅ 𝑠−2 ⋅ 𝐾−1 is Boltzmann constant, 𝛼 = +2Δ(0) +𝑘𝐵∙𝑇𝑐 is +the gap-to-transition temperature ratio and this is the only unknown parameter in Eq. 12. + +14 +I-43d-TaH3 (P = 197 GPa) +12 +(T) +E(0) = 5.45 ± 0.10 nm +10 +T. = 21.8 ± 0.2 K +8 +6 +4 +2 +raw Bc2(T) data +fit to B-WHH model +0 +0 +4 +8 +12 +16 +20 +24 +28 +temperature (K)6 + +To estimate 𝛼 = +2Δ(0) +𝑘𝐵∙𝑇𝑐 in the compressed tantalum hydride we propose the following +approach. Carbotte [58] collected various parameters for 32 electron-phonon mediated +superconductors, which exhibit 0.43 ≤ 𝜆𝑒−𝑝ℎ ≤ 3.0 and 3.53 ≤ +2Δ(0) +𝑘𝐵∙𝑇𝑐 ≤ 5.19. In Figure 3 we +presented the dataset reported by Carbotte in his Table IV [61]. The dependence +2Δ(0) +𝑘𝐵∙𝑇𝑐 vs 𝜆𝑒−𝑝ℎ +can be approximate by linear function (Fig. 3) [62]: +2Δ(0) +𝑘𝐵𝑇𝑐 = 𝐶 + 𝐷 × 𝜆𝑒−𝑝ℎ + + + + + + + +(13) +where 𝐶 = 3.26 ± 0.06, and 𝐷 = 0.74 ± 0.04. The substitution of deduced 𝜆𝑒−𝑝ℎ = 1.53 +for tantalum hydride (hydride (P = 197 GPa) in Eq. 13 gives +2Δ(0) +𝑘𝐵𝑇𝑐 = 4.39 ± 0.12 (Fig. 3). In +the result, calculated Fermi temperature is 𝑇𝐹 = 1324 ± 74 𝐾. + + + +Figure 3. The gap-to-transition temperature ratio, +2⋅Δ(0) +𝑘𝐵⋅𝑇𝑐 , vs the electron-phonon coupling constant, +𝜆𝑒−𝑝ℎ, dataset reported by Carbotte in the Table IV [61]. Linear fit is shown by dash-line. Parameters +for several superconductors, including compressed tantalum hydride (I-43d-phase of TaH3 at P = 197 +GPa), are indicated by arrows. 95% confidence bands for the linear fit are shown by pink shadow area. + + +6.0 +2△(0) += C+D2 +5.5 +kgT. +Pbo.65Bio.35 +5.0 +c +Pbo.4 Tlo.6 +Pbo.5Bio.5 +4.5 +Nb3Al +Nb3Sn +La +4.0 +I-43d-TaH3 +80 +3.5 +Nb +data (Carbotte (1990)) +linear fit +V +Al +/-43d-TaH3 (P=197 GPa) +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Le-ph7 + +2.4. Location in Uemura plot and the strength of nonadiabaticity +We used the calculated Fermi temperature, 𝑇𝐹 = 1324 ± 74 𝐾, and superconducting +transition temperature, 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.6 𝐾, to locate I-43d-phase of TaH3 (P = 197 GPa) in the +Uemura plot [63,64] (Fig. 4). With its ratio of +𝑇𝑐 +𝑇𝐹 = 0.019 ± 0.001, the I-43d-phase of TaH3 +(P = 197 GPa) is located in the unconventional superconductors band (Figure 4). + +Figure 4. Uemura plot, where the I-43d-phase of TaH3 (𝑃 = 197 𝐺𝑃𝑎) is shown together with +several families of superconductors: metals, iron-based superconductors, diborides, cuprates, Laves +phases, near-room-temperature superconductors and others. References on original data can be found +in Refs. 50,59,65-68. + +Superconductors also can be classified by the ratio of maximum phonon energy, ℏ𝜔𝐷 +(where 𝜔𝐷 is Debye frequency) to the charge carrier energy at the Fermi level, +ℏ𝜔𝐷 +𝑘𝐵𝑇𝐹. In so- +called adiabatic superconductors +ℏ𝜔𝐷 +𝑘𝐵𝑇𝐹 < 10−3, which implies that these materials exhibit very +fast charge carriers and relatively slow phonons. The condition is satisfied in pure metals +[69], superconducting alloys [61], and 2H-TaSeS [50] (Figs. 5,6). + + H,S (from (O)) +O H,S (from 2(0)) +区一metals +A15alloys +(La,Nd)H. +LaH +Heusler alloys +noncentrosymmetric +Laves phase +100 +cuprates. +H.S +intermetallics +SrTiO3 +α-MoB +Ba1-xK,BiO3 +MgB. +iron-based + 0.4 + + + + + + + + +(15) +These materials are: diborides Nb0.75Mo0.25B2 [73], noncentrosymmetric Nb0.5Os0.5 [74], +highly compressed metalized oxygen [75,76], magic-angle twisted bilayer graphene [67,77], +SrTiO3 [67], and highly compressed metalized ionic salt CsI [78,79]. It should be stressed +that all these superconductors exhibit low or super-low transition temperature, 𝑇𝑐 < 8 𝐾. + +H,S (from E(0) +H,S (from 2(0) +100 +区一metals +■SrTiO3 +A15 alloys +Csl +Heusleralloys +noncentrosymmetric +10 +Laves phase +Nbo.75Mo0.25B2 +intermetallics +-02 +一 +SrTiO3 +TaH, +WB, +Ba1-xKxBiO3 +noncentrosymmetric +α-MoB, +(La,Nd)H10 +iron-based SCs +LaH10 (from (0) +aH +人α +LaH10 (from 2(0)) +LiC +T +H,S +■ +0.1 +(La,Nd)H10 (from (O)) +Heuslerz +Laves +Csl (P=209 GPa) +MgB2 +V +A15 +S-O2 (P=115 GPa) +区 +0.01 +LiC6 +metals +Te/T- = 0.025 +Te/T- = 0.4 +0.001 +肉 +α-MoB, (P =110 GPa) +Nbo.75Moo.25B2 +0.25 +0.5 +1 +2 +MgB2 +WB, (P =121 GPa) +electron-phonon coupling strength, 2 +/-43d-TaH3 (P=197 GPa)9 + + +Figure 6. Plot +𝑇𝜃 +𝑇𝐹 vs 𝑇𝑐 for several families of superconductors and where the I-43d-phase of TaH3 +(𝑃 = 197 𝐺𝑃𝑎) is also shown. References on original data can be found in Refs. 50,59,65-68. + +In all three plots (Figs. 4-6), the I-43d-phase of TaH3 (P = 197 GPa) is located in close +proximity to -MoB2 [40], iron-based superconductors [67], and Ba1-xKxBiO3 [80-82]. This +manifests that the I-43d-phase of TaH3 (P = 197 GPa) can be classified as typical high- +temperature superconductor. + +III. Conclusions +In this work, we analyzed experimental data reported by He et al [43] for the I-43d-phase +of TaH3 (P = 197 GPa) and deduced several parameters of this superconductor: +(1) the Debye temperature, 𝑇𝜃 = 263 𝐾, +(2) the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.53 ± 0.13; +(3) the ground state coherence length, 𝜉(0) = 1.53 ± 0.13; + + H,S (from E(O)) +O H,S (from 2(0) +区一metals +A15alloys +100 +吕 MATBG +Heusler alloys +■ SrTiO3 +noncentrosymmetric +Laves phase +Csl +intermetallics +SrTiO3 +10 +-02 +Nbo.75Moo.25B2 +< iron-based SCs +★ +cuprates (2(0) and J.(sf, D) +WB2 +(La,Nd)H, +LaH1o (from E(0) +1 +LaH10 (from 2(0) +-10 +TaH, +α-MoB. +H,S +noncentrosymmetric +口 +(La,Nd)H1o (from E(0)) +intermetallicso +Csl (P=209 GPa) +MATBG (from J.(sf, ) +0.1 +Heusler +MgB, +口 +MATBG (from E(O) +aves +2H-TaSeS +cuprates +-O, (P=115 GPa) +2H- +A15 +0.01 +T/T- = 0.025 +metals +TaSeS +- Te/Te = 0.4 +Ba1-xK,BiO3 +LiC6 +0.001 +α-MoB, (P=110 GPa) +Nbo.75Moo.25B2 +0.1 +1 +10 +100 +MgB2 +WB, (P=121 GPa) +transition temperature, T. (K) +I-43d-TaH3 (P=197 GPa)10 + +(4) the Fermi temperature 𝑇𝐹 = 1324 ± 74 𝐾; +Deduced parameters indicate that the I-43d-phase of TaH3 (P = 197 GPa) can be +classified as typical unconventional high-temperature superconductor. + +Acknowledgement +The author thanks financial support provided by the Ministry of Science and Higher +Education of Russia (theme “Pressure” No. АААА-А18-118020190104-3). The research +funding from the Ministry of Science and Higher Education of the Russian Federation (Ural +Federal University Program of Development within the Priority-2030 Program) is gratefully +acknowledged. + +Data availability statement +The data that support the findings of this study are available from the corresponding author +upon reasonable request. + +Declaration of interests +The author declares that he has no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. + +References +[1] A.P. Drozdov, M. I. 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Fermi-liquid nonadiabatic highly compressed cesium iodide +superconductor. Condens. Matter 7, 65 (2022). +[80] R. J. Cava, B. Batlogg, J. J. Krajewski, R. Farrow, L. W. Rupp, A. E. White, K. Short, +W. F. Peck, T. Kometani. Superconductivity near 30 K without copper: The Ba0.6K0.4BiO3 +perovskite. Nature 332, 814-816. (1988). +[81] Y. J. Uemura, Universal correlations between Tc and nS/meff (carrier density over +effective mass) in high-Tc cuprate. Phys. Rev. Lett. 62, 2317-2320 (1989). +[82] D. Szczesniak, A. Z. Kaczmarek, E. A. Drzazga-Szczesniak, R. Szczesniak. Phonon- +mediated superconductivity in bismuthates by nonadiabatic pairing. Phys. Rev. B 104, +094501 (2021). + + diff --git a/O9E4T4oBgHgl3EQfkA3U/content/tmp_files/load_file.txt b/O9E4T4oBgHgl3EQfkA3U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d7e523917922fed6b0cac1744e0ebef1c2de0b9 --- /dev/null +++ b/O9E4T4oBgHgl3EQfkA3U/content/tmp_files/load_file.txt @@ -0,0 +1,918 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf,len=917 +page_content='1 Debye temperature, electron-phonon coupling constant, and moderate nonadiabaticity in highly-compressed I-43d-TaH3 superconductor Evgueni F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Talantsev1,2,*,** 1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Miheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, 18, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Kovalevskoy St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=', Ekaterinburg, 620108, Russia 2NANOTECH Centre, Ural Federal University, 19 Mira St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=', Ekaterinburg, 620002, Russia *corresponding author **corresponding author’s E-mail: evgeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='talantsev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='com Abstract Recently He et al (2022 arXiv2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13739) reported on the discovery of high-temperature superconductivity in highly-compressed polyhydride of tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' At pressure 𝑃 = 197 𝐺𝑃𝑎, the polyhydride I-43d-phase of TaH3 exhibits zero-resistance transition temperature 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Measurements of the low-temperature magnetoresistance showed that this superconductor has the ground state upper critical field (defined by the zero resistance criterion) 𝐵𝑐2(0) = 11 ± 1 𝑇𝑒𝑠𝑙𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Here, we performed detailed analysis of the reported experimental data by He et al (2022 arXiv2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13739) and deduced several parameters of the I-43d-phase of TaH3: (a) the Debye temperature, 𝑇𝜃 = 263 𝐾, (b) the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (c) the Fermi temperature 𝑇𝐹 = 1324 ± 74 𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (d) the strength of nonadiabaticity, 𝑇𝜃 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (e) and the ratio of 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0185 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='010 which implies that I-43d-phase of TaH3 falls in unconventional superconductors band in the Uemura plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Deduced parameters indicate that the I-43d-phase of TaH3 (P = 197 GPa) can be classified as typical unconventional high-temperature superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 2 Debye temperature, electron-phonon coupling constant, and moderate nonadiabaticity in highly-compressed I-43d-TaH3 superconductor I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Introduction The discovery of near-room temperature superconductivity in highly compressed sulphur hydride by Drozdov et al [1] manifested a new era in superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' This research field represents one of the most fascinating exploration in modern physics where advanced first principles calculations is essential part of the quest [2-42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' From 2015 till now, more than two dozens of high-temperature and near-room temperature superconducting polyhydride phases have been discovered [1-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Recently, He et al [40] reported on the discovery of a new highly-compressed polyhydride of tantalum which exhibits the zero-resistance superconducting transition temperature 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 𝐾 at pressure of 𝑃 = 197 𝐺𝑃𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The superconducting critical temperature is close to the boiling point of neon, 𝑇𝐵,𝑛𝑒𝑜𝑛 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='1 𝐾, and well above the boiling point of hydrogen, 𝑇𝐵,𝐻2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='3 𝐾, and thus, this polyhydride can be classified as high-temperature superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The analysis of X-ray diffraction (XRD) pattern showed that the superconducting state is attributed to the I-43d-phase of TaH3 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Here, we performed detailed analysis of the temperature dependent magnetoresistance 𝑅(𝑇, 𝐵) data reported by He et al [43] and showed that the I-43d-phase of TaH3 exhibits conventional resistive transition width broadening vs applied magnetic field, 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Also, from 𝑅(𝑇, 𝐵) data several fundamental parameters of the I-43d-phase of TaH3 (𝑃 = 197 𝐺𝑃𝑎) have been extracted: (1) the Debye temperature, 𝑇𝜃 = 263 𝐾, (2) the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (3) the ground state coherence length, 𝜉(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (4) the Fermi temperature 𝑇𝐹 = 1324 ± 74 𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 3 (5) the strength of nonadiabaticity, 𝑇𝜃 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (6) 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='01, which implies that the I-43d-phase of TaH3 falls in unconventional superconductors band in the Uemura plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Results and Discussion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Debye temperature Debye temperature, 𝑇𝜃, can be deduced as a free-fitting parameter of the fit of temperature dependent resistance, 𝑅(𝑇), to the saturated resistance model within conventional phenomenology of the Bloch-Grüneisen (BG) equation [44-50]: 𝑅(𝑇) = 1 1 𝑅𝑠𝑎𝑡+ 1 𝑅0+𝐴×( 𝑇 𝑇𝜃 ) 5 ×∫ 𝑥5 (𝑒𝑥−1)(1−𝑒−𝑥) 𝑇𝜃 𝑇 0 𝑑𝑥 (1) where 𝑅𝑠𝑎𝑡, 𝑅0, 𝑇𝜃 and 𝐴 are free-fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' In Figure 1 we showed the fit of the 𝑅(𝑇) dataset measured by He et al [43] for the tantalum hydride compressed at 𝑃 = 197 𝐺𝑃𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Temperature dependent resistance data, R(T), for compressed tantalum hydride (I-43d- phase of TaH3 at P = 197 GPa) and data fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 1 (raw data reported by He et al [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Green balls indicate the bounds for which R(T) data was used for the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Deduced 𝑇𝜃 = 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='3 𝐾, 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 𝐾, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53, fit quality is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='99998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 95% confidence bands are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='35 I-43d-TaH (P = 197 GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='30 2e-ph = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='25 T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='= 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='3 K (U) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' resistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='10 raw R(T) fit to BG model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='05 fitted R(T) range T = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 K c,zero 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='00 0 40 80 120 160 200 240 280 320 temperature (K)4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The electron phonon coupling constant From the deduced 𝑇𝜃 and measured 𝑇𝑐, which we defined by strict resistance criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚 → 0 (in the given case, the ratio was 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='02), where 𝑅𝑛𝑜𝑟𝑚 is the sample resistance at the onset of the superconducting transition, the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ, can be calculated as the root of McMillan equation [51-54]: 𝑇𝑐 = ( 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='45) × 𝑇𝜃 × 𝑒 −( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='04(1+𝜆𝑒−𝑝ℎ) 𝜆𝑒−𝑝ℎ−𝜇∗(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='62𝜆𝑒−𝑝ℎ)) × 𝑓1 × 𝑓2 ∗ (2) where 𝑓1 = (1 + ( 𝜆𝑒−𝑝ℎ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='46(1+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='8𝜇∗)) 3 2 ⁄ ) 1 3 ⁄ (3) 𝑓2 ∗ = 1 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0241 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0735 × 𝜇∗) × 𝜆𝑒−𝑝ℎ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' (4) where 𝜇∗ is the Coulomb pseudopotential, which it was assumed to be 𝜇∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='13 (which is typical value utilized in the first principles calculation for many electron-phonon mediated superconductors [38,40-42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' We deduced 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 1), which is in a conventional range for other highly compressed hydride superconductors [21,54,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Ground state coherence length To deduce the ground state coherence length, 𝜉(0), we fitted the upper critical field datatset, 𝐵𝑐2(𝑇), to analytical approximant of the Werthamer-Helfand-Hohenberg model [56,57], which was proposed by Baumgartner et al [58]: 𝐵𝑐2(𝑇) = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='693 × 𝜙0 2𝜋𝜉2(0) × ((1 − 𝑇 𝑇𝑐) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='153 × (1 − 𝑇 𝑇𝑐) 2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='152 × (1 − 𝑇 𝑇𝑐) 4 ) (11) where 𝜙0 = ℎ 2𝑒 is the superconducting flux quantum, ℎ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='626 × 10−34 𝐽 ⋅ 𝑠 is Planck constant, 𝑒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='602 × 10−19 𝐶, and 𝜉(0) and 𝑇𝑐 ≡ 𝑇𝑐(𝐵 = 0) are free fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 11 is designated as B-WHH model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The upper critical field data, Bc2(T), for compressed tantalum hydride (I-43d-phase of TaH3 at P = 197 GPa) and data fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Raw R(T,B) dataset reported by He et al [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Definition criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='02 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Deduced parameters are: 𝜉(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='02 𝑛𝑚, 𝑇𝑐 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='2 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Fit quality is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='9943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 95% confidence bands are shown by pink shadow areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 𝐵𝑐2(𝑇) dataset was deduced from R(T,B) curves reported by He et al [43] in their Figure 2,a [43], for which we utilized the same criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='02, which was used to define 𝑇𝑐 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Obtained 𝐵𝑐2(𝑇) data and data fit are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Deduced 𝜉(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='10 𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The Fermi temperature To calculate the Fermi temperature, we utilized the equation [59,60]: 𝑇𝐹 = 𝜋2𝑚𝑒 8∙𝑘𝐵 × (1 + 𝜆𝑒−𝑝ℎ) × 𝜉2(0) × ( 𝛼∙𝑘𝐵∙𝑇𝑐 ℏ ) 2 , (12) where 𝑚𝑒 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='109 × 10−31 𝑘𝑔 is bare electron mass, ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='055 × 10−34 𝐽 ⋅ 𝑠 is reduced Planck constant, 𝑘𝐵 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='381 × 10−23 𝑚2 ⋅ 𝑘𝑔 ⋅ 𝑠−2 ⋅ 𝐾−1 is Boltzmann constant, 𝛼 = 2Δ(0) 𝑘𝐵∙𝑇𝑐 is the gap-to-transition temperature ratio and this is the only unknown parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 14 I-43d-TaH3 (P = 197 GPa) 12 (T) E(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='10 nm 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='2 K 8 6 4 2 raw Bc2(T) data fit to B-WHH model 0 0 4 8 12 16 20 24 28 temperature (K)6 To estimate 𝛼 = 2Δ(0) 𝑘𝐵∙𝑇𝑐 in the compressed tantalum hydride we propose the following approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Carbotte [58] collected various parameters for 32 electron-phonon mediated superconductors, which exhibit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='43 ≤ 𝜆𝑒−𝑝ℎ ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 ≤ 2Δ(0) 𝑘𝐵∙𝑇𝑐 ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' In Figure 3 we presented the dataset reported by Carbotte in his Table IV [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The dependence 2Δ(0) 𝑘𝐵∙𝑇𝑐 vs 𝜆𝑒−𝑝ℎ can be approximate by linear function (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 3) [62]: 2Δ(0) 𝑘𝐵𝑇𝑐 = 𝐶 + 𝐷 × 𝜆𝑒−𝑝ℎ (13) where 𝐶 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='06, and 𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The substitution of deduced 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='53 for tantalum hydride (hydride (P = 197 GPa) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 13 gives 2Δ(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='12 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' In the result, calculated Fermi temperature is 𝑇𝐹 = 1324 ± 74 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The gap-to-transition temperature ratio, 2⋅Δ(0) 𝑘𝐵⋅𝑇𝑐 , vs the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ, dataset reported by Carbotte in the Table IV [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Linear fit is shown by dash-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Parameters for several superconductors, including compressed tantalum hydride (I-43d-phase of TaH3 at P = 197 GPa), are indicated by arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 95% confidence bands for the linear fit are shown by pink shadow area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 2△(0) = C+D2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 kgT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Pbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='65Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 c Pbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='4 Tlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 Pbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 Nb3Al Nb3Sn La 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 I 43d TaH3 80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 Nb data (Carbotte (1990)) linear fit V Al / 43d TaH3 (P=197 GPa) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='0 Le ph7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Location in Uemura plot and the strength of nonadiabaticity We used the calculated Fermi temperature, 𝑇𝐹 = 1324 ± 74 𝐾, and superconducting transition temperature, 𝑇𝑐,𝑧𝑒𝑟𝑜 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='6 𝐾, to locate I-43d-phase of TaH3 (P = 197 GPa) in the Uemura plot [63,64] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' With its ratio of 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='001, the I-43d-phase of TaH3 (P = 197 GPa) is located in the unconventional superconductors band (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Uemura plot, where the I-43d-phase of TaH3 (𝑃 = 197 𝐺𝑃𝑎) is shown together with several families of superconductors: metals, iron-based superconductors, diborides, cuprates, Laves phases, near-room-temperature superconductors and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' References on original data can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 50,59,65-68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' Superconductors also can be classified by the ratio of maximum phonon energy, ℏ𝜔𝐷 (where 𝜔𝐷 is Debye frequency) to the charge carrier energy at the Fermi level, ℏ𝜔𝐷 𝑘𝐵𝑇𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' In so- called adiabatic superconductors ℏ𝜔𝐷 𝑘𝐵𝑇𝐹 < 10−3, which implies that these materials exhibit very fast charge carriers and relatively slow phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' The condition is satisfied in pure metals [69], superconducting alloys [61], and 2H-TaSeS [50] (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' 5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' H,S (from (O)) O H,S (from 2(0)) 区一metals A15alloys (La,Nd)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' LaH Heusler alloys noncentrosymmetric Laves phase 100 cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content='S intermetallics SrTiO3 α-MoB Ba1-xK,BiO3 MgB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQfkA3U/content/2301.05148v1.pdf'} +page_content=' iron-based +-10 +-20 +0.000 +-20 +-10 +0 +10 +20 +x (nm)0.012 +20 +10 +0.008 +(wu) +0 +-10 +0.004 +-20 +0.000 +-20 +-10 +0 +10 +20 +x (nm)20 +0.012 +10 +0.008 +(wu) +0 +-10 +0.004 +-20 +0.000 +-20 +-10 +0 +10 +20 +x (nm)6 +4 +2 +0 +6 +4 +2 +0 +40 +20 +K +K +K +K5 +对给定nu, 能带分成两组，一个的为（+，up）， +其他三个为另一组 +Δintra +Δ′intra +Δinter +(+↑) +(+↓) +(-↑) +(- ↓) +ν=-3 +ν=-1 +ν=1 +ν=3 +FIG. 5. +(Color online) The full HF band structures of +TBG/BN at θ′ = 0.54◦ for each flavor at different ν. The +magnitude of the projection of each HF band state on the +single-particle flat bands is represented by the size of the red +circle. The band gap ∆intra between the intra-flavor flat-like +bands, the ∆inter between the inter-flavor flat-like bands, and +the ∆′ +intra between the remote and flat-like bands are labeled +by the green shades. The inter-flavor valley-wave excitation +mainly between the flat-like bands, the intra-flavor exciton +excitation mainly between the flat-like bands, and the intra- +flavor exciton excitation mainly between the flat-like bands +and the remote bands are indicated by the dashed green lines. +ν from -3 to 3, with the gap still reaching about 16 meV +at ν = 3. +In comparison to the HF band gaps, the ˜∆inter of +the valley wave modes are much smaller than the ∆inter +for both TBG and TBG/BN, while the ˜∆intra of the +exciton modes of TBG/BN reaches about half of the +∆intra, which is a significant contrast to the much smaller +˜∆intra/∆intra for TBG, as shown in Fig 2(b). For the +exciton modes, the two-body exciton wavefunction as a +function of the electron (re) and hole (rh) positions can +be calculated as +Ψ(re, rh) = +1 +Nk +� +k +ukψpk(re)ψ∗ +hk(rh), +(2) +where ψpk and ψhk are the HF conduction and valence +band states corresponding to the exciton excitation. Fig- +ure 3 exhibits the wavefunction of the lowest exciton +mode at q = 0 with rh at the origin of a supercell where +the bilayer graphene is locally AA-stacked. The parti- +cle and hole are strongly bound at all the odd ν for the +pristine TBG with the particle localized around the ori- +gin. +The spatial map of the exciton wavefunctions in +TBG/BN spread a much larger range with the particle +mainly distributed around the nonzero smallest superlat- +tice vectors. Unlike TBG, TBG/BN at ν = 3 has a quite +different exciton wavefunction from that at ν = −3 with +the wavefunction at ν = 3 less spatially localized. +The quantitative behavior of the excitation spectrum +varies with θ′, as seen in Fig. 4. The systems with the +negative θ′ of −0.56◦ tend to have a smaller valley-wave +excitation energy, while their exciton energies are much +higher than those at positive θ′ for the positive ν. For +the valley-wave modes, the ˜∆inter at the two positive θ′ +have similar values for the negative ν, while they differ +by about 1 meV for the positive ν. The exciton energy +at θ′ = 0.54◦ is higher than that at θ′ = 1.64◦ for the +negative ν but has similar values for the positive ν. +III. +FULL HF BANDS AND EXCITATIONS +Since the remote bands are frozen in the active-band +approximation of the SCHF calculations, the excitation +processes between the remote and flat bands have been +ignored, and the quantitative properties of the flat bands +can be modified when the remote bands are allowed to +be updated in the SCHF calculations. Full SCHF cal- +culations have also been performed to obtain the full +HF bands of TBG/BN, and the excitation spectrums +are computed by considering the excitation processes be- +tween the five highest valence HF bands and the five +lowest conduction HF bands. It is noted that the conver- +gent spin-wave spectrum requires the possible excitation +processes between all the HF bands, which are beyond +our calculation capability, so only the inter-flavor valley- +wave and the intra-flavor exciton modes are considered +based on the full SCHF ground states. +To compare the active-band approximation with the +full SCHF description of the central HF bands, the pro- +jection of each HF band state on the single-particle +flat bands is computed as � +m |⟨ψ0 +m(σ, k)|ψi(σ, k)⟩|2 with +|ψ0 +m(σ, k)⟩ representing the two single-particle flat-band +states of flavor σ and |ψi⟩ a HF band state. We find that + +K +M +K80 +60 +40 +(meV) +20 +nergy +-20 +零 +-40 +-60 +-80 +K +M +KM +KK +M +K80 +60 +40 +(meV) +20 +nergy +-20 +零 +-40 +-60 +-80 +K +M +KM +K80 +60 +40 +nergy (meV) +20 +-20 +-40 +-60 +-80 +K +M +KK +M +KK +M +K80 +60 +40 +(meV) +20 +hergy( +-20 +-40 +-60 +-80 +K +M +KM +KM +K6 +Energy (meV) +ν +valley wave +ν=-3 +exciton +ν=-3 +exciton′ +ν=-3 +TBG/BN +θ′=0.54° +(a) +(b) +FIG. 6. (Color online) (a) The band gaps ∆intra, ∆inter, and ∆′ +intra and the corresponding excitation gaps ˜∆intra, ˜∆inter, +and ˜∆′ +intra from the full HF calculations at different ν for TBG/BN with θ′ = 0.54◦. +(b) The energy spectrums for the +valley-wave excitation, the exciton excitation mainly between flat-like bands, and the exciton excitation (dented by exciton′) +mainly between empty flat-like bands and the filled remote bands at ν = −3. +at a k-point rather away from the ¯Γ point, only two low- +energy HF band states of a flavor are mainly contributed +by the single-particle flat-band states, as shown in Fig. +5 for TBG/BN with θ′ = 0.54◦. These HF bands are +termed as flat-like bands to distinguish them from the +single-particle flat bands. In contrast, several other HF +bands near the ¯Γ point can have substantial contribution +from the flat-band states, especially for the flavor with +one flat-like band occupied and the other flat-like band +empty. In particular, the flat-band contribution becomes +very small for some low-energy HF states at ¯Γ. When +the flat-like bands of a flavor are both occupied or empty, +they are generally well separated from the remote bands, +and the intra-flavor gap around EF between the remote +and flat-like bands is denoted by ∆′ +intra. The flat-like +bands become entangled with the remote bands when +EF lies between them, and the intra-flavor gap between +these flat-like bands is denoted by ∆intra. Similar to the +active-band approximation, the inter-flavor gap ∆inter is +also between the flat-like bands. For the full HF bands, +the global gap among all flavors is just ∆′ +intra. ∆intra +has large and similar values for all the negative and posi- +tive ν, which is similar to the active-band approximation. +However, the systems at positive ν have much smaller +∆′ +intra and ∆inter than those at negative ν, which indi- +cates the strong breaking of the particle-hole symmetry +for the full HF band structures. At each ν, there are also +three flavors with the same quantitative band properties, +as seen in Fig. 5. +We consider the inter-flavor valley-wave excitation +modes corresponding to ∆inter, and the intra-flavor exci- +ton modes corresponding to ∆intra and ∆′ +intra, based on +the full HF ground states. The excitation spectrum and +the excitation gaps of these modes are displayed in Fig. +6 for TBG/BN with θ′ = 0.54◦. The valley-wave exci- +tation gap ˜∆inter becomes slightly higher than that ob- +tained from the active-band approximation and reaches +about 3 meV, but is still rather small compared with +∆inter. The excitation gaps ˜∆intra of the exciton modes +between the flat-like bands have similar values as those +from the active-band approximation and are below half +of ∆intra. In contrast, The gaps ( ˜∆′ +intra) of the exciton +modes between the flat-like bands and the remote bands +are just slightly smaller than ∆′ +intra. This indicates that +the exciton modes between the flat-like bands and the +remote bands are composed of weak-bound particle-hole +pairs, while strong binding of the particle-hole pairs oc- +curs in the exciton modes between the flat-like bands. +At ν = −3, 1, 3, ˜∆intra is higher than ˜∆′ +intra and even +the gap ∆′ +intra. Only at ν = 1, ˜∆intra has a lower value +than ˜∆′ +intra. In addition, the excitation energies of the +lowest modes for the exciton modes between the flat-like +bands and the remote bands are much more dispersive as +a function of q than those of the valley-wave modes and +the exciton modes between the flat-like bands, as shown +in Fig. 6(b). +The optical properties of the Chern insulators are de- +termined by the intra-flavor exciton modes, and the opti- +cal conductivity within the TDHF method is given by64 +Reσxx = +γ +ℏωNkΩ0 +� +i +1 +(ℏω − ℏωi)2 + γ2 +� +Ik,I′k′ +J∗ +x,Ikui,Iku∗ +i,I′k′Jx,I′k′ +(3) +where ω is the frequency of the incident light, ℏωi is +the energy of an exciton mode labeled by i, ui is the +state vector of the exciton mode, Jx,Ik = ⟨ψpIk| − + +70 +intra +Ainter +'intra +60 +XXx +50 +intra +40 +30 +20 +10 +0 +3 +370 +60 +50 +40 +30 +20 +10 +K +K +K +K7 +e/ℏ∂Hk/∂kx|ψhIk⟩ is the element of the current density +operator between the empty and occupied states of the +excitation process I, γ is a small energy for broadening of +the excitation energy, Ω0 is the area of the moire super- +cell, and Nk is the number of k-points. So at ω = ωi, the +contribution of the exciton mode i to σxx is proportional +to σi ≡ ℏ2/(e2Nk) � +Ik,I′k′ J∗ +x,Ikui,Iku∗ +i,I′k′Jx,I′k′. +We +find that the σi of the lowest exciton mode between the +remote and flat-like bands at ν = −3 reaches 0.102 eV ˚A2 +and is even much larger than that of the lowest exciton +mode between the flat-like bands, which is just 0.022 eV +˚A2. +Therefore, the lowest-frequency optical properties +associated with the intra-flavor excitations are mainly +determined by the exciton modes between the remote +bands and the flat-like bands at ν = −3, 1, 3, while they +are mainly contributed by the exciton mode between the +flat-like bands at ν = −1. +At the other two θ′ of 1.64◦ and −0.56◦, ∆′ +intra +from the full SCHF calculations can become larger than +∆inter, but are all much smaller than ∆intra, as seen +in Figs. +S2 and S3 of the SM. For θ′ = −0.56◦, the +system at ν = −3 becomes metallic with the highest oc- +cupied band of the (+, ↑) flavor slightly overlapping with +the lowest empty bands of other flavors. +The systems +at θ′ = 1.64◦ generally have smaller ∆′ +intra than those +at other θ′. For the exciton modes, the excitation gaps +˜∆′ +intra are also much smaller than ˜∆intra, and the sys- +tems with θ′ = −0.56◦ have the largest ˜∆intra, as seen +in Fig. S3 of SM. In addition, ˜∆′ +intra can even become +larger than the indirect gap ∆′ +intra for some systems with +θ′ of 1.64◦ and −0.56◦. The ˜∆inter for the valley-wave +modes all have similar values of about 3 meV. +IV. +SUMMARY AND CONCLUSIONS +In the 1×1 commensurate supercells of TBG/BN, the +single-particle flat bands around EF are gaped due to +the broken C2z symmetry, and the SCHF ground states +at odd ν are the Chern insulators with flavor-polarized +HF bands. In the active-band approximation, the two +active HF bands in the same flavor are well separated in +TBG/BN when they are both filled or empty, and the +intra-flavor gap ∆intra in TBG/BN is much larger than +that in the pristine TBG. The energy spectrums of the +collective excitation modes for the Chern insulator states +are obtained with the TDHF method. +The spin-wave +modes in both TBG/BN and TBG have a zero excita- +tion gap, while the gaps of the valley-wave and exciton +modes in TBG/BN are much larger than those in TBG. +The excitation gap ˜∆inter and ˜∆intra in TBG/BN reach +about 2.5 meV and 20 meV, respectively, with ˜∆intra +almost a half of the intra-flavor band gap ∆intra. In con- +trast to TBG with almost particle-hole symmetric exci- +tation modes for positive and negative ν, the excitation +spectrums and gaps of TBG/BN at positive ν are rather +different from those at negative ν. The exciton wavefunc- +tions in TBG are also much more spatially localized than +those in TBG/BN. Full SCHF calculations show that +more HF bands besides the two central bands can have +rather large contribution from the single-particle flat- +band states in TBG/BN, and the intra-flavor gap ∆intra +between the flat-like bands is much larger than the ∆′ +intra +between the remote and flat-like bands. The excitation +gap ˜∆′ +intra of the exciton modes between the remote and +flat-like bands is just slightly smaller than ∆′ +intra, but +is generally lower than the ˜∆intra between the flat-like +bands, so the optical properties of the Chern insulator +states are mainly determined by the exciton modes be- +tween the remote and flat-like bands. The valley-wave +modes from full HF calculations have similar energies as +those in the active-band approximation. In addition, the +quantitative behavior of the excitation spectrums varies +with θ′ of TBG/BN. +ACKNOWLEDGMENTS +We gratefully acknowledge valuable discussions with +D. Tom´anek, Y. Yin, and X. Xiong. This research was +supported by the National Natural Science Foundation of +China (Grants No. 11974312 and No. 92270104) and the +Open Research Fund of CNMGE Platform & NSCC-TJ. +∗ E-mail: xqlin@zjut.edu.cn +1 R. Bistritzer and A. H. MacDonald, “Moir´e bands in +twisted double-layer graphene,” Proc. Natl. Acad. Sci. +U.S.A. 108, 12233 (2011). +2 E. Su´arez Morell, J. D. Correa, P. Vargas, M. Pacheco, +and Z. Barticevic, “Flat bands in slightly twisted bilayer +graphene: Tight-binding calculations,” Phys. Rev. B 82, +121407 (2010). +3 J. M. B. Lopes dos Santos, N. M. R. Peres, +and A. H. +Castro Neto, “Continuum model of the twisted graphene +bilayer,” Phys. Rev. B 86, 155449 (2012). +4 S. Fang and E. Kaxiras, “Electronic structure theory of +weakly interacting bilayers,” Phys. Rev. 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B 104, 035119 (2021). + diff --git a/OtE4T4oBgHgl3EQf9w7z/content/tmp_files/load_file.txt b/OtE4T4oBgHgl3EQf9w7z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68c5e1bd5a4602eb4adee89e29c682ddf47d7bde --- /dev/null +++ b/OtE4T4oBgHgl3EQf9w7z/content/tmp_files/load_file.txt @@ -0,0 +1,1019 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf,len=1018 +page_content='Collective excitations of the Chern-insulator states in commensurate double moir´e superlattices of twisted bilayer graphene on hexagonal boron nitride Xianqing Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ∗ Quan Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='1 Cheng Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='1 and Jun Ni2 1College of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Zhejiang University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Hangzhou 310023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' China 2State Key Laboratory of Low-Dimensional Quantum Physics and Frontier Science Center for Quantum Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 2023) We study the collective excitation modes of the Chern insulator states in magic-angle twisted bilayer graphene aligned with hexagonal boron nitride (TBG/BN) at odd integer fillings (ν) of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the 1×1 commensurate double moir´e superlattices in TBG/BN at three twist angles (θ′) between BN and graphene, self-consistent Hartree-Fock calculations show that the electron- electron interaction and the broken C2z symmetry lead to the Chern-insulator ground states with valley-spin flavor polarized HF bands at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In the active-band approximation, the HF bands in the same flavor of TBG/BN are much more separated than those of the pristine TBG with TBG/BN having a larger intra-flavor band gap so that the energies of the lowest intra-flavor exciton modes of TBG/BN computed within the time-dependent HF method are much higher than those of TBG and reach about 20 meV, and the exciton wavefunctions of TBG/BN become less localized than those of TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The inter-flavor valley-wave modes in TBG/BN have excitation energies higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='5 meV which is also much larger than that of TBG, while the spin-wave modes all have zero excitation gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In contrast to TBG with particle-hole symmetric excitation modes for positive and negative ν, the excitation spectrums and gaps of TBG/BN at positive ν are rather different from those at negative ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The quantitative behavior of the excitation spectrum of TBG/BN also varies with θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Full HF calculations demonstrate that more HF bands besides the two central bands can have rather large contributions from the single-particle flat-band states, then the lowest exciton modes that determine the optical properties of the Chern insulator states in TBG/BN are generally the ones between the remote and flat-like bands, while the valley-wave modes have similar energies as those in the active-band approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' INTRODUCTION Flat bands with vanishing band widths and well sep- arated from other remote bands occur around the Fermi level in magic-angle twisted bilayer graphene (TBG)1–7, and the experimental realization of such TBG intrigued great interest in exploring various electronic, transport and optical properties associated with the flat bands8–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The emergence of correlated insulator states at integer filling of the flat bands in TBG and the superconductiv- ity in the vicinity of these insulating states have been observed and theoretically comprehended8–22,24,31–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' There are eight single-particle flat bands taking into ac- count the spin and valley degrees in TBG, then the elec- tron filling of the flat bands per moir´e supercell rela- tive to charge neutrality point (CNP) is in the range of 4 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At odd ν, the ground states are Chern insu- lators with spontaneously broken symmetry in the val- ley and spin degrees due to the electron-electron (e-e) interaction39–41,43,46–48,50,51,55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The alignment of TBG with BN breaks the C2z symmetry in the relaxed atomic structure and the single-particle Hamiltonian51,56–61 and thus enhances the energy gaps of such Chern insula- tor states39,43,46,51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In particular, the quantum anoma- lous Hall effect associated with their finite Chern num- bers has been experimentally realized in TBG aligned with BN (TBG/BN)62,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For such insulating correlated states, low-energy collective excitation states may appear within the gap due to the Coulomb interaction between the particle and hole states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In experiments, the ob- served Pomeranchuk effect from the measured electron compressibility in TBG at extremely low temperatures implies the presence of the low-energy collective excita- tions for the correlated insulator states24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The optical excitations in the infrared regime have also been observed in twisted graphene systems around the integer fillings of the flat bands23,25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the pristine TBG or the TBG with a sublat- tice potential difference46,53,54,64, theoretical analysis or Hartree-Fock (HF) calculations indeed demonstrated the occurrence of the low-energy collective excitation modes of inter-flavor spin wave, valley wave and intra-flavor ex- citon at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The spin-wave excitation states are Gold- stone modes with a zero excitation gap46,53,54,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The valley-wave modes have an extremely small excitation gap for the pristine TBG64, and a sublattice potential difference increases their excitation energies46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the pristine TBG, low energy exciton states of a few meV also appear64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We note that all the previous calcula- tions focused on one odd ν of -3 or 3 and adopted the active-band approximation that considers only the exci- tations between flat bands46,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' A full HF calculation of the excitation states at all odd ν may provide more ex- citation modes and can influence the excitation energy spectrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For TBG/BN with enhanced Chern insula- tor states at odd ν, previous studies have established that BN induces not only the sublattice potential differ- ence in graphene but also spatially varying effective moir´e arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='05359v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='mes-hall] 13 Jan 2023 2 θ (+↑) (+↓) (-↓) (-↑) (+↑) (+↓) (-↓) (-↑) TBG TBG/BN TBG/BN θ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='08° θ′=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54° TBG θ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='08° ν=-3 ν=-1 ν=1 ν=3 Δinter Δintra θ′ BN TBG (a) (b) (c) Energy (meV) ��� ��� Γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) The single-particle and HF band structures of TBG and TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (a) The schematic view of the commensurate double moir´e superlattices in TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The twist angle (θ) between the graphene layers and that (θ′) between graphene and BN are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (b) The single-particle band structures of TBG at the magic θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='08◦ and a commensurate supercell of TBG/BN at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The red and blue lines represent bands in the ξ = + valley and the ξ = − valley, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (c) The HF band structures of the Chern-insulator ground states at odd ν in the active-band approximation for the TBG and TBG/BN systems in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The HF bands of each flavor are plotted separately along the k-point path same as that in (b), and the flavor is labeled by the valley (+ or -) and spin (↑ or ↓) indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The conduction and valence bands are represented by the red and blue lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The intra-flavor energy gap (∆intra) between the conduction and valence bands in the same flavor and the inter-favor energy gap (∆inter) between the highest valence band in one flavor and the lowest conduction band in another flavor are labeled by the green shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The spin-wave excitation between the valence and conduction bands in two different flavors with the same valley index but the opposite spin indices, and the valley-wave excitation between the bands in two flavors with the same spin index but the opposite valley indices, and the exciton excitation between bands in the same flavor are indicated by the dashed green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' potentials, and the structural deformation due to the in- terlayer vdW interaction between BN and graphene also strongly breaks the C2z symmetry of the single-particle Hamiltonian51,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Moreover, the correlated band struc- ture of TBG/BN changes with the twist angle (θ′) be- tween TBG and BN51,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Therefore, it is desirable to explore systematically the collective excitation modes at all odd ν for all the possible commensurate configurations of TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Here, we demonstrate that the energies of the lowest intra-flavor exciton modes of TBG/BN are much higher than those of TBG and reach about 20 meV, the inter- flavor valley-wave modes have excitation energies higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='5 meV which is also much larger than that of TBG, while the spin-wave modes all have zero excitation gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The excitation spectrums and gaps of TBG/BN at positive ν are rather different from those at negative ν, which contrasts with the particle-hole symmetric excita- tion modes for positive and negative ν in TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Full HF calculations indicate that the lowest exciton modes that determine the optical properties of the Chern insulator states in TBG/BN are generally the ones between the remote and flat-like bands, while the valley-wave modes have similar energies as those in the active-band approx- imation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Moreover, the quantitative behavior of the ex- citation spectrum of TBG/BN also varies with θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' HF BANDS AND EXCITATIONS IN THE ACTIVE-BAND APPROXIMATION For TBG with the magic twist angle of θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='08◦ aligned with BN, we consider the 1×1 commensurate su- percells of TBG/BN at three twist angles θ′ between BN and its adjacent graphene layer of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(a), and their structural parameters are detailed in the Supplemental Material (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At the ori- gin of the TBG/BN supercell, both the local stackings between the graphene layers and between graphene and BN are taken to be the AA stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The moir´e superlat- tices of TBG/BN and the pristine TBG are fully relaxed based on the continuum elastic theory to obtain their stable atomic structures51,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the fully relaxed TBG/BN, an effective single- particle tight-binding model (H0) of the graphene lay- ers can be built taking into account the relaxation effect and the full moir´e Hamiltonian induced by BN51,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The parameters in H0 and its expression in the planewave- like basis are detailed in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The single-particle flat bands around the Fermi level (EF ) in TBG are well sep- arated by the effective moir´e potentials induced by BN, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The C2z symmetry in the pristine TBG is broken in TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In a rigid TBG/BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='effective Hamiltonian induced by BN lacks the C2z sym- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='metry as reflected in the in-plane inversion asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='50K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='M K80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='MKM3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='ν=-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='ν=-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='ν=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='ν=3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='spin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='wave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='valley ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='wave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='exciton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='TBG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='TBG/BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���intra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���inter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���inter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���intra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Δinter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Δintra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���inter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='���intra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Δinter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Δintra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='TBG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='TBG/BN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) The collective excitation modes of TBG/BN and the pristine TBG at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (a) The energy spectrums of the two or one lowest excitation modes as a function of the wave vector q for the spin-wave, valley-wave and exciton excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The blue and red lines represent the excitation bands of the pristine TBG and the TBG/BN at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ¯K′ in the k-point path is just the opposite of ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The valley-wave excitation gap ( ˜∆inter) and the exciton excitation gap ( ˜∆intra) at q = 0 are labeled by the green shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (b) The HF band gaps and the corresponding excitation gaps at different ν for TBG/BN and TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' terms of the moir´e potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The structural relaxation of TBG/BN also leads to the in-plane atomic deforma- tion without the C2z symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The strength of the ef- fective moir´e potential by BN varies with θ′, giving rising to θ′-dependent flat bands, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' S1(a) of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The widths of the flat bands are much larger at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦ than those at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦, while the valence and conduction bands are more separated at θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The system at θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦ also has a much smaller en- ergy difference between the flat and remote bands due to the wider flat bands and the larger gap at EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Upon inclusion of the e-e interaction, TBG/BN and TBG become Chern insulators at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We employ the self-consistent HF (SCHF) method41,51 to obtain the mean-field ground states of the systems at odd ν, then the time dependent HF (TDHF) approach46,64 is adopted to explore the collective excitations of TBG/BN and TBG based on the SCHF ground states as detailed in the Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We first perform the HF calculations in the active-band approximation, and the computationally ex- pensive full HF calculations are then done for further ex- ploration of the collective excitations as presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the active-space approximation, only the two central HF bands of each flavor are updated dur- ing the SCHF iterations and they are only expanded in the basis of the single-particle flat bands, and the lower remote bands are kept frozen but still contribute to the mean-field Hartree and Fock operators of the active-band Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In addition, the HF operators contributed by the isolated fixed and rotated graphene layers with EF at CNP are subtracted from the HF Hamiltonian to avoid double-counting of the e-e interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The HF band structures of the Chern-insulator ground states at odd ν are exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(c) for TBG/BN with θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ and the pristine TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In TBG, sublat- tice polarization within one layer spontaneously occurs at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In the ξ = + valley, the lower band has a Chern number of +1 and the higher band has a Chern number of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The Chern numbers of the bands in the ξ = − valley are just opposite to those in the ξ = + valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At each odd ν, the ground states of TBG and TBG/BN are Chern insulators with the total Chern numbers of ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For each ν, three of the four flavors have the same quan- titative band properties, such as the intra-flavor band gaps and the band widths, and one flavor has different properties, which is taken to be the (+, ↑) valley-spin fla- vor, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At a flavor with the two bands both filled or empty, the two bands of TBG/BN are well separated, while those of the pristine TBG have close en- ergies around the ¯K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For TBG/BN at ν = −3, 20 10 10 0 20 10 K K K KIntra 50 Inter Intra Inter 40 30 20 10 0 Y 3Intra 50 Inter Intra Inter 40 30 20 10 0 34 TBG ν=-3 TBG ν=3 TBG/BN ν=-3 TBG/BN ν=3 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) The spatial distribution (|Ψ(re, rh)|2) of the exciton wavefunction of the lowest mode at q = 0 as a function of the electron position re with the hole position rh at the origin of a supercell where the bilayer is locally AA- stacked for TBG at ν = −3 (a), ν = 3 (b), and TBG/BN with θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ at ν = −3 (c), ν = 3 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' the two empty bands in the same flavor are separated by 17 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' When one flat band is filled and the other one is empty in a flavor, the intra-flavor band gap (∆intra) between them in TBG/BN is much larger than that in the pristine TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Compared to ∆intra, the inter-favor band gap (∆inter) between the highest valence band in one flavor and the lowest conduction band in another fla- vor generally has a smaller value, so the global band gap is just ∆inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The HF bands at ν = 3 and ν = 1 appear to be the particle-hole symmetric correspondences of the bands at ν = −3 and ν = −1, respectively, while the band gaps can still be quite different between positive and negative ν, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(c) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The θ′ of TBG/BN influences the quantitative properties of the HF bands, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' S1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At ν = ±3, ∆inter at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ is much larger than those at other θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The ∆intra at θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦ is the largest for ν = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In addition, when two bands in a flavor are both filled or empty, they have a much larger energy difference at θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We employ the TDHF method to obtain the collective excitation modes based on the HF ground states at odd ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We consider the collective modes with the momentum q expressed as46,64 |Ψ{I}(q)⟩ = � I,k uI,k(q)f † pI,k+qfhI,k|0⟩, (1) where |0⟩ is the HF ground state, I represents an excita- tion process from the occupied band with index hI to the empty band with the index pI, and the operator f anni- hilates an electron in the HF band states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' A collective mode is characterized by its set of excitation processes, θ′= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64° θ′= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54° θ′= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56° ν=-3 ν=-1 ν=1 ν=3 spin wave valley wave exciton Energy (meV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) The energy spectrums of the lowest spin-wave, valley-wave and exciton excitation modes at dif- ferent ν for TBG/BN with θ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' which are labeled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 1(c) for the inter-flavor spin- wave, valley-wave, and intra-flavor exciton modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the pristine TBG, all the excitation spectrums ex- hibit approximate particle-hole symmetry and are almost the same for all odd ν, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The spin-wave mode has a zero excitation gap, the valley- wave mode has an extremely small gap of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='5 meV, and the exciton mode has a gap of about 5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Such finite gaps of the valley-wave and exciton modes are slightly larger than those predicted for the TBG de- scribed by the Bistritzer-MacDonald model64, which can be attributed to the in-plane structural deformation in the relaxed TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the pristine TBG, both the spin- wave and valley-wave excitations have two low-energy collective modes in the gap at each q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In contrast, the excitation spectrum of TBG/BN at positive ν are rather different from those at negative ν, and those with the same sign of ν are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The lowest spin-wave mode at positive ν has a larger spectrum width than that at negative ν but they all have zero excitation gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the valley-wave, all the excitation energies are higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='5 meV, which is much larger than that of the pris- tine TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' This is consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' [46] for TBG with a sublattice potential difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The valley wave has a higher excitation gap ( ˜∆inter) at positive ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For both the spin wave and the valley wave, the lowest modes become much more apart than those in TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The lowest exci- ton modes of TBG/BN have much higher energies than those of TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The exciton gap ( ˜∆intra) decreases with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='004 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='003 10 (wu) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='002 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='001 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='000 20 10 0 10 20 x (nm)20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='002 10 (wu) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='001 > 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='000 20 10 0 10 20 x (nm)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='012 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='008 (wu) 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='004 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='000 20 10 0 10 20 x (nm)20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='012 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='008 (wu) 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='004 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='000 20 10 0 10 20 x (nm)6 4 2 0 6 4 2 0 40 20 K K K K5 对给定nu, 能带分成两组，一个的为（+，up）， 其他三个为另一组 Δintra Δ′intra Δinter (+↑) (+↓) (-↑) (- ↓) ν=-3 ν=-1 ν=1 ν=3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) The full HF band structures of TBG/BN at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ for each flavor at different ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The magnitude of the projection of each HF band state on the single-particle flat bands is represented by the size of the red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The band gap ∆intra between the intra-flavor flat-like bands, the ∆inter between the inter-flavor flat-like bands, and the ∆′ intra between the remote and flat-like bands are labeled by the green shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The inter-flavor valley-wave excitation mainly between the flat-like bands, the intra-flavor exciton excitation mainly between the flat-like bands, and the intra- flavor exciton excitation mainly between the flat-like bands and the remote bands are indicated by the dashed green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ν from -3 to 3, with the gap still reaching about 16 meV at ν = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In comparison to the HF band gaps, the ˜∆inter of the valley wave modes are much smaller than the ∆inter for both TBG and TBG/BN, while the ˜∆intra of the exciton modes of TBG/BN reaches about half of the ∆intra, which is a significant contrast to the much smaller ˜∆intra/∆intra for TBG, as shown in Fig 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the exciton modes, the two-body exciton wavefunction as a function of the electron (re) and hole (rh) positions can be calculated as Ψ(re, rh) = 1 Nk � k ukψpk(re)ψ∗ hk(rh), (2) where ψpk and ψhk are the HF conduction and valence band states corresponding to the exciton excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Fig- ure 3 exhibits the wavefunction of the lowest exciton mode at q = 0 with rh at the origin of a supercell where the bilayer graphene is locally AA-stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The parti- cle and hole are strongly bound at all the odd ν for the pristine TBG with the particle localized around the ori- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The spatial map of the exciton wavefunctions in TBG/BN spread a much larger range with the particle mainly distributed around the nonzero smallest superlat- tice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Unlike TBG, TBG/BN at ν = 3 has a quite different exciton wavefunction from that at ν = −3 with the wavefunction at ν = 3 less spatially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The quantitative behavior of the excitation spectrum varies with θ′, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The systems with the negative θ′ of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦ tend to have a smaller valley-wave excitation energy, while their exciton energies are much higher than those at positive θ′ for the positive ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the valley-wave modes, the ˜∆inter at the two positive θ′ have similar values for the negative ν, while they differ by about 1 meV for the positive ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The exciton energy at θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦ is higher than that at θ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦ for the negative ν but has similar values for the positive ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' FULL HF BANDS AND EXCITATIONS Since the remote bands are frozen in the active-band approximation of the SCHF calculations, the excitation processes between the remote and flat bands have been ignored, and the quantitative properties of the flat bands can be modified when the remote bands are allowed to be updated in the SCHF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Full SCHF cal- culations have also been performed to obtain the full HF bands of TBG/BN, and the excitation spectrums are computed by considering the excitation processes be- tween the five highest valence HF bands and the five lowest conduction HF bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' It is noted that the conver- gent spin-wave spectrum requires the possible excitation processes between all the HF bands, which are beyond our calculation capability, so only the inter-flavor valley- wave and the intra-flavor exciton modes are considered based on the full SCHF ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' To compare the active-band approximation with the full SCHF description of the central HF bands, the pro- jection of each HF band state on the single-particle flat bands is computed as � m |⟨ψ0 m(σ, k)|ψi(σ, k)⟩|2 with |ψ0 m(σ, k)⟩ representing the two single-particle flat-band states of flavor σ and |ψi⟩ a HF band state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We find that K M K80 60 40 (meV) 20 nergy 20 零 40 60 80 K M KM KK M K80 60 40 (meV) 20 nergy 20 零 40 60 80 K M KM K80 60 40 nergy (meV) 20 20 40 60 80 K M KK M KK M K80 60 40 (meV) 20 hergy( 20 40 60 80 K M KM KM K6 Energy (meV) ν valley wave ν=-3 exciton ν=-3 exciton′ ν=-3 TBG/BN θ′=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54° (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (Color online) (a) The band gaps ∆intra, ∆inter, and ∆′ intra and the corresponding excitation gaps ˜∆intra, ˜∆inter, and ˜∆′ intra from the full HF calculations at different ν for TBG/BN with θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' (b) The energy spectrums for the valley-wave excitation, the exciton excitation mainly between flat-like bands, and the exciton excitation (dented by exciton′) mainly between empty flat-like bands and the filled remote bands at ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' at a k-point rather away from the ¯Γ point, only two low- energy HF band states of a flavor are mainly contributed by the single-particle flat-band states, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 5 for TBG/BN with θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' These HF bands are termed as flat-like bands to distinguish them from the single-particle flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In contrast, several other HF bands near the ¯Γ point can have substantial contribution from the flat-band states, especially for the flavor with one flat-like band occupied and the other flat-like band empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In particular, the flat-band contribution becomes very small for some low-energy HF states at ¯Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' When the flat-like bands of a flavor are both occupied or empty, they are generally well separated from the remote bands, and the intra-flavor gap around EF between the remote and flat-like bands is denoted by ∆′ intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The flat-like bands become entangled with the remote bands when EF lies between them, and the intra-flavor gap between these flat-like bands is denoted by ∆intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Similar to the active-band approximation, the inter-flavor gap ∆inter is also between the flat-like bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the full HF bands, the global gap among all flavors is just ∆′ intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ∆intra has large and similar values for all the negative and posi- tive ν, which is similar to the active-band approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' However, the systems at positive ν have much smaller ∆′ intra and ∆inter than those at negative ν, which indi- cates the strong breaking of the particle-hole symmetry for the full HF band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At each ν, there are also three flavors with the same quantitative band properties, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We consider the inter-flavor valley-wave excitation modes corresponding to ∆inter, and the intra-flavor exci- ton modes corresponding to ∆intra and ∆′ intra, based on the full HF ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The excitation spectrum and the excitation gaps of these modes are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 6 for TBG/BN with θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='54◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The valley-wave exci- tation gap ˜∆inter becomes slightly higher than that ob- tained from the active-band approximation and reaches about 3 meV, but is still rather small compared with ∆inter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The excitation gaps ˜∆intra of the exciton modes between the flat-like bands have similar values as those from the active-band approximation and are below half of ∆intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In contrast, The gaps ( ˜∆′ intra) of the exciton modes between the flat-like bands and the remote bands are just slightly smaller than ∆′ intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' This indicates that the exciton modes between the flat-like bands and the remote bands are composed of weak-bound particle-hole pairs, while strong binding of the particle-hole pairs oc- curs in the exciton modes between the flat-like bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At ν = −3, 1, 3, ˜∆intra is higher than ˜∆′ intra and even the gap ∆′ intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Only at ν = 1, ˜∆intra has a lower value than ˜∆′ intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In addition, the excitation energies of the lowest modes for the exciton modes between the flat-like bands and the remote bands are much more dispersive as a function of q than those of the valley-wave modes and the exciton modes between the flat-like bands, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The optical properties of the Chern insulators are de- termined by the intra-flavor exciton modes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' and the opti- cal conductivity within the TDHF method is given by64 Reσxx = γ ℏωNkΩ0 � i 1 (ℏω − ℏωi)2 + γ2 � Ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='I′k′ J∗ x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Ikui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='Iku∗ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='I′k′Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='I′k′ (3) where ω is the frequency of the incident light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ℏωi is the energy of an exciton mode labeled by i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ui is the state vector of the exciton mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Jx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content="Ik = ⟨ψpIk| − 70 intra Ainter 'intra 60 XXx 50 intra 40 30 20 10 0 3 370 60 50 40 30 20 10 K K K K7 e/ℏ∂Hk/∂kx|ψhIk⟩ is the element of the current density operator between the empty and occupied states of the excitation process I," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' γ is a small energy for broadening of the excitation energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Ω0 is the area of the moire super- cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' and Nk is the number of k-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' So at ω = ωi, the contribution of the exciton mode i to σxx is proportional to σi ≡ ℏ2/(e2Nk) � Ik,I′k′ J∗ x,Ikui,Iku∗ i,I′k′Jx,I′k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' We find that the σi of the lowest exciton mode between the remote and flat-like bands at ν = −3 reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='102 eV ˚A2 and is even much larger than that of the lowest exciton mode between the flat-like bands, which is just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='022 eV ˚A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Therefore, the lowest-frequency optical properties associated with the intra-flavor excitations are mainly determined by the exciton modes between the remote bands and the flat-like bands at ν = −3, 1, 3, while they are mainly contributed by the exciton mode between the flat-like bands at ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' At the other two θ′ of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦ and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦, ∆′ intra from the full SCHF calculations can become larger than ∆inter, but are all much smaller than ∆intra, as seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' S2 and S3 of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦, the system at ν = −3 becomes metallic with the highest oc- cupied band of the (+, ↑) flavor slightly overlapping with the lowest empty bands of other flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The systems at θ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦ generally have smaller ∆′ intra than those at other θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' For the exciton modes, the excitation gaps ˜∆′ intra are also much smaller than ˜∆intra, and the sys- tems with θ′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦ have the largest ˜∆intra, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' S3 of SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In addition, ˜∆′ intra can even become larger than the indirect gap ∆′ intra for some systems with θ′ of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='64◦ and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='56◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The ˜∆inter for the valley-wave modes all have similar values of about 3 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In the 1×1 commensurate supercells of TBG/BN, the single-particle flat bands around EF are gaped due to the broken C2z symmetry, and the SCHF ground states at odd ν are the Chern insulators with flavor-polarized HF bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In the active-band approximation, the two active HF bands in the same flavor are well separated in TBG/BN when they are both filled or empty, and the intra-flavor gap ∆intra in TBG/BN is much larger than that in the pristine TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The energy spectrums of the collective excitation modes for the Chern insulator states are obtained with the TDHF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The spin-wave modes in both TBG/BN and TBG have a zero excita- tion gap, while the gaps of the valley-wave and exciton modes in TBG/BN are much larger than those in TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The excitation gap ˜∆inter and ˜∆intra in TBG/BN reach about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='5 meV and 20 meV, respectively, with ˜∆intra almost a half of the intra-flavor band gap ∆intra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In con- trast to TBG with almost particle-hole symmetric exci- tation modes for positive and negative ν, the excitation spectrums and gaps of TBG/BN at positive ν are rather different from those at negative ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The exciton wavefunc- tions in TBG are also much more spatially localized than those in TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Full SCHF calculations show that more HF bands besides the two central bands can have rather large contribution from the single-particle flat- band states in TBG/BN, and the intra-flavor gap ∆intra between the flat-like bands is much larger than the ∆′ intra between the remote and flat-like bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The excitation gap ˜∆′ intra of the exciton modes between the remote and flat-like bands is just slightly smaller than ∆′ intra, but is generally lower than the ˜∆intra between the flat-like bands, so the optical properties of the Chern insulator states are mainly determined by the exciton modes be- tween the remote and flat-like bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' The valley-wave modes from full HF calculations have similar energies as those in the active-band approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' In addition, the quantitative behavior of the excitation spectrums varies with θ′ of TBG/BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ACKNOWLEDGMENTS We gratefully acknowledge valuable discussions with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Tom´anek, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Yin, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' This research was supported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 11974312 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' 92270104) and the Open Research Fund of CNMGE Platform & NSCC-TJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' ∗ E-mail: xqlin@zjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content='cn 1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Bistritzer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' MacDonald, “Moir´e bands in twisted double-layer graphene,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} +page_content=' Acad.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE4T4oBgHgl3EQf9w7z/content/2301.05359v1.pdf'} diff --git a/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/2301.12951v1.pdf.txt b/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/2301.12951v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..81b8e95e5166d597f1edae9a060174a9c6f79122 --- /dev/null +++ b/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/2301.12951v1.pdf.txt @@ -0,0 +1,1203 @@ +On the Interaction between Node Fairness and Edge Privacy +in Graph Neural Networks +He Zhang1 , Xingliang Yuan1 , Quoc Viet Hung Nguyen2 and Shirui Pan3 +1Faculty of Information Technology, Monash University +2Institute for Integrated and Intelligent Systems, Griffith University +3School of Information and Communication Technology, Griffith University +{he.zhang1, xingliang.yuan}@monash.edu, {henry.nguyen, s.pan}@griffith.edu.au +Abstract +Due to the emergence of graph neural networks +(GNNs) and their widespread implementation in +real-world scenarios, the fairness and privacy of +GNNs have attracted considerable interest since +they are two essential social concerns in the era of +building trustworthy GNNs. Existing studies have +respectively explored the fairness and privacy of +GNNs and exhibited that both fairness and privacy +are at the cost of GNN performance. However, the +interaction between them is yet to be explored and +understood. In this paper, we investigate the in- +teraction between the fairness of a GNN and its +privacy for the first time. We empirically identify +that edge privacy risks increase when the individ- +ual fairness of nodes is improved. Next, we present +the intuition behind such a trade-off and employ the +influence function and Pearson correlation to mea- +sure it theoretically. To take the performance, fair- +ness, and privacy of GNNs into account simultane- +ously, we propose implementing fairness-aware re- +weighting and privacy-aware graph structure per- +turbation modules in a retraining mechanism. Ex- +perimental results demonstrate that our method is +effective in implementing GNN fairness with lim- +ited performance cost and restricted privacy risks. +1 +Introduction +In recent years, in addition to competent performance, there +has been an increasing desire for fairness [Agarwal et al., +2021; Li et al., 2021; Dong et al., 2021] and private infor- +mation security [Wu et al., 2022a; Wu et al., 2021] in GNNs, +because both privacy and fairness are two essential concerns +of GNN users [Zhang et al., 2022]. In the context of GNNs, +existing studies [He et al., 2021; Kang et al., 2020] only focus +on performance and privacy/fairness; however, privacy and +fairness are not isolated from each other in practical scenar- +ios. For example, in e-commerce platforms [Guo et al., 2022; +Zhu et al., 2019], where users and items are considered as +nodes and their interactions are regarded as edges in a graph, +a recommender system should serve all users with compe- +tent performance. Meanwhile, user bias [Wu et al., 2022d] +should be avoided and similar users (i.e., nodes) should re- +ceive similar item recommendations to boost individual fair- +ness; the purchase records of a customer (i.e., edges between +nodes) should not be exposed to others without permission. +Therefore, it is necessary to study the interaction among per- +formance, fairness, and privacy to build trustworthy GNNs +comprehensively [Zhang et al., 2022; Sharma et al., 2022]. +Existing literature has studied the interaction between per- +formance and fairness/privacy of GNNs. +For example, to +boost individual fairness in GNNs, a method called RE- +DRESS [Dong et al., 2021] proposes to add a regularisation +term concerning fairness from the ranking perspective into +the loss function of GNNs. Moreover, InFoRM [Kang et al., +2020] uses the Lipschitz property and proposes a metric to +evaluate the individual fairness of nodes. This metric can also +be involved in the loss function of a target model to reduce the +bias existing in the GNN predictions. To promote edge pri- +vacy, for instance, Wu et al. [Wu et al., 2022b] explore the +vulnerability of edges in the training graph and introduce dif- +ferential privacy (DP) mechanisms [Sajadmanesh and Gatica- +Perez, 2021] to protect edges from leakage. These studies +also demonstrate that both individual fairness and edge pri- +vacy are at cost of GNN performance. +To comprehensively build trustworthy GNNs, it is in- +evitable to study the interaction between fairness and privacy. +Currently, a few works have explored the impact of improv- +ing model privacy on fairness [Zhang et al., 2021] or promot- +ing algorithm fairness on privacy [Chang and Shokri, 2021], +which are in the context of general machine learning models +for Independent Identically Distribution (IID) data. In con- +trast, this paper will study the interaction of node fairness and +edge privacy of GNNs for complex graph data, which has not +yet been explored and measured to the best of our knowl- +edge. Moreover, given the trade-off between fairness/privacy +and performance and unexplored interaction between fairness +and privacy, a challenging research topic of building trustwor- +thy GNNs is how to ensure the privacy and fairness of a GNN +model simultaneously with keeping competent model perfor- +mance (i.e., with limited performance cost). +In this paper, we empirically verify the adverse effect of +individual fairness of nodes on edge privacy in GNN mod- +els. To understand this observation, we employ the influ- +ence functions of training samples to measure and explain +this trade-off. Furthermore, we propose a Privacy-aware Per- +arXiv:2301.12951v1 [cs.LG] 30 Jan 2023 + +turbations and Fairness-aware Re-weighting (PPFR) method +to implement GNN fairness with limited performance cost +and restricted privacy risks. The contributions of this paper +are summarised as follows: +• In this paper, we explore the interaction between fair- +ness and privacy of GNN for the first time. We empiri- +cally show that the edge privacy risk increases when en- +hancing individual fairness of nodes, i.e., there exists a +trade-off between fairness and privacy of GNNs. +• To understand GNN behaviours concerning different +trustworthiness aspects (e.g., fairness and privacy), we +propose employing the influence function and Pearson +correlation coefficient to quantitatively measure the in- +teraction (e.g., trade-off) between them. +• Based on a re-weighting method and a graph structure +perturbation method, we propose a novel method to de- +vise competent GNNs with reduced bias and edge leak- +age risks, whose effectiveness has been demonstrated by +our experimental evaluations. +2 +Background +Graphs. A graph G = {V, E} includes a node set V = +� +v1, . . . , v|V| +� +and an edge set E. E characterises the rela- +tionship information in G. The set of edges can also be de- +noted by an adjacency matrix A ∈ {0, 1}|V|×|V|, in which +Ai,j = 1 when eij = (vi, vj) ∈ E, otherwise Ai,j = 0. +Matrix X ∈ R|V|×k (k indicates the dimensionality of fea- +tures) denotes node features, the i-th row of X represents the +feature of node vi. Without loss of generality, another de- +scription form of a graph is G = {A, X}. In this paper, we +focus on the undirected graph, i.e. Ai,j = Aj,i. +Node Classification and GCN. For a graph G = {V, E}, the +set of labelled nodes is denoted by Vl ⊂ V, where yv is the +label of v ∈ Vl. The set of unlabelled nodes in G is indicated +by Vu = V \ Vl. Given G and node labels, node classifica- +tion aims to train a GNN model f, which can predict labels +for nodes in Vu. In this paper, we consider the graph con- +volutional network (GCN) model [Kipf and Welling, 2017], +which is a typical GNN model for node classification. Given +a L layer GCN model, we assume that E(l) and W(l) rep- +resent the output node embeddings and the weight matrix of +the l-th hidden layer, respectively. The graph convolution at +the l-th layer can be formulated as E(l) = σ( ˆAE(l−1)W(l)), +where σ is the nonlinear activation, ˆA = ˜D− 1 +2 ( A + I) ˜D− 1 +2 , +and ˜D being the degree matrix of (A + I). +Individual Fairness. To be fair to all users, GNNs should +ensure that everyone is treated equally and receives the same +quality of service regardless of their background. In the con- +text of node classification, individual fairness requires that +any two similar nodes receive similar GNN predictions [Kang +et al., 2020]. Specifically, given the similarity matrix S of +nodes and GNN predictions Y, the bias with respect individ- +ual fairness is measured by Bias(Y, S) = Tr(YT LSY), +where YT represents the transpose of Y, LS indicates the +Laplacian matrix of S [Kang et al., 2020]. To improve indi- +vidual fairness of GNNs, a method called InFoRM proposed +to involve Bias(Y, S) into the loss function during the train- +ing phase of GNNs [Kang et al., 2020]. +Link Stealing Attacks. Existing studies [He et al., 2021; +Wu et al., 2022b] have demonstrated that attackers are ca- +pable of inferring the existence of a link between any two +nodes in the training graph of a GNN model. For example, +by querying node predictions of the target GNN model, at- +tackers can use the prediction similarity of node pairs to in- +fer whether two nodes are connected in a specific node pair +[He et al., 2021]. This attack is based on the intuition that +if two nodes share more similar predictions from the target +GNN model then there is a greater likelihood of the nodes +being linked together [He et al., 2021]. +Currently, exist- +ing methods employ the AUC score to evaluate the vulner- +ability of a GNN model to link-stealing attacks, i.e., leak- +age risk of edges in the training graph [He et al., 2021; +Wu et al., 2022b]. +3 +Interaction Between Fairness and Privacy: +A Preliminary Study +This section presents a preliminary study on the node clas- +sification task to assess the effect of promoting individual +fairness on the risk of edge leakage. Introducing the item +concerning fairness into the loss function of a GNN model +is effective in reducing bias, while it potentially affects the +privacy risk of edges in the training graph. +3.1 +Preliminary Study Settings +Datasets and Models. +In our preliminary experiments, +we employ Cora [Kipf and Welling, 2017], Citeseer [Kipf +and Welling, 2017], and Pubmed [Kipf and Welling, 2017] +datasets, which are commonly used in evaluating GNNs in +node classification. Models selected for this study are GCNs +[Kipf and Welling, 2017] with 16 hidden layers that employ +ReLU and softmax activation functions. We use accuracy as +the metric for evaluating the performance of GCNs. +Fairness. +Following previous studies [Kang et al., 2020; +Chang and Shokri, 2021], we combine the Bias(Y, S) and +original loss function together in the training phase to pro- +mote fairness. In this paper, the similarity matrix S is defined +as the Jaccard index [Kang et al., 2020], and the bias in GNN +predictions is measured by Bias(Y, S). The smaller the bias +value, the fairer the GNN prediction Y. +Privacy. In this paper, we assume that attackers can only +query target models to obtain node predictions from target +GNNs, which is the most practical link-stealing attack (i.e., +Attack-0 in [He et al., 2021]). Based on prediction similarity, +attackers attempt to infer the existence of an edge in any node +pairs in the training graph. The edge leakage risk is measured +by AUC (area under the ROC curve) score, where larger val- +ues indicate higher privacy risks. Following a previous study +[He et al., 2021], the prediction similarity is calculated using +Cosine, Euclidean, Correlation, Chebyshev, Braycurtis, Can- +berra, Cityblock and Sqeuclidean distances. +3.2 +Observations +In our preliminary studies, we focus on the change of pri- +vacy risk on edges when boosting individual fairness in GNN + +Table 1: Comparison of the accuracy, bias, and privacy risks of GCN models. +Datasets +Acc↑ +Bias↓ +Privacy Risks ↓ +Cosi +Eucl +Corr +Cheb +Bray +Canb +City +Sqeu +Cora +Vanilla +86.12 +0.0766 +92.34 +91.98 +92.13 +92.41 +92.57 +90.41 +92.57 +91.98 +Reg +85.38 +0.0494 +93.42 +93.67 +93.14 +93.73 +94.10 +93.81 +94.10 +93.67 +Citeseer +Vanilla +63.66 +0.0445 +92.82 +92.68 +92.74 +93.00 +93.10 +94.15 +93.10 +92.68 +Reg +63.11 +0.0301 +94.31 +94.64 +94.00 +94.77 +95.03 +96.10 +95.03 +94.64 +Pubmed +Vanilla +85.37 +0.0706 +88.59 +89.26 +87.04 +89.37 +89.37 +92.85 +89.37 +89.26 +Reg +83.37 +0.0108 +92.66 +92.90 +89.44 +92.95 +92.95 +93.67 +92.95 +92.90 +* In this table, “Vanilla” represents the generic training of GNNs, and “Reg” indicates that the fairness regularisation is introduced into the +loss function of GNNs during their vanilla training. “Cosi”, “Eucl”, “Corr”, “Cheb”, “Bray”, “Canb”, “City”, and “Sqeu” represent the +Cosine, Euclidean, Correlation, Chebyshev, Braycurtis, Canberra, Cityblock and Sqeuclidean distances, respectively. +predictions. As shown in Table 1, we observe that boost- +ing fairness comes at the cost of model performance, i.e., the +prediction accuracy is sacrificed when improving bias on all +datasets. However, performance reduction is not the only ad- +verse effect. Changes in AUC scores indicate that edge leak- +age risks increase when GNN fairness is promoted. +This phenomenon can be explained by the definition of S +and its different influences on different node pairs. In ho- +mophily graphs (e.g., Cora, Citeseer, Pubmed), similar nodes +(i.e., nodes with the same label) are more likely to connect +to each other [Zhu et al., 2021; Zheng et al., 2022]. In these +graphs, calculating Jaccard similarity [Zhang et al., 2020] be- +tween nodes leads S to assign higher values (e.g., 1) for node +pairs in which two nodes own a higher proportion of the same +1-hop neighbour nodes, while lower values (e.g., 0) for other +node pairs. Therefore, boosting individual fairness based on +Jaccard similarity has limited even zero influence on the latter +node pairs (i.e., almost unchanged large distances), but en- +courages nodes in the former to obtain more similar predic- +tions (i.e., smaller distances). As a byproduct of promoting +fairness, the distinction between connected and unconnected +node pairs is increased. Although we can explain the inter- +action between fairness and privacy intuitively, comprehen- +sively building trustworthy GNNs yearns for quantitatively +measuring this trade-off and balancing the performance, fair- +ness, and privacy of GNNs. +4 +Method +This section presents our method for promoting the fairness +of GNN models while restricting edge privacy risks. We first +introduce how to evaluate the influence of training samples +and then employ a correlation index to measure the inter- +action between fairness and privacy. Finally, we propose a +method that can restrict edge privacy risks when promoting +GNN fairness. +4.1 +The Influence of Training Samples +In this section, we first present how the weight of training +samples influences the parameters of models, and then dis- +cuss its influence on interested functions. +Influence of Training Samples on GNN Parameters +Generally, given a set of labelled nodes in graph G, we can +train a GCN model by minimising the following loss function +[Wu et al., 2022c]: +θ∗(1) = arg min +θ +� +v∈Vl +L(ˆyv, yv; θ), +(1) +where yv represents the ground truth label of node v ∈ Vl, +and ˆyv indicates the predicted label from the GCN model with +parameter θ. The 1 (i.e., all-one vector) here represents that +all nodes in Vl are treated equally during the training of the +GCN model. When changing the weight of training samples, +the obtained parameter can be expressed as +θ∗(1 + w) = arg min +θ +� +v∈Vl +(1 + wv)L(ˆyv, yv; θ), +(2) +where wv ∈ [−1, 1] and (1 + wv) represents the weight of +node v in the loss function when training a GCN model (e.g., +wv = −1 indicates leaving node v out of training phase). +To estimate θ∗(1 + w) without retraining the GCN model, +here we employ the influence function [Koh and Liang, 2017; +Kang et al., 2022] and Taylor expansion to conduct the fol- +lowing first-order approximation: +θ∗(1 + w) ≈ θ∗(1) + +� +v∈Vl +wvIθ∗(1)(v), +(3) +where Iθ∗(1)(v) = dθ∗(1) +dwv |wv=0 is the influence function with +respect to node v. According to the classical analysis [Koh +and Liang, 2017], it can be calculated as +Iθ∗(1)(v) = H−1 +θ∗(1)∇θL(ˆyv, yv; θ∗(1)), +(4) +where Hθ∗(1) = +1 +|Vl| +� +v∈Vl ∇2 +θL(ˆyv, yv; θ∗(1)) is the Hes- +sian matrix of loss function with respect to parameter θ∗(1). +Influence on Interested Functions +To study the behaviour of GNN models, different functions +have been proposed to evaluate GNN outputs. To evaluate + +Table 2: Effectiveness of frisk in evaluating link leakage risks on +Citeseer dataset. The risks of link leakage increase when promoting +fairness and frisk is effective in measuring these risk changes. +Acc +Bias +frisk +Cosi +Eucl +Corr +Vanilla +63.66 +0.0445 +7.99 +92.82 +92.68 +92.74 +Reg +63.11 +0.0301 +9.40 +94.31 +94.64 +94.00 +Reg +58.15 +0.005 +20.80 +96.89 +96.64 +95.04 +whether GNNs treat all users equally, individual fairness re- +quires that similar individuals are treated equally. As men- +tioned in Section 2, the existing bias [Kang et al., 2020] in a +GNN model can be measured by +fbias(θ) = Bias(Y, S) = Tr(YT LSY). +(5) +To steal edges in the training graph, existing attacks [He et +al., 2021] first calculate the distance of any two nodes with re- +spect to their GNN predictions, then categorise all distances +into two clusters with the KNN method. The authors pro- +pose to employ AUC scores between the connected and un- +connected node pairs to evaluate the vulnerability of GNNs. +In this paper, we use the following function to measure link +leakage risks of the training graph, i.e., +frisk(θ) = +|E[d0(ˆyi, ˆyj)] − E[d1(ˆyi, ˆyj)]| +(var(d0(ˆyi, ˆyj)) + var(d1(ˆyi, ˆyj)))/2, +(6) +where d(ˆyi, ˆyj) represents the distance between the GNN pre- +dictions on i-th and j-th node in graph G, and the subscript +in d0(·, ·)/d1(·, ·) (i.e., 0 or 1) indicates that the input comes +from unconnected/connected nodes. E[·] and var(·) represent +the mean and variance operations, respectively. We empir- +ically verify the effectiveness of frisk in measuring privacy +risks and only show its value on the Citeseer dataset in Table +2 due to limited space. +In this paper, we use the Taylor expansion of f with re- +spect to the parameters θ and the equation (3) to calculate the +influence of training samples on f. Specifically, we assume +that both fbias and frisk are induced from the parameter θ +of target GNN since the input of these functions is the final +prediction of target GNN. Therefore, the influence of train- +ing samples [Koh and Liang, 2017; Kang et al., 2022] on a +interested function f can be expressed as +If(w) ≈ ∇θf(θ∗(1))T [θ∗(1 + w) − θ∗(1)] +≈ ∇θf(θ∗(1))T +� � +v∈Vl +wvIθ∗(1)(v) +� +(7) +Remarks. In this paper, any derivable function that takes the +GNN prediction Y as input can be considered as f. For ex- +ample, f can be instantiated as the loss function that concerns +the utility (i.e., performance) of the target model [Li and Liu, +2022], i.e., +Iutil(w) = +� +v∈Vl +∇θL(ˆyv, yv; θ∗(1))T +� � +v∈Vl +wvIθ∗(1)(v) +� +. +(8) +4.2 +Measuring Interactions by Influence Functions +Measuring the interaction between fairness and privacy is not +trivial. This is because individual fairness is proposed from +the node view, while the privacy risk lies in the edge perspec- +tive. Thus, considering fairness and privacy in the same coor- +dinate space is essential for measuring their interaction when +building trustworthy GNNs. According to recent research [Li +and Liu, 2022], after the vanilla training of Neural Networks, +considering the influence of training samples on model fair- +ness and utility at the same time can achieve fairness at no +utility cost, which inspires us to measure the interaction be- +tween fairness and privacy. +First, we respectively calculate the influence on fbias and +frisk, which helps analyse their interaction at the same coor- +dinate space. According to Equation 7, we can estimate how +training samples impact fbias and frisk by calculating If(w). +Specifically, the influence on fairness can be expressed as fol- +lows, +Ifbias(w) = ∇θfbias(θ∗(1))T +� � +v∈Vl +wvIθ∗(1)(v) +� +, +(9) +and the influence on privacy risk can be calculated as +Ifrisk(w) = ∇θfrisk(θ∗(1))T +� � +v∈Vl +wvIθ∗(1)(v) +� +. +(10) +In this paper, we use wv to denote an all-zero vector except +wv is -1. Thus, If(wv) represents leaving node v out of the +training of a GCN model, i.e., +If(wv) = −∇θf(θ∗(1))T H−1 +θ∗(1)∇θL(ˆyv, yv; θ∗(1)). +(11) +Concatenating all If(wv) (v ∈ Vl, f = fbias/risk) together, +we obtain the influence vector Ifbias/risk ∈ R1×|Vl|. +Next, we employ the Pearson correlation coefficient r be- +tween Ifbias and Ifrisk to measure the interactions between +fairness and privacy. Specifically, +r = Pearson(Ifbias, Ifrisk), +(12) +where r ∈ [−1, 1]. In the context of measuring interactions +between fairness and privacy, r = 1 represents a mutual pro- +motion, while r = −1 indicates there is a conflict between +them. +4.3 +Boosting Fairness with Restricted Privacy +Risks +Building trustworthy GNNs requires considering perfor- +mance and several aspects of trustworthiness simultaneously +[Zhang et al., 2022], however, taking performance, fairness, +and privacy into consideration at the same time is not trivial +[Gu et al., 2022]. First, the existing literature shows that both +fairness [Dong et al., 2021] and privacy [Wu et al., 2022b] +of GNNs are at the cost of performance. Furthermore, our +empirical study (i.e., Tables 1 and 2) shows that promoting +fairness (i.e., reducing bias) in GNN predictions results in the +increase of privacy risk of edges in the training graph. +In this paper, we argue that the performance of GNNs lies +in their central position when they serve users. Our goal is + +to devise a method that can boost fairness with limited per- +formance costs and restricted privacy risks. Next, we will +present our approach to satisfying this design goal, including +fairness-aware re-weighting and privacy-aware perturbations. +Fairness-aware Re-weighting. +Inspired by the previous study [Li and Liu, 2022], we pro- +pose to re-weight the training samples to boost fairness with +a limited performance cost. Specifically, the weight can be +obtained by solving the following Quadratically Constrained +Linear Programming (QCLP), +min +� +v∈Vl +wvIfbias(wv) +s.t. +� +v∈Vl +w2 +v ≤ α|Vl|, +� +v∈Vl +wvIfutil(wv) ≤ β � I+ +futil(wv), +wv ∈ [−1, 1]. +(13) +In this optimisation, wv ∈ [−1, 1] is the variable. The objec- +tive tends to minimise the total bias existing in GNN predic- +tions. The first constraint is designed to control the degree of +re-weighting. The second constraint represents that obtained +weights only cost limited model utility, where I+ +futil(wv) in- +dicates Ifutil(wv) with positive value. α and β are hyperpa- +rameters. In this paper, after solving this QCLP with Gurobi +optimiser [Gur, 2022], the obtained weight of training sample +wfair = [w1, w2, ..., w|Vl|] can be engaged into re-training of +the target GNN model with weighted loss in equation (2). +Privacy-aware Perturbations +Given the proposed re-weighting method, a straightforward +approach of taking privacy and fairness into account simulta- +neously is that involving an item that concerns Ifrisk into the +QCLP (e.g., playing a role as a constraint or part of the objec- +tive). However, the negative correlation between Ifbias and +Ifrisk leads to the weak effectiveness of this simple method. +Hence, it is necessary to locate the cause of edge leakage risks +and devise a method to restrain it. Next, we will first present +the analysis of link-stealing attacks, and then show our per- +turbation method for inhibiting leakage risks. +(1) Modeling Edge Leakage Risks. In this paper, we take +the most practical link-stealing attacks (i.e., Attack-0 in [He +et al., 2021], black-box setting) as our object. Specifically, +in these attacks, after querying the target GNN, attackers +can calculate a link score to infer if two nodes are con- +nected based on their prediction similarity. Instead of directly +analysing privacy risks in the link score space, we will study +it in the embedding space and show how the existence of an +edge impacts the learned embedding. +In the embedding space, a well-trained GCN model clus- +ters similar nodes together so that they have similar predic- +tions. According to a previous study [Pan et al., 2018], we +assume the learned node embedding follows the normal dis- +tribution. For simplicity, we employ the left normalisation +ˆA = ˜D−1( A + I) in GCN models and consider the bi- +nary node classification task. Assuming that µi and σ indi- +cate the mean and standard deviation of the node embedding +from class yi (i = 0, 1) at the t-th layer, we obtain that the +learned embedding E(t,yi) ∼ N(µi, σ2). +In this paper, we focus on analysing the intra-class +node pairs (i.e., nodes with the same label), since they +are the majority of all node pairs. +In a GCN model, +edges are involved in the one-hop mean-aggregation op- +eration, i.e., ˆAE. +From the view of an individual node +vi, the one-hop mean-aggregation operation is [ ˆAE(t)]i = +1 +di+1(E(t) +i ++ � +vj ∈N (vi) +mj=0 +E(t,y0) +j ++ � +vj ∈N (vi) +mj=1 +E(t,y1) +j +), where +N(vi) represents the neighbour node set of vi, and mi = +1 indicates vi is from class y1, otherwise mn += +0. +Given m += +[m1, ..., m|V|], E(t,y1) += +diag(m)E(t), +E(t,y0) += +(I − diag(m))E(t), and E(t) += +E(t,y0) + +E(t,y1). Therefore, � +vj∈N (vi) +mj=0 +E(t,y0) +j +∼ N(dy0 +i µ0, dy0 +i σ2) +and � +vj∈N (vi) +mj=1 +E(t,y1) +j +∼ N(dy1 +i µ1, dy1 +i σ2), where dy0/y1 +i +represent the number of nodes from Ey0/y1 and di = dy0 +i ++ +dy1 +i +indicate the degree of vi. Without loss of generality, we +assume that vi and vj in the node pair (vi, vj) come from +class 0. +As shown below, the node distance sensitivity in embed- +ding space can be calculated as the difference between cases +where vi and vj are connected or unconnected. +Case 0: When vi and vj is unconnected, +� +ˆAE(t)�0 +i and +� +ˆAE(t)�0 +j can be approximately expressed as +� +ˆAE(t)�0 +i ≈ +1 +di + 1 +� +E(t) +i ++ dy0 +i µ0 + dy1 +i µ1 +� +, +� +ˆAE(t)�0 +j ≈ +1 +dj + 1 +� +E(t) +j ++ dy0 +j µ0 + dy1 +j µ1 +� +. +(14) +Case 1: +When vi and vj is connected, +� +ˆAE(t)�1 +i and +� +ˆAE(t)�1 +j can be approximately expressed as +� +ˆAE(t)�1 +i ≈ +1 +di + 2 +� +E(t) +i ++ E(t) +j ++ dy0 +i µ0 + dy1 +i µ1 +� +, +� +ˆAE(t)�1 +j ≈ +1 +dj + 2 +� +E(t) +j ++ E(t) +i ++ dy0 +j µ0 + dy1 +j µ1 +� +. +(15) +Given (14) and (15), the distance of vi and vj in the embed- +ding space is calculated as +d0(vi, vj) = +� +ˆAE(t)�0 +i − +� +ˆAE(t)�0 +j , +d1(vi, vj) = +� +ˆAE(t)�1 +i − +� +ˆAE(t)�1 +j . +(16) +Thus, the sensitivity of d(vi, vj) with respect to the existence +of edge eij is +∆d(vi, vj) =∥d0(vi, vj) − d1(vi, vj)∥ += +������� +� +ˆAE(t)�0 +i − E(t) +j +di + 2 +− +� +ˆAE(t)�0 +j − E(t) +i +dj + 2 +������� +, +(17) + +Vanilla Training +GNN +Predictions +Privacy-aware +Perturbations +Influence +Functions +Quadratically Constrained +Linear Programming +Fairness-aware +Re-weighting +Re-training +Introducing +Heterogeneous Edges +GNN +Model +Perturbated Graph Structure +Figure 1: The Framework of Our Privacy-aware Perturbations and Fairness-aware Re-weighting (PPFR) Method. After the vanilla training of +a GNN model, the privacy-aware module introduces heterophily edges to generate the perturbed graph structure, fairness-aware re-weighting +module employs the influence function and quadratically constrained linear programming to obtain fairness-aware sample weights. After +that, the perturbed graph structure and fairness-aware sample weights are involved in the re-training of the target GNN model to promote +fairness with limited performance cost and restricted privacy risks. +and its expectation is E [∆d(vi, vj)] = ∥(µ1 − µ0)δ∥, where +δ = +dy1 +i +(di+1)(di+2) − +dy1 +j +(dj+1)(dj+2), due to +E +�� +ˆAE(t)�0 +i − E(t) +j +� += (1 + dy0 +i )µ0 + dy1 +i µ1 +di + 1 +− µ0 += dy1 +i (µ1 − µ0) +di + 1 +, +E +�� +ˆAE(t)�0 +j − E(t) +i +� += +(1 + dy0 +j )µ0 + dy1 +j µ1 +dj + 1 +− µ0 += +dy1 +j (µ1 − µ0) +dj + 1 +. +(18) +(2) Perturbation-based Method. Although our modeling +process cannot fully depict privacy risks, this coarse grain +analysis provides us with insights into understanding risk +sources and designing methods to restrict edge leakage. For +example, the item µ0 − µ1 in E [∆d(vi, vj)] indicates a GNN +model with higher performance (i.e., higher discrimination +when owning a larger µ0 − µ1 value) has higher edge leak- +age risks due to the homophily of graph data (i.e., connected +nodes are more likely to have similar attributes and the same +label). Another item δ implies that, for nodes with similar +degree values, the heterophily difference between nodes is +positively correlated to the privacy risk. +According to these insights, we introduce heterophily edge +noises into the graph structure to reduce edge leakage risks +with a limited performance cost. Specifically, after the vanilla +training (i.e., with (1)) of target models, we employ its pre- +dictions on all nodes to add heterophily neighbours for each +node, i.e., +N(vi) = N(vi) ∪ {vi1, ..., vik}, +(19) +where the GNN prediction on v ∈ {vi1, ..., vik} is different +from that on vi, k = γ|N(vi)| and γ is a hype-parameter. +With Equation (19) we obtain the perturbed adjacency matrix +A′, which is involved in the retraining of target GNNs. In +addition to reducing heterophily differences between nodes +of the same class (i.e., δ in E [∆d(vi, vj)]), the introduction of +heterophily edges can help close the distance of nodes across +classes (i.e., µ0 − µ1 in E [∆d(vi, vj)]). +Target Model Re-training +Fig. 1 shows the whole framework of our Privacy-aware Per- +turbations and Fairness-aware Re-weighting (PPFR) method, +whose goal is to promote fairness with limited performance +costs and restricted privacy risks. The entire training of the +target model includes two phases: vanilla training and PPFR +retraining. Specifically, we first conduct vanilla training to +obtain a competent GNN model. +After that, we continue +to train (i.e., re-training) the target GNN with the perturbed +graph structure and weighted loss function derived from the +fairness-aware weights. In PPFR, the epoch number of re- +training phase ere is defined as ere = seva, where eva is +the epoch number of vanilla training and s = 0.1 is a hyper- +parameter. +5 +Experiments +In this section, we conduct experiments to evaluate our pro- +posed PPFR method. +We first introduce the experimental +setup, followed by our experimental results and discussion. +5.1 +Experimental Setup +Datasets, Models, and Metrics. The datasets, model setup, +and fairness and privacy metrics follow that in Section 3.1. +Note that the privacy result in this section is the average AUC +derived from 8 different distances. Moreover, we use the fol- +lowing metric to evaluate the effect of a method Ω on both + +L(y, Yu; 0 +arg min +0 +VEViWfair +[W1, W2, ..., WIVilarg min +0 +UEViTable 3: Effectiveness of our PPFR method and Correlation between +Ifbias and Ifrisk. +Datasets +∆bias ↓ +∆risk ↓ +∆ ↑ +r +Reg +-35.51 +1.80 +-7.44×10−1 +DPReg +29.11 +-17.07 +-1.87×10−1 +DPFR +-1.17 +0.34 +-6.55×10−3 +Cora +PPFR +-11.75 +-0.73 +1.46×10−2 +-0.66 +Reg +-32.36 +1.91 +-7.17×10−1 +DPReg +290.34 +-14.95 +-1.71×100 +DPFR +0.22 +-0.08 +-7.01×10−4 +Citeseer +PPFR +-14.16 +-0.31 +8.97×10−3 +-0.51 +Reg +-84.70 +3.54 +-1.28×100 +DPReg +-64.45 +-32.71 +8.81×10−1 +DPFR +-29.60 +0.94 +-1.98×10−1 +Pubmed +PPFR +-31.30 +-0.18 +1.25×10−2 +-0.41 +* In this table, columns ∆bias and ∆risk show evaluation results +in the percentage (i.e.,%) form. In the ∆ column, blue colour +represents desired results, while red colour indicates undesired +results. “r” in the last column indicates the Pearson correlation +coefficient between Ifbias and Ifrisk. +fairness and privacy, i.e., +∆ = ∆bias∆risk +|∆acc| +, +(20) +where ∆(·) = +Ω(·)−w/o(·) +w/o(·) +takes bias/risk/accuracy values of +methods Ω and w/o as inputs to evaluate the change ratio on +bias/risk/accuracy when using method Ω, w/o represents the +GNN model obtained by vanilla training. According to the +definition, ∆ measures the cost-effectiveness with respect to +GNN performance when promoting both fairness and privacy. +A positive ∆ indicates that Ω can boost fairness and privacy +simultaneously, otherwise ∆ is negative. +Baselines. To verify the effectiveness of our method (i.e., +PPFR), we combine privacy and fairness methods together +as the baseline methods. In this paper, we consider differen- +tial privacy (DP) (i.e., EdgeRand and LapGraph [Wu et al., +2022b]) as the method to improve edge privacy of GNNs, +and both EdgeRand and LapGraph are engaged in generating +perturbed adjacency matrix. Specifically, to boost edge pri- +vacy, EdgeRand/LapGraph uses a randomisation/Laplacian +mechanism to introduce edge noises into the original graph +structure. +We follow the parameter setting in [Wu et al., +2022b] to introduce ϵ-edge DP (i.e., EdgeRand/LapGraph) +into our evaluation. According to the previous study [Wu et +al., 2022b], EdgeRand and LapGraph have similar effective- +ness when ϵ is small, while LapGraph is more applicable to +large graphs. Thus, we apply EdgeRand on Cora and Citeseer +datasets and LapGraph on the Pubmed dataset. +In this paper, Reg represents that the fairness regularisa- +tion is introduced into the loss function of GNN models dur- +ing their vanilla training. DPReg represents using the edge +DP method and adding the fairness regularisation into loss si- +multaneously. DPFR indicates the combination of edge DP +and our fairness-aware re-weighting (FR) in this paper, where +a perturbed graph is engaged in the retraining phase. +5.2 +Experimental Results +As shown in Table 3, we evaluate the effectiveness of +our Privacy-aware Perturbations and Fairness-aware Re- +weighting (PPFR) method in boosting fairness with limited +performance costs and restricted privacy risks. Results in Ta- +ble 3 include two parts: correlation results between Ifbias +and Ifrisk, and comparison between our PPFR and baseline +methods. +Correlation. In the last column of Table 3, we quantitatively +measure the correlation between fairness and privacy with our +proposed index (i.e., Eq. 12). The negative correlation results +between Ifbias and Ifrisk show that there exists a negative +correlation between node individual fairness and edge leak- +age risks. These correlation results are consistent with our +observations (i.e., the trade-off between fairness and privacy) +in Table 1, and also underpin the design of our PPFR method, +that is, it does not involve Ifbias and Ifrisk simultaneously in +linear programming. +Effectiveness. Other evaluation results also verify the effec- +tiveness of our PPFR method. (1) Due to involving GNN +performance into consideration, our PPFR method can main- +tain the same level of performance as vanilla GNNs on all +datasets (i.e., 81.09, 60.50, 81.57 on Cora, Citeseer, and +Pubmed, respectively), which is vital in serving GNN users. +However, the low performance of DPReg (i.e., 63.26, 47.47, +and 64.95 on three datasets, respectively) potentially leads to +the impracticability of GNN systems. (2) When comparing +the DPFR and PPFR methods, our privacy-aware perturba- +tion (PP) method is more effective than the edge DP meth- +ods. Although edge DP (i.e., DPFR rows) is effective in con- +trolling edge leakage risks compared to current fairness pro- +motion methods (e.g., Reg rows), it still poses a higher pri- +vacy risk than vanilla GNNs. In contrast, our method is more +effective since it presents equal even lower privacy risks in +almost all cases, which indicates PPFR can achieve fairness +with non-negative privacy impacts. (3) According to the ∆ re- +sults, PPFR is the only method that has a positive sign across +all datasets, indicating that it can boost fairness and privacy +while maintaining competent accuracy. +6 +Conclusion +In this paper, we investigate the interaction between the fair- +ness and privacy of GNNs, which is indispensable in compre- +hensively building trustworthy GNNs. We empirically ob- +serve the adverse effects of node fairness on edge privacy +risks and propose to quantitatively measure their trade-off +through influence functions and Pearson correlation. Finally, +we devise a retraining method to increase GNN fairness with +limited performance cost and restricted privacy risk, whose +effectiveness is demonstrated by our experimental evalua- +tions on real-world datasets. In the future, we will conduct a +theoretical analysis between the fairness and privacy of GNNs +and explore other methods to balance the different aspects of +trustworthy GNNs. + +References +[Agarwal et al., 2021] Chirag +Agarwal, +Himabindu +Lakkaraju, and Marinka Zitnik. +Towards a unified +framework for fair and stable graph representation +learning. In UAI, pages 2114–2124. PMLR, 2021. +[Chang and Shokri, 2021] Hongyan Chang and Reza Shokri. +On the privacy risks of algorithmic fairness. In EuroS&P, +pages 292–303. IEEE, 2021. +[Dong et al., 2021] Yushun Dong, Jian Kang, Hanghang +Tong, and Jundong Li. Individual fairness for graph neu- +ral networks: A ranking based approach. In KDD, pages +300–310. 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AAAI Press, 2021. + diff --git a/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/load_file.txt b/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b188cfffd03fc69f8da6f8c70c42b915b48ae065 --- /dev/null +++ b/PtFOT4oBgHgl3EQf4zQ-/content/tmp_files/load_file.txt @@ -0,0 +1,679 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf,len=678 +page_content='On the Interaction between Node Fairness and Edge Privacy in Graph Neural Networks He Zhang1 , Xingliang Yuan1 , Quoc Viet Hung Nguyen2 and Shirui Pan3 1Faculty of Information Technology, Monash University 2Institute for Integrated and Intelligent Systems, Griffith University 3School of Information and Communication Technology, Griffith University {he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='zhang1, xingliang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='yuan}@monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='edu, {henry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='nguyen, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='pan}@griffith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='au Abstract Due to the emergence of graph neural networks (GNNs) and their widespread implementation in real-world scenarios, the fairness and privacy of GNNs have attracted considerable interest since they are two essential social concerns in the era of building trustworthy GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Existing studies have respectively explored the fairness and privacy of GNNs and exhibited that both fairness and privacy are at the cost of GNN performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' However, the interaction between them is yet to be explored and understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we investigate the in- teraction between the fairness of a GNN and its privacy for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We empirically identify that edge privacy risks increase when the individ- ual fairness of nodes is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Next, we present the intuition behind such a trade-off and employ the influence function and Pearson correlation to mea- sure it theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To take the performance, fair- ness, and privacy of GNNs into account simultane- ously, we propose implementing fairness-aware re- weighting and privacy-aware graph structure per- turbation modules in a retraining mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Ex- perimental results demonstrate that our method is effective in implementing GNN fairness with lim- ited performance cost and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 1 Introduction In recent years, in addition to competent performance, there has been an increasing desire for fairness [Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] and private infor- mation security [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] in GNNs, because both privacy and fairness are two essential concerns of GNN users [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the context of GNNs, existing studies [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020] only focus on performance and privacy/fairness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' however, privacy and fairness are not isolated from each other in practical scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For example, in e-commerce platforms [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2019], where users and items are considered as nodes and their interactions are regarded as edges in a graph, a recommender system should serve all users with compe- tent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Meanwhile, user bias [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022d] should be avoided and similar users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', nodes) should re- ceive similar item recommendations to boost individual fair- ness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' the purchase records of a customer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', edges between nodes) should not be exposed to others without permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Therefore, it is necessary to study the interaction among per- formance, fairness, and privacy to build trustworthy GNNs comprehensively [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Existing literature has studied the interaction between per- formance and fairness/privacy of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For example, to boost individual fairness in GNNs, a method called RE- DRESS [Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] proposes to add a regularisation term concerning fairness from the ranking perspective into the loss function of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Moreover, InFoRM [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020] uses the Lipschitz property and proposes a metric to evaluate the individual fairness of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' This metric can also be involved in the loss function of a target model to reduce the bias existing in the GNN predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To promote edge pri- vacy, for instance, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b] explore the vulnerability of edges in the training graph and introduce dif- ferential privacy (DP) mechanisms [Sajadmanesh and Gatica- Perez, 2021] to protect edges from leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' These studies also demonstrate that both individual fairness and edge pri- vacy are at cost of GNN performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To comprehensively build trustworthy GNNs, it is in- evitable to study the interaction between fairness and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Currently, a few works have explored the impact of improv- ing model privacy on fairness [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] or promot- ing algorithm fairness on privacy [Chang and Shokri, 2021], which are in the context of general machine learning models for Independent Identically Distribution (IID) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In con- trast, this paper will study the interaction of node fairness and edge privacy of GNNs for complex graph data, which has not yet been explored and measured to the best of our knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Moreover, given the trade-off between fairness/privacy and performance and unexplored interaction between fairness and privacy, a challenging research topic of building trustwor- thy GNNs is how to ensure the privacy and fairness of a GNN model simultaneously with keeping competent model perfor- mance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', with limited performance cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we empirically verify the adverse effect of individual fairness of nodes on edge privacy in GNN mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To understand this observation, we employ the influ- ence functions of training samples to measure and explain this trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Furthermore, we propose a Privacy-aware Per- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='12951v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='LG] 30 Jan 2023 turbations and Fairness-aware Re-weighting (PPFR) method to implement GNN fairness with limited performance cost and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The contributions of this paper are summarised as follows: In this paper, we explore the interaction between fair- ness and privacy of GNN for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We empiri- cally show that the edge privacy risk increases when en- hancing individual fairness of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', there exists a trade-off between fairness and privacy of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To understand GNN behaviours concerning different trustworthiness aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', fairness and privacy), we propose employing the influence function and Pearson correlation coefficient to quantitatively measure the in- teraction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', trade-off) between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Based on a re-weighting method and a graph structure perturbation method, we propose a novel method to de- vise competent GNNs with reduced bias and edge leak- age risks, whose effectiveness has been demonstrated by our experimental evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 2 Background Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' A graph G = {V, E} includes a node set V = � v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' , v|V| � and an edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' E characterises the rela- tionship information in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The set of edges can also be de- noted by an adjacency matrix A ∈ {0, 1}|V|×|V|, in which Ai,j = 1 when eij = (vi, vj) ∈ E, otherwise Ai,j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Matrix X ∈ R|V|×k (k indicates the dimensionality of fea- tures) denotes node features, the i-th row of X represents the feature of node vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Without loss of generality, another de- scription form of a graph is G = {A, X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we focus on the undirected graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Ai,j = Aj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Node Classification and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For a graph G = {V, E}, the set of labelled nodes is denoted by Vl ⊂ V, where yv is the label of v ∈ Vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The set of unlabelled nodes in G is indicated by Vu = V \\ Vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Given G and node labels, node classifica- tion aims to train a GNN model f, which can predict labels for nodes in Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we consider the graph con- volutional network (GCN) model [Kipf and Welling, 2017], which is a typical GNN model for node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Given a L layer GCN model, we assume that E(l) and W(l) rep- resent the output node embeddings and the weight matrix of the l-th hidden layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The graph convolution at the l-th layer can be formulated as E(l) = σ( ˆAE(l−1)W(l)), where σ is the nonlinear activation, ˆA = ˜D− 1 2 ( A + I) ˜D− 1 2 , and ˜D being the degree matrix of (A + I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Individual Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To be fair to all users, GNNs should ensure that everyone is treated equally and receives the same quality of service regardless of their background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the con- text of node classification, individual fairness requires that any two similar nodes receive similar GNN predictions [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, given the similarity matrix S of nodes and GNN predictions Y, the bias with respect individ- ual fairness is measured by Bias(Y, S) = Tr(YT LSY), where YT represents the transpose of Y, LS indicates the Laplacian matrix of S [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To improve indi- vidual fairness of GNNs, a method called InFoRM proposed to involve Bias(Y, S) into the loss function during the train- ing phase of GNNs [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Link Stealing Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Existing studies [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b] have demonstrated that attackers are ca- pable of inferring the existence of a link between any two nodes in the training graph of a GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For example, by querying node predictions of the target GNN model, at- tackers can use the prediction similarity of node pairs to in- fer whether two nodes are connected in a specific node pair [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' This attack is based on the intuition that if two nodes share more similar predictions from the target GNN model then there is a greater likelihood of the nodes being linked together [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Currently, exist- ing methods employ the AUC score to evaluate the vulner- ability of a GNN model to link-stealing attacks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', leak- age risk of edges in the training graph [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 3 Interaction Between Fairness and Privacy: A Preliminary Study This section presents a preliminary study on the node clas- sification task to assess the effect of promoting individual fairness on the risk of edge leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Introducing the item concerning fairness into the loss function of a GNN model is effective in reducing bias, while it potentially affects the privacy risk of edges in the training graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='1 Preliminary Study Settings Datasets and Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In our preliminary experiments, we employ Cora [Kipf and Welling, 2017], Citeseer [Kipf and Welling, 2017], and Pubmed [Kipf and Welling, 2017] datasets, which are commonly used in evaluating GNNs in node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Models selected for this study are GCNs [Kipf and Welling, 2017] with 16 hidden layers that employ ReLU and softmax activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We use accuracy as the metric for evaluating the performance of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Following previous studies [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Chang and Shokri, 2021], we combine the Bias(Y, S) and original loss function together in the training phase to pro- mote fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, the similarity matrix S is defined as the Jaccard index [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020], and the bias in GNN predictions is measured by Bias(Y, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The smaller the bias value, the fairer the GNN prediction Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we assume that attackers can only query target models to obtain node predictions from target GNNs, which is the most practical link-stealing attack (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Attack-0 in [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Based on prediction similarity, attackers attempt to infer the existence of an edge in any node pairs in the training graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The edge leakage risk is measured by AUC (area under the ROC curve) score, where larger val- ues indicate higher privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Following a previous study [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021], the prediction similarity is calculated using Cosine, Euclidean, Correlation, Chebyshev, Braycurtis, Can- berra, Cityblock and Sqeuclidean distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='2 Observations In our preliminary studies, we focus on the change of pri- vacy risk on edges when boosting individual fairness in GNN Table 1: Comparison of the accuracy, bias, and privacy risks of GCN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Datasets Acc↑ Bias↓ Privacy Risks ↓ Cosi Eucl Corr Cheb Bray Canb City Sqeu Cora Vanilla 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0766 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='34 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='98 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='13 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='41 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='57 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='41 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='57 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='98 Reg 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0494 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='42 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='67 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='14 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='73 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='10 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='81 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='10 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='67 Citeseer Vanilla 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0445 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='82 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='68 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='74 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='00 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='10 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='15 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='10 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='68 Reg 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0301 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='31 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='64 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='00 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='77 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='03 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='10 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='03 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='64 Pubmed Vanilla 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0706 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='59 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='26 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='04 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='37 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='37 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='85 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='37 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='26 Reg 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0108 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='66 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='90 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='44 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='95 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='95 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='67 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='95 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='90 In this table, “Vanilla” represents the generic training of GNNs, and “Reg” indicates that the fairness regularisation is introduced into the loss function of GNNs during their vanilla training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' “Cosi”, “Eucl”, “Corr”, “Cheb”, “Bray”, “Canb”, “City”, and “Sqeu” represent the Cosine, Euclidean, Correlation, Chebyshev, Braycurtis, Canberra, Cityblock and Sqeuclidean distances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' As shown in Table 1, we observe that boost- ing fairness comes at the cost of model performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', the prediction accuracy is sacrificed when improving bias on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' However, performance reduction is not the only ad- verse effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Changes in AUC scores indicate that edge leak- age risks increase when GNN fairness is promoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' This phenomenon can be explained by the definition of S and its different influences on different node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In ho- mophily graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Cora, Citeseer, Pubmed), similar nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', nodes with the same label) are more likely to connect to each other [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In these graphs, calculating Jaccard similarity [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020] be- tween nodes leads S to assign higher values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 1) for node pairs in which two nodes own a higher proportion of the same 1-hop neighbour nodes, while lower values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 0) for other node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Therefore, boosting individual fairness based on Jaccard similarity has limited even zero influence on the latter node pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', almost unchanged large distances), but en- courages nodes in the former to obtain more similar predic- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', smaller distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' As a byproduct of promoting fairness, the distinction between connected and unconnected node pairs is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Although we can explain the inter- action between fairness and privacy intuitively, comprehen- sively building trustworthy GNNs yearns for quantitatively measuring this trade-off and balancing the performance, fair- ness, and privacy of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 4 Method This section presents our method for promoting the fairness of GNN models while restricting edge privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We first introduce how to evaluate the influence of training samples and then employ a correlation index to measure the inter- action between fairness and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Finally, we propose a method that can restrict edge privacy risks when promoting GNN fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='1 The Influence of Training Samples In this section, we first present how the weight of training samples influences the parameters of models, and then dis- cuss its influence on interested functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Influence of Training Samples on GNN Parameters Generally, given a set of labelled nodes in graph G, we can train a GCN model by minimising the following loss function [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022c]: θ∗(1) = arg min θ � v∈Vl L(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ), (1) where yv represents the ground truth label of node v ∈ Vl, and ˆyv indicates the predicted label from the GCN model with parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', all-one vector) here represents that all nodes in Vl are treated equally during the training of the GCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' When changing the weight of training samples, the obtained parameter can be expressed as θ∗(1 + w) = arg min θ � v∈Vl (1 + wv)L(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ), (2) where wv ∈ [−1, 1] and (1 + wv) represents the weight of node v in the loss function when training a GCN model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', wv = −1 indicates leaving node v out of training phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To estimate θ∗(1 + w) without retraining the GCN model, here we employ the influence function [Koh and Liang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022] and Taylor expansion to conduct the fol- lowing first-order approximation: θ∗(1 + w) ≈ θ∗(1) + � v∈Vl wvIθ∗(1)(v), (3) where Iθ∗(1)(v) = dθ∗(1) dwv |wv=0 is the influence function with respect to node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to the classical analysis [Koh and Liang, 2017], it can be calculated as Iθ∗(1)(v) = H−1 θ∗(1)∇θL(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ∗(1)), (4) where Hθ∗(1) = 1 |Vl| � v∈Vl ∇2 θL(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ∗(1)) is the Hes- sian matrix of loss function with respect to parameter θ∗(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Influence on Interested Functions To study the behaviour of GNN models, different functions have been proposed to evaluate GNN outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To evaluate Table 2: Effectiveness of frisk in evaluating link leakage risks on Citeseer dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The risks of link leakage increase when promoting fairness and frisk is effective in measuring these risk changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Acc Bias frisk Cosi Eucl Corr Vanilla 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0445 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='99 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='82 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='68 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='74 Reg 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='0301 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='40 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='31 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='64 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='00 Reg 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='005 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='80 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='89 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='64 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='04 whether GNNs treat all users equally, individual fairness re- quires that similar individuals are treated equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' As men- tioned in Section 2, the existing bias [Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2020] in a GNN model can be measured by fbias(θ) = Bias(Y, S) = Tr(YT LSY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (5) To steal edges in the training graph, existing attacks [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] first calculate the distance of any two nodes with re- spect to their GNN predictions, then categorise all distances into two clusters with the KNN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The authors pro- pose to employ AUC scores between the connected and un- connected node pairs to evaluate the vulnerability of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we use the following function to measure link leakage risks of the training graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', frisk(θ) = |E[d0(ˆyi, ˆyj)] − E[d1(ˆyi, ˆyj)]| (var(d0(ˆyi, ˆyj)) + var(d1(ˆyi, ˆyj)))/2, (6) where d(ˆyi, ˆyj) represents the distance between the GNN pre- dictions on i-th and j-th node in graph G, and the subscript in d0(·, ·)/d1(·, ·) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 0 or 1) indicates that the input comes from unconnected/connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' E[·] and var(·) represent the mean and variance operations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We empir- ically verify the effectiveness of frisk in measuring privacy risks and only show its value on the Citeseer dataset in Table 2 due to limited space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we use the Taylor expansion of f with re- spect to the parameters θ and the equation (3) to calculate the influence of training samples on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, we assume that both fbias and frisk are induced from the parameter θ of target GNN since the input of these functions is the final prediction of target GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Therefore, the influence of train- ing samples [Koh and Liang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022] on a interested function f can be expressed as If(w) ≈ ∇θf(θ∗(1))T [θ∗(1 + w) − θ∗(1)] ≈ ∇θf(θ∗(1))T � � v∈Vl wvIθ∗(1)(v) � (7) Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, any derivable function that takes the GNN prediction Y as input can be considered as f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For ex- ample, f can be instantiated as the loss function that concerns the utility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', performance) of the target model [Li and Liu, 2022], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Iutil(w) = � v∈Vl ∇θL(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ∗(1))T � � v∈Vl wvIθ∗(1)(v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='2 Measuring Interactions by Influence Functions Measuring the interaction between fairness and privacy is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' This is because individual fairness is proposed from the node view, while the privacy risk lies in the edge perspec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Thus, considering fairness and privacy in the same coor- dinate space is essential for measuring their interaction when building trustworthy GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to recent research [Li and Liu, 2022], after the vanilla training of Neural Networks, considering the influence of training samples on model fair- ness and utility at the same time can achieve fairness at no utility cost, which inspires us to measure the interaction be- tween fairness and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' First, we respectively calculate the influence on fbias and frisk, which helps analyse their interaction at the same coor- dinate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to Equation 7, we can estimate how training samples impact fbias and frisk by calculating If(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, the influence on fairness can be expressed as fol- lows, Ifbias(w) = ∇θfbias(θ∗(1))T � � v∈Vl wvIθ∗(1)(v) � , (9) and the influence on privacy risk can be calculated as Ifrisk(w) = ∇θfrisk(θ∗(1))T � � v∈Vl wvIθ∗(1)(v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (10) In this paper, we use wv to denote an all-zero vector except wv is -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Thus, If(wv) represents leaving node v out of the training of a GCN model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', If(wv) = −∇θf(θ∗(1))T H−1 θ∗(1)∇θL(ˆyv, yv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' θ∗(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (11) Concatenating all If(wv) (v ∈ Vl, f = fbias/risk) together, we obtain the influence vector Ifbias/risk ∈ R1×|Vl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Next, we employ the Pearson correlation coefficient r be- tween Ifbias and Ifrisk to measure the interactions between fairness and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, r = Pearson(Ifbias, Ifrisk), (12) where r ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the context of measuring interactions between fairness and privacy, r = 1 represents a mutual pro- motion, while r = −1 indicates there is a conflict between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='3 Boosting Fairness with Restricted Privacy Risks Building trustworthy GNNs requires considering perfor- mance and several aspects of trustworthiness simultaneously [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022], however, taking performance, fairness, and privacy into consideration at the same time is not trivial [Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' First, the existing literature shows that both fairness [Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] and privacy [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b] of GNNs are at the cost of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Furthermore, our empirical study (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Tables 1 and 2) shows that promoting fairness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', reducing bias) in GNN predictions results in the increase of privacy risk of edges in the training graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we argue that the performance of GNNs lies in their central position when they serve users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Our goal is to devise a method that can boost fairness with limited per- formance costs and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Next, we will present our approach to satisfying this design goal, including fairness-aware re-weighting and privacy-aware perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Fairness-aware Re-weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Inspired by the previous study [Li and Liu, 2022], we pro- pose to re-weight the training samples to boost fairness with a limited performance cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, the weight can be obtained by solving the following Quadratically Constrained Linear Programming (QCLP), min � v∈Vl wvIfbias(wv) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' � v∈Vl w2 v ≤ α|Vl|, � v∈Vl wvIfutil(wv) ≤ β � I+ futil(wv), wv ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (13) In this optimisation, wv ∈ [−1, 1] is the variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The objec- tive tends to minimise the total bias existing in GNN predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The first constraint is designed to control the degree of re-weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The second constraint represents that obtained weights only cost limited model utility, where I+ futil(wv) in- dicates Ifutil(wv) with positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' α and β are hyperpa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, after solving this QCLP with Gurobi optimiser [Gur, 2022], the obtained weight of training sample wfair = [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', w|Vl|] can be engaged into re-training of the target GNN model with weighted loss in equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Privacy-aware Perturbations Given the proposed re-weighting method, a straightforward approach of taking privacy and fairness into account simulta- neously is that involving an item that concerns Ifrisk into the QCLP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', playing a role as a constraint or part of the objec- tive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' However, the negative correlation between Ifbias and Ifrisk leads to the weak effectiveness of this simple method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Hence, it is necessary to locate the cause of edge leakage risks and devise a method to restrain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Next, we will first present the analysis of link-stealing attacks, and then show our per- turbation method for inhibiting leakage risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (1) Modeling Edge Leakage Risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we take the most practical link-stealing attacks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Attack-0 in [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021], black-box setting) as our object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, in these attacks, after querying the target GNN, attackers can calculate a link score to infer if two nodes are con- nected based on their prediction similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Instead of directly analysing privacy risks in the link score space, we will study it in the embedding space and show how the existence of an edge impacts the learned embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the embedding space, a well-trained GCN model clus- ters similar nodes together so that they have similar predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to a previous study [Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2018], we assume the learned node embedding follows the normal dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For simplicity, we employ the left normalisation ˆA = ˜D−1( A + I) in GCN models and consider the bi- nary node classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Assuming that µi and σ indi- cate the mean and standard deviation of the node embedding from class yi (i = 0, 1) at the t-th layer, we obtain that the learned embedding E(t,yi) ∼ N(µi, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we focus on analysing the intra-class node pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', nodes with the same label), since they are the majority of all node pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In a GCN model, edges are involved in the one-hop mean-aggregation op- eration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', ˆAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' From the view of an individual node vi, the one-hop mean-aggregation operation is [ ˆAE(t)]i = 1 di+1(E(t) i + � vj ∈N (vi) mj=0 E(t,y0) j + � vj ∈N (vi) mj=1 E(t,y1) j ), where N(vi) represents the neighbour node set of vi, and mi = 1 indicates vi is from class y1, otherwise mn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Given m = [m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', m|V|], E(t,y1) = diag(m)E(t), E(t,y0) = (I − diag(m))E(t), and E(t) = E(t,y0) + E(t,y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Therefore, � vj∈N (vi) mj=0 E(t,y0) j ∼ N(dy0 i µ0, dy0 i σ2) and � vj∈N (vi) mj=1 E(t,y1) j ∼ N(dy1 i µ1, dy1 i σ2), where dy0/y1 i represent the number of nodes from Ey0/y1 and di = dy0 i + dy1 i indicate the degree of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Without loss of generality, we assume that vi and vj in the node pair (vi, vj) come from class 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' As shown below, the node distance sensitivity in embed- ding space can be calculated as the difference between cases where vi and vj are connected or unconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Case 0: When vi and vj is unconnected, � ˆAE(t)�0 i and � ˆAE(t)�0 j can be approximately expressed as � ˆAE(t)�0 i ≈ 1 di + 1 � E(t) i + dy0 i µ0 + dy1 i µ1 � , � ˆAE(t)�0 j ≈ 1 dj + 1 � E(t) j + dy0 j µ0 + dy1 j µ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (14) Case 1: When vi and vj is connected, � ˆAE(t)�1 i and � ˆAE(t)�1 j can be approximately expressed as � ˆAE(t)�1 i ≈ 1 di + 2 � E(t) i + E(t) j + dy0 i µ0 + dy1 i µ1 � , � ˆAE(t)�1 j ≈ 1 dj + 2 � E(t) j + E(t) i + dy0 j µ0 + dy1 j µ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (15) Given (14) and (15), the distance of vi and vj in the embed- ding space is calculated as d0(vi, vj) = � ˆAE(t)�0 i − � ˆAE(t)�0 j , d1(vi, vj) = � ˆAE(t)�1 i − � ˆAE(t)�1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (16) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' the sensitivity of d(vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' vj) with respect to the existence of edge eij is ∆d(vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' vj) =∥d0(vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' vj) − d1(vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' vj)∥ = ������� � ˆAE(t)�0 i − E(t) j di + 2 − � ˆAE(t)�0 j − E(t) i dj + 2 ������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (17) Vanilla Training GNN Predictions Privacy-aware Perturbations Influence Functions Quadratically Constrained Linear Programming Fairness-aware Re-weighting Re-training Introducing Heterogeneous Edges GNN Model Perturbated Graph Structure Figure 1: The Framework of Our Privacy-aware Perturbations and Fairness-aware Re-weighting (PPFR) Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' After the vanilla training of a GNN model, the privacy-aware module introduces heterophily edges to generate the perturbed graph structure, fairness-aware re-weighting module employs the influence function and quadratically constrained linear programming to obtain fairness-aware sample weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' After that, the perturbed graph structure and fairness-aware sample weights are involved in the re-training of the target GNN model to promote fairness with limited performance cost and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' and its expectation is E [∆d(vi, vj)] = ∥(µ1 − µ0)δ∥, where δ = dy1 i (di+1)(di+2) − dy1 j (dj+1)(dj+2), due to E �� ˆAE(t)�0 i − E(t) j � = (1 + dy0 i )µ0 + dy1 i µ1 di + 1 − µ0 = dy1 i (µ1 − µ0) di + 1 , E �� ˆAE(t)�0 j − E(t) i � = (1 + dy0 j )µ0 + dy1 j µ1 dj + 1 − µ0 = dy1 j (µ1 − µ0) dj + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (18) (2) Perturbation-based Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Although our modeling process cannot fully depict privacy risks, this coarse grain analysis provides us with insights into understanding risk sources and designing methods to restrict edge leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' For example, the item µ0 − µ1 in E [∆d(vi, vj)] indicates a GNN model with higher performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', higher discrimination when owning a larger µ0 − µ1 value) has higher edge leak- age risks due to the homophily of graph data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', connected nodes are more likely to have similar attributes and the same label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Another item δ implies that, for nodes with similar degree values, the heterophily difference between nodes is positively correlated to the privacy risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to these insights, we introduce heterophily edge noises into the graph structure to reduce edge leakage risks with a limited performance cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, after the vanilla training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', with (1)) of target models, we employ its pre- dictions on all nodes to add heterophily neighbours for each node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', N(vi) = N(vi) ∪ {vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', vik}, (19) where the GNN prediction on v ∈ {vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', vik} is different from that on vi, k = γ|N(vi)| and γ is a hype-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' With Equation (19) we obtain the perturbed adjacency matrix A′, which is involved in the retraining of target GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In addition to reducing heterophily differences between nodes of the same class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', δ in E [∆d(vi, vj)]), the introduction of heterophily edges can help close the distance of nodes across classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', µ0 − µ1 in E [∆d(vi, vj)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Target Model Re-training Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 1 shows the whole framework of our Privacy-aware Per- turbations and Fairness-aware Re-weighting (PPFR) method, whose goal is to promote fairness with limited performance costs and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The entire training of the target model includes two phases: vanilla training and PPFR retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, we first conduct vanilla training to obtain a competent GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' After that, we continue to train (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', re-training) the target GNN with the perturbed graph structure and weighted loss function derived from the fairness-aware weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In PPFR, the epoch number of re- training phase ere is defined as ere = seva, where eva is the epoch number of vanilla training and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='1 is a hyper- parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 5 Experiments In this section, we conduct experiments to evaluate our pro- posed PPFR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We first introduce the experimental setup, followed by our experimental results and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='1 Experimental Setup Datasets, Models, and Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The datasets, model setup, and fairness and privacy metrics follow that in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Note that the privacy result in this section is the average AUC derived from 8 different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Moreover, we use the fol- lowing metric to evaluate the effect of a method Ω on both L(y, Yu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 0 arg min 0 VEViWfair [W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', WIVilarg min 0 UEViTable 3: Effectiveness of our PPFR method and Correlation between Ifbias and Ifrisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Datasets ∆bias ↓ ∆risk ↓ ∆ ↑ r Reg 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='44×10−1 DPReg 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='11 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='87×10−1 DPFR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='55×10−3 Cora PPFR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='46×10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='66 Reg 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='17×10−1 DPReg 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='34 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='71×100 DPFR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='01×10−4 Citeseer PPFR 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='97×10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='51 Reg 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='28×100 DPReg 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='45 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='71 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='81×10−1 DPFR 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='98×10−1 Pubmed PPFR 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='25×10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='41 In this table, columns ∆bias and ∆risk show evaluation results in the percentage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=',%) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the ∆ column, blue colour represents desired results, while red colour indicates undesired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' “r” in the last column indicates the Pearson correlation coefficient between Ifbias and Ifrisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' fairness and privacy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', ∆ = ∆bias∆risk |∆acc| , (20) where ∆(·) = Ω(·)−w/o(·) w/o(·) takes bias/risk/accuracy values of methods Ω and w/o as inputs to evaluate the change ratio on bias/risk/accuracy when using method Ω, w/o represents the GNN model obtained by vanilla training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to the definition, ∆ measures the cost-effectiveness with respect to GNN performance when promoting both fairness and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' A positive ∆ indicates that Ω can boost fairness and privacy simultaneously, otherwise ∆ is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' To verify the effectiveness of our method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', PPFR), we combine privacy and fairness methods together as the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, we consider differen- tial privacy (DP) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', EdgeRand and LapGraph [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b]) as the method to improve edge privacy of GNNs, and both EdgeRand and LapGraph are engaged in generating perturbed adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Specifically, to boost edge pri- vacy, EdgeRand/LapGraph uses a randomisation/Laplacian mechanism to introduce edge noises into the original graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We follow the parameter setting in [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b] to introduce ϵ-edge DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', EdgeRand/LapGraph) into our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' According to the previous study [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2022b], EdgeRand and LapGraph have similar effective- ness when ϵ is small, while LapGraph is more applicable to large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Thus, we apply EdgeRand on Cora and Citeseer datasets and LapGraph on the Pubmed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In this paper, Reg represents that the fairness regularisa- tion is introduced into the loss function of GNN models dur- ing their vanilla training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' DPReg represents using the edge DP method and adding the fairness regularisation into loss si- multaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' DPFR indicates the combination of edge DP and our fairness-aware re-weighting (FR) in this paper, where a perturbed graph is engaged in the retraining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='2 Experimental Results As shown in Table 3, we evaluate the effectiveness of our Privacy-aware Perturbations and Fairness-aware Re- weighting (PPFR) method in boosting fairness with limited performance costs and restricted privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Results in Ta- ble 3 include two parts: correlation results between Ifbias and Ifrisk, and comparison between our PPFR and baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the last column of Table 3, we quantitatively measure the correlation between fairness and privacy with our proposed index (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' The negative correlation results between Ifbias and Ifrisk show that there exists a negative correlation between node individual fairness and edge leak- age risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' These correlation results are consistent with our observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', the trade-off between fairness and privacy) in Table 1, and also underpin the design of our PPFR method, that is, it does not involve Ifbias and Ifrisk simultaneously in linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Other evaluation results also verify the effec- tiveness of our PPFR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (1) Due to involving GNN performance into consideration, our PPFR method can main- tain the same level of performance as vanilla GNNs on all datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='09, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='50, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='57 on Cora, Citeseer, and Pubmed, respectively), which is vital in serving GNN users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' However, the low performance of DPReg (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='26, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='47, and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='95 on three datasets, respectively) potentially leads to the impracticability of GNN systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (2) When comparing the DPFR and PPFR methods, our privacy-aware perturba- tion (PP) method is more effective than the edge DP meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Although edge DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', DPFR rows) is effective in con- trolling edge leakage risks compared to current fairness pro- motion methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', Reg rows), it still poses a higher pri- vacy risk than vanilla GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In contrast, our method is more effective since it presents equal even lower privacy risks in almost all cases, which indicates PPFR can achieve fairness with non-negative privacy impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' (3) According to the ∆ re- sults, PPFR is the only method that has a positive sign across all datasets, indicating that it can boost fairness and privacy while maintaining competent accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' 6 Conclusion In this paper, we investigate the interaction between the fair- ness and privacy of GNNs, which is indispensable in compre- hensively building trustworthy GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' We empirically ob- serve the adverse effects of node fairness on edge privacy risks and propose to quantitatively measure their trade-off through influence functions and Pearson correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' Finally, we devise a retraining method to increase GNN fairness with limited performance cost and restricted privacy risk, whose effectiveness is demonstrated by our experimental evalua- tions on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' In the future, we will conduct a theoretical analysis between the fairness and privacy of GNNs and explore other methods to balance the different aspects of trustworthy GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=' References [Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFOT4oBgHgl3EQf4zQ-/content/2301.12951v1.pdf'} +page_content=', 2021] Chirag Agarwal, Himabindu Lakkaraju, and Marinka Zitnik.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Elster,1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Launey,2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Maris,3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Popa,1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Weppner4 1Institute of Nuclear and Particle Physics, and Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA 2Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA 3Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 4Natural Sciences, Eckerd College, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Petersburg, FL 33711, USA The effective interaction between a nucleon and a nucleus is one of the most important ingredients for reaction theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Theoretical formulations were introduced early by Feshbach and Watson, and efforts of deriving and computing those ‘optical potentials’ in a microscopic fashion have a long tradition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, only recently the leading order term in the Watson multiple scattering approach could be calculated fully ab initio, meaning that the same nucleon-nucleon (NN) interaction enters both the structure as well as the reaction pieces on equal footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This allows the uncertainties from the underlying chiral effective NN interaction to be systematically explored in nucleon-nucleus elastic scattering observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this contribution the main ingredients for arriving at the ab initio leading order of the effective nucleon-nucleus interaction in the Watson approach will be reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Concentrating on one specific chiral NN interaction from the LENPIC collaboration and light nuclei with a 0+ ground state, the leading order nucleon-nucleus interaction is calculated using up to the third chiral order (N2LO) in the nucleon-nucleon potential, and elastic scattering observables are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Then pointwise as well as correlated uncertainty quantification is used for the estimation of the chiral truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Elastic scattering observables for 4He, 12C, and 16O for between 65 and 200 MeV projectile energy will be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' INTRODUCTION Simplifying the many-body problem posed by scattering of a proton or neutron from a nucleus to a two-body problem with an effective (optical) potential was introduced already by Bethe [1] in the 1930s, and its justification summarized by Feshbach [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Since then differential cross sections as well as spin observables for elastic scattering played an important role in either determining the parameters in phenomenological optical models for proton or neutron scattering from nuclei or in testing validity and accuracy of microscopic models thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The theoretical approach to elastic scattering from a nuclear target presented in this article is based on the ansatz of a multiple scattering expansion that was pioneered by Watson [3, 4], made familiar by Kerman, McManus, and Thaler (KMT) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' and refined further as spectator expansion [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Specifically, elastic scattering from stable nuclei has led in the 1990s to a large body of work on microscopic optical potentials in which the nucleon-nucleon interaction and the density of the nucleus were taken as input to rigorous calculations of first-order potentials, in either a Kerman-McManus-Thaler (KMT) or a Watson expansion of the multiple scattering series (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [9–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Here the primary goal was a deeper understanding of the reaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, a main disadvantage of that work was the lack of sophisticated nuclear structure input compared to what is available today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Recent developments of the nucleon-nucleon (NN) and three-nucleon (3N) interactions, derived from chiral effective field theory, have yielded major progress [15–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' These, together with the utilization of massively parallel computing resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', see [23–27]), have placed ab initio large-scale simulations at the frontier of nuclear structure and reaction explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Among other successful many-body theories, the ab initio no-core shell-model (NCSM) approach (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', [28–31]), has over the last decade taken center stage in the development of microscopic tools for studying the structure of atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The NCSM concept combined with a symmetry-adapted (SA) basis in the ab initio SA-NCSM [32] has further expanded the reach to the structure of intermediate-mass nuclei [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Following the developments in nuclear structure theory, it is very natural to again consider rigorous calculations of effective folding nucleon-nucleus (NA) potentials, since now the nuclear densities required as input for the folding with the NN scattering amplitudes can be based on the same chiral NN interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This development also allows to investigate effects of truncation uncertainties in the chiral expansion on NA scattering observables in a similar fashion as already successfully performed in NN scattering (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [34–36]), nucleon-deuteron scattering [37], or structure observables for light nuclei [31, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The theoretical and computational developments leading to ab initio NA effective interactions (in leading order in the spectator expansion) are described in a serious of publications by the authors [39–43] and others (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [44–47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus the aim of this review is to shed light on truncation uncertainties in the chiral expansion, and within that context give a perspective on intricacies of the spectator expansion as well as the explicit content of its leading order term, which can now be calculated ab initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='04293v1 [nucl-th] 11 Jan 2023 2 Deriving ab initio optical potentials within a multiple scattering approach focuses on projectile energies at energies about 80 MeV or higher, since the expectation is that at those energies the leading order term may already capture the most important physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Another recent ab initio approach starts from a formulation introduced by Feshbach [48] and constructs optical potentials and elastic scattering observables within a Green’s function approach [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For elastic scattering from medium-mass nuclei the coupled-cluster method [51] and the SA-NCSM [52] approach have been successfully implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' These approaches are by design better suited for calculating scattering observables at energies below about 20-30 MeV due to restrictions on the size of the model spaces which increase with increasing projectile energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [53] an extensive overview of the status of the field of optical potentials and their need in the rare-isotope era is given and the current status of ab initio approaches is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We want to encourage the reader to refer to this work, for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' WATSON OPTICAL POTENTIAL WITHIN THE SPECTATOR EXPANSION The standard starting point for describing elastic scattering of a single projectile from a target of A particles within a multiple scattering approach is the separation of the Lippmann-Schwinger (LS) equation for the transition operator T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' T = V + V G0(E)T (1) into two parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' namely an integral equation for T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' T = U + UG0(E)PT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (2) where U is the effective potential operator defined by a second integral equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' U = V + V G0(E)QU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (3) Here P is a projection onto the ground state of the target, P = |Φ0⟩⟨Φ0| ⟨Φ0|Φ0⟩ , with P +Q = 1 and [G0(E), P] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The free propagator for the projectile and target system is given by G0(E) = (E − h0 − HA + iϵ)−1 where h0 is the kinetic energy of the projectile and HA is the Hamiltonian of the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The general solutions of the nuclear bound state problem HA|Φ⟩ include the ground state, excited states and continuum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the scattering problem given by the transition amplitude T the reference energy separating bound and continuum states is chosen such that the ground state energy is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus energies referring to the target Hamiltonian in G0 are excitation energies of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' With these definitions the transition operator for elastic scattering may be redefined as Tel = PTP, in which case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (2) can be written as Tel = PUP + PUPG0(E)Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (4) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Spectator expansion of the operator U The transition operator for elastic scattering is given by a straightforward one-body integral equation, which of course requires the knowledge of PUP, which is a many-body operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For a brief review we follow the spectator expansion of PUP as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [54] in contrast to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [6] where the expansion of T is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Following those references, we assume the presence of two-body forces only for the present discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The extension to many- body forces is not precluded by the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' With this assumption the operator U can be expanded as U = A � i=1 Ui, (5) where Ui is given by Ui = v0i + v0iG0(E)Q A � j=1 Uj, (6) provided that V = �A i=1 v0i, where the two-body potential v0i acts between the projectile and the ith target nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Through the introduction of an operator τi which satisfies τi = v0i + v0iG0(E)Qτi, (7) 3 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (6) can be rearranged as Ui = τi + τiG0(E)Q � j̸=i Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (8) This rearrangement process can be continued for all A target particles, so that the operator for the optical potential can be expanded in a series of A terms of the form U = A � i=1 τi + A � i,j̸=i τij + A � i,j̸=i,k̸=i,j τijk + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (9) This is the Spectator Expansion for U , where each term is treated in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The separation of the interactions according to the number of interacting nucleons has a certain latitude, due to the many-body nature of G0(E), which needs to be considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In the following we will concentrate on the leading-order term, which is still a many-body operator due the the presence of G0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The next-to-leading order term in this spectator expansion for U has been formally derived and connected to standard three-body equations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Propagator expansion in the leading-order term of U When using the leading-order term of the spectator expansion as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (7), for elastic scattering only PτiP, or equivalently ⟨Φ0|τi|Φ0⟩ needs be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' With this in mind, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (7) can be re-expressed as τi = v0i + v0iG0(E)τi − v0iG0(E)Pτi = ˆτi − ˆτiG0(E)Pτi, (10) or ⟨Φ0|τi|Φ0⟩ = ⟨Φ0|ˆτi|Φ0⟩ − ⟨Φ0|ˆτi|Φ0⟩ 1 (E − EA) − h0 + iε⟨Φ0|τi|Φ0⟩, (11) where ˆτi is defined as the solution of ˆτi = v0i + v0iG0(E)ˆτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (12) The combination of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (10) and (2) corresponds to the leading-order Watson optical potential [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In ab initio structure calculations the one-body densities or ground state wave functions for protons and neutrons are calculated separately, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (11) allows to combine e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' for proton scattering of a nucleus the proton-neutron interaction (ˆτi=pn) with the neutron one-body density and the proton-proton interaction with the proton one-body density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The sum over i then adds both to obtain the driving term ⟨Φ0|ˆτi|Φ0⟩ the integral equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' If the projectile-target-nucleon interaction is assumed to be the same for all target nucleons and if iso-spin ef- fects are neglected then the KMT approximation ( A−1 A ⟨Φ0|ˆτi|Φ0⟩) can be derived from the leading-order Watson potential [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' When working with momentum space integral equations, the numerical implementation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (11) is straightforward [40, 41, 45, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Working in coordinate space with differential equations does not allow an equally straightforward implementation, and thus the KMT prescription is the most favorable alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A comparison between leading-order Watson potential and the KMT prescription is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 1 for elastic proton scattering from 8He at 71 MeV laboratory kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Despite the relatively large difference between the proton and neutron densities for this nucleus the KMT prescription agrees with the exact Watson description very well up to momentum transfers of about 2 fm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (11) is a one-body integral equation, the principal problem is to find a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (12), which due to many-body character of G0(E) is still a many-body integral equation, and in fact no more easily solved than the starting point of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For most practical calculations the so-called closure approximation to G0(E) is implemented [56] turning Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (12) into a one-body integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This approximation replaces HA by a constant that is interpreted as an average excitation energy, and is justified when the projectile energy is large compared to typical excitation energies of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The closure approximation is very successfully applied for elastic scattering around 80 MeV and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Going beyond the closure approximation in the spirit of the spectator expansion we want to single out one target nucleon i and write G0(E) as G0(E) = (E − h0 − HA + iε)−1 = (E − h0 − hi − � j̸=i vij − Hi + iε)−1, (13) 4 where the target Hamiltonian is expanded as HA = hi + � j̸=i vij + Hi with vij being the interaction between target nucleons i and j, and Hi being an (A-1)-body operator containing all higher order effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Realizing that � j̸=i vij ≡ Wi and thus Hi = HA − hi − Wi does not have an explicit dependence on the ith particle, then Hi may be replaced by an average energy Ei which is akin to the effective binding energy between the ith nucleon and the A − 1 spectator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This is not an approximation since G0(E) may be regarded as G0(E) = [(E − Ei) − h0 − hi − Wi − (Hi − Ei) + iε]−1 (14) and (Hi − Ei) should be set aside to be treated in the next order of the expansion of the propagator G0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this order of the expansion G0(E) becomes Gi(E) = [(E − Ei) − h0 − hi − Wi + iε]−1, (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (12) reads ˆτi = v0i + v0iGi(E)ˆτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (16) In order to connect the above expression with the free NN amplitude t0i = v0i + v0igit0i (17) with gi = [(E − Ei) − h0 − hi + iε]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (18) algebraic relations between the resolvents lead to ˆτi = t0i + t0iGiWigi(E)ˆτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (19) Defining GiWi = giTi with Ti = Wi + WigiTi leads to ˆτi = t0i + t0igiTigi ˆτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (20) The three-body character of the above expression becomes more evident if one defines it as a set of coupled equations as ˆτi = t0i + t0igiXi Xi = Tigi ˆτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (21) Though the spectator expansion of the operator U in terms of active particles is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (9), we see that this expansion is performed in terms of quantities which contain many-body propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Each of the ingredients τi, τij, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' may themselves be expanded in a spectator expansion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' expanding the many-body propagator also according to the number of active participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The corrections to the propagator in the leading-order term of U contributions that arise from the Q space, whereas the terms arising from the propagator remain in the P space at first order level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus their contribution may be more relevant for elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In an explicit treatment of Gi(E) it is necessary to consider the explicit form of � j̸=i vij = Wi, which is a priori a two-body operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In the framework of ab initio nuclear structure calculations this will involve two-body densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In earlier work [54, 57, 58] the quantity Wi was treated as one-body operator, specifically a mean-field potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This was a physically reasonable choice, though being outside the strict demands of the spectator expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, those studies revealed that the next order in the propagator expansion has little effect on elastic scattering observables at energies larger than 100 MeV, while the description of differential cross section and spin-observables for elastic scattering from 40Ca at 48 MeV showed considerable improvement with respect to experiment [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Obviously this type of calculation will need to be explored within an ab initio approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [57] the energy Ei of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (18) was set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As illustrated in this section, deriving a multiple scattering expansion for elastic NA scattering means projecting on the ground state of the target in order to obtain a Lippman-Schwinger type equation for the transition amplitude and obtaining an operator U for the effective interaction, which is defined in the space Q = 1 − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this spirit, the spectator expansion contains therefore two pieces, namely the expansion of the operator U in terms of active particles in the scattering process as well as the expansion of target Hamiltonian HA in the propagator G0(E) in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus it is very difficult to define a single expansion parameter which governs the convergence of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' LEADING ORDER AB INITIO OPTICAL POTENTIAL BASED ON A CHIRAL NN INTERACTION The leading order of the spectator expansion involves two active nucleons, the projectile and a target nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' the leading order is driven by the NN amplitude M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' which in its most general form can be parameterized in terms of Wolfenstein amplitudes [59–61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' M(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) = A(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ)1 ⊗ 1 + iC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) � σ(0) · ˆn � ⊗ 1 + iC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) 1 ⊗ � σ(i) · ˆn � + M(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ)(σ(0) · ˆn) ⊗ (σ(i) · ˆn) + [G(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) − H(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ)] (σ(0) · ˆq) ⊗ (σ(i) · ˆq) + [G(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) + H(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ)] (σ(0) · ˆK) ⊗ (σ(i) · ˆK) + D(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ) � (σ(0) · ˆq) ⊗ (σ(i) · ˆK) + (σ(0) · ˆK) ⊗ (σ(i) · ˆq) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (22) where σ(0) describes the spin of the projectile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' and σ(i) the spin of the struck nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The average momentum in the NN frame is defined as KNN = 1 2 (k′ NN + kNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The scalar functions A, C, M, G, H, and D are referred to as Wolfenstein amplitudes and only depend on the scattering momenta and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Each term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (22) has two components, namely a scalar function of two vector momenta and an energy and the coupling between the operators of the projectile and the struck nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The linear independent unit vectors ˆq, ˆK, and ˆn are defined in terms of the momentum transfer and the average momentum as ˆq = q |q| , ˆK = K |K| , ˆn = K × q |K × q|, (23) and span the momentum vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' With the exception of the momentum transfer q, which is invariant under frame transformation, the vectors in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (23) need to be considered in their respective frame in explicit calculations [41, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the struck target nucleon the expectation values of the operator 1 and the scalar products of σ(i) with the linear independent unit vectors of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (23) need to be evaluated with the ground state wave functions of the respective nucleus when calculating the leading-order NA effective interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Evaluating the expectation value of the operator 1 in the ground state of the nucleus results in the scalar nonlocal, translationally invariant one-body density that has traditionally been used as input to microscopic or ab initio calculations of leading order effective interactions [11, 12, 40, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The other operators from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (23), namely (σ(i) · ˆn), (σ(i) · ˆq), and (σ(i) · ˆK) need to also be evaluated for a leading-order ab initio NA effective interaction, in which the NN interaction is treated on equal footing in the reaction and structure calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus, the general expression for a nonlocal density needs to include the spin operator σ(i) explicitly, ρKs qs (p, p′) = � Φ′ 0 ����� A � i=1 δ3(pi − p)δ3(p′ i − p′)σ(i)Ks qs ����� Φ0 � , (24) where σ(i)Ks qs is the spherical representation of the spin operator and the wavefunction Φ0 (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', pA) = ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', pA|Φ0⟩ is defined in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Evaluating this expression for Ks = 0 gives the nonlocal one-body scalar density and Ks = 1 becomes a nonlocal one-body spin density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The Wolfenstein parameterization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (22) requires the evaluation of scalar products of the one-body spin density with unit momentum vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Since those only depend on the momenta p and p′, those can be calculated as ρKs (p, p′) · ˆn, ρKs (p, p′) · ˆq, and ρKs (p, p′) · ˆK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the explicit calculation of ρKs (p, p′) · ˆn, we refer the reader to [41, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The scalar products (σ(i) · ˆq) and (σ(i) · ˆK) represent scalar products of a pseudo-vector and a vector, a construct that is not invariant under parity transformations, and thus vanish when sandwiched between ground state wave functions, which is explicitly shown in [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus the tensor contributions of the NN force only enter the leading order effective NA interaction through the Wolfenstein amplitude M as long as elastic scattering is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' When e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' transition amplitudes between states of different parity would be considered, the other tensor amplitudes will contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Currently contributions to elastic scattering observables due to the spin-projected one-body densities have only been calculated for light nuclei with 0+ ground states, and it was found that this contribution is very small for nuclei with equal proton and neutron numbers [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This is likely different for nuclei with ground states of nonzero spin, 6 which was explored for 10B polarization transfer observables in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [63, 64], where the authors assume a nuclear structure which consists of a core and valence nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [45] extends the standard leading order calculation to nonzero spin nuclei, however does not consider the inherent tensor contributions from the NN force in their formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This leaves the importance of a consistent treatment of the NN force on elastic scattering from nonzero spin nuclei still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The complete calculation of the leading-order effective interaction describing the scattering of a proton from a nucleus in a 0+ ground state and which enters the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (11) as driving term is given by �Up(q, KNA, ϵ) = (25) � α=n,p � d3Kη (q, K, KNA) Apα � q, 1 2 �A + 1 A KNA − K � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ � ρKs=0 α � P′, P � + i(σ(0) · ˆn) � α=n,p � d3Kη (q, K, KNA) Cpα � q, 1 2 �A + 1 A KNA − K � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ � ρKs=0 α � P′, P � + i � α=n,p � d3Kη (q, K, KNA) Cpα � q, 1 2 �A + 1 A KNA − K � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ � Sn,α � P′, P � cos β + i(σ(0) · ˆn) � α=n,p � d3Kη (q, K, KNA) (−i)Mpα � q, 1 2 �A + 1 A KNA − K � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ϵ � Sn,α � P′, P � cos β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The term η (q, K, KNA) is the Møller factor [65] describing the transformation from the NN frame to the NA frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The functions Apα, Cpα, and Mpα represent the NN interaction through Wolfenstein amplitudes [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Since the incoming proton can interact with either a proton or a neutron in the nucleus, the index α indicates the neutron (n) and proton (p) contributions, which are calculated separately and then summed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' With respect to the nucleus, the operator i(σ(0) · ˆn) represents the spin-orbit operator in momentum space with respect to the projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As such, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (25) exhibits the expected form of an interaction between a spin- 1 2 projectile and a target nucleus in a J = 0 state [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The momentum variables in the problem are given as q = p′ − p = k′ − k, (26) K = 1 2 (p′ + p) , KNA = A A + 1 � (k′ + k) + 1 2 (p′ + p) � , P = K + A − 1 A q 2 , P′ = K − A − 1 A q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The two quantities representing the structure of the nucleus are the scalar one-body density ρKs=0 α � P′, P � and the spin-projected momentum distribution Sn,α � P′, P � = ρKs=1 � P′, P � ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Both distributions are nonlocal and translationally invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The reduced matrix elements entering the one-body densities are obtained within the NCSM (SA-NCSM) in the center-of-mass frame of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In order to employ them in calculating the leading-order effective NA interaction, this center-of-mass variable must be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Within the framework of NCSM (SA-NCSM) the technique for obtaining nonlocal and translationally invariant one-body densities is well developed [40, 44, 67–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Lastly, the term cos β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (25) results from projecting ˆn from the NN frame to the NA frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For further details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' CHIRAL TRUNCATION UNCERTAINTIES IN THE LEADING ORDER OPTICAL POTENTIAL With the emergence of nuclear forces based on chiral effective field theory (EFT), we are presented with an opportu- nity to study the nucleon-nucleus effective interaction as it develops order-by-order in a chiral EFT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Given the hierarchical nature of chiral EFT, we can combine these order-by-order results to reliably estimate truncation uncertainties associated with the higher chiral orders not included in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To this end, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [35–37] first implemented uncertainty quantification for the cases of NN and Nd scattering by assuming a quantity y(x) at a chiral order k can be written as yk(x) = yref(x) k � n=0 cn(x)Qn(x) (27) 7 where yref(x) is a reference value that sets the scale of the problem and also includes the dimensions of the quantity y(x) of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' By construction, the coefficients cn(x) are dimensionless and are expected to be of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The remaining quantity Q(x) is the expansion parameter associated with the chiral EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The expansion parameter is usually defined as Q = 1 Λb max(Mπ, p) (28) where Λb is the breakdown scale of the EFT, Mπ is the pion mass, and p is the relevant momentum for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Various works [35–37] have identified the relevant momentum in different ways, but keeping with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [43] we choose the relevant momentum as the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') momentum in the nucleon-nucleus system p2 NA = ElabA2m2(Elab + 2m) m2(A + 1)2 + 2AmElab (29) where Elab is the kinetic energy of the projectile in the laboratory frame, A is the target nucleus’s mass number, and m is the mass of the nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Previous scattering works [36, 43] have noted that various results indicate, when identifying the relevant momentum, the momentum transfer q should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' That is, the expansion parameter would be more appropriately defined as Q = 1 Λb max(Mπ, pNA, q) (30) The momentum transfer in elastic scattering is defined as q = 2pNA sin �θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 2 � (31) where θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' is the scattering angle in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Notably, including the momentum transfer in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (30) makes the expansion parameter a function of θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', even though the other momentum scales in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (30) are independent of the scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' When considering observables such as the differential cross section or analyzing power that are functions of θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', this implies the expansion parameter will be larger at backward angles than at forward angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Furthermore, since the leading order of the spectator expansion is not applicable at low energies, we only consider scattering at lab energies of 65 MeV or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As a result, the chiral expansion parameter becomes Q = max(pNA, q)/Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This expansion parameter is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 2 for the case of A = 4 and Λb = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Because of the factorization of the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' momentum, there is a universal scattering angle at which the momentum transfer q begins to dominate the expansion parameter, regardless of the chosen Elab or nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We will exploit this behavior in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nuclear structure calculations Prior to our detailed study of truncation uncertainties of a chiral NN interaction in elastic NA scattering observables we need to choose a specific chiral NN interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Here we want to focus on the EKM chiral NN interaction [18, 19] with a semi-local coordinate space regulator of R = 1 fm, which has a breakdown scale of Λb = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This interaction gives a slightly better description of the ground state energies in the upper p-shell than a similar, more recent interaction with a semi-local momentum space regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For consistency with the leading-order optical we only use the NN potentials, omitting three-nucleon forces, which appear at N2LO in the chiral expansion, both in the structure and the scattering part of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Including three-nucleon forces consistently in both, the structure and scattering calculations requires going beyond the leading-order optical potential, and is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Though initial attempts of incorporating three-nucleon forces as an effective density-dependent NN force in the scattering part have been presented [46], they can not yet be considered as systematic consideration of three-nucleon forces in NA scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For similar reasons, we restrict most of our results to N2LO since three-nucleon force contributions at N3LO and N4LO are significant [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Next, the translationally-invariant one-body density needed for the scattering calculation can be obtained using the NCSM approach, in which the nuclear wavefunction is expanded in Slater determinants of harmonic oscillator basis functions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ideally, one uses a sufficiently large basis to ensure convergence of this expansion, but in practice observables depend on both the many-body basis truncation, Nmax (defined as the total number of harmonic oscillator quanta in the many-body system above the minimal configuration), and on the harmonic oscillator scale ¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In Table I we give the ground state binding energies and point-proton radii of 4He, 12C, and 16O obtained with the EKM chiral 8 NN potential [18, 19] with a semi-local coordinate space regulator of R = 1 fm (note that at N2LO we did not include any three-nucleon forces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For 4He we can obtain nearly converged results for both the binding energy and the proton radius, and these results agree, to within their estimated numerical uncertainties (the first set of uncertainties in Table I), with Yakubovsky calculations using the same NN potential [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, for larger nuclei such as 12C and 16O we are more limited in the Nmax values that can be reached on current computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Pointwise truncation uncertainties To assess the relative size of chiral truncation uncertainties compared to other known uncertainties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' the har- monic oscillator parameters Nmax and ¯hΩ, we employ a pointwise truncation procedure and study reaction observables that are not functional quantities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' reaction cross sections at a specified laboratory energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This pointwise approach was previously implemented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [36, 43] and it starts by assuming the expansion parameter Q and reference scale yref are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' From there, we can apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (27) to calculate the coefficients cn, which are treated as independent draws from the same underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The properties of this distribution can be learned from Bayesian tech- niques and the posterior distribution for the prediction can be readily calculated with its associated credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For more details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In order to estimate the chiral truncation uncertainties of the obtained ground state binding energies and radii, we apply the pointwise approach with Q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='3 as the effective expansion parameter, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' These uncertainties are listed as the second set of uncertainties in Table I, starting from NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Here we see that for the energies, the chiral uncertainties are at least of the same order as the estimated numerical uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' however, the uncertainties of the radii of 12C and 16O are clearly dominated by their systematic dependence on the basis parameter ¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To illustrate the pointwise approach for scattering observables, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 3 shows the reaction cross sections for proton scattering from 4He at 65 MeV and 16O at 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For each case, the result is shown as a function of Nmax, and variations with respect to ¯hΩ are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' While more obvious for the smaller nucleus where the NCSM can better converge, in both cases the uncertainty resulting from the chiral truncation remains larger than the uncertainty arising from the many-body method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To better illustrate this point, we present the reaction cross section for 4He with a scale starting from 115 mb and with a range of only 45 mb, while using the full range of 600 mb for 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' While larger model spaces will better converge the NCSM results, smaller truncation uncertainties will only be achieved by higher chiral orders, despite the noticeable dependence of the radii on the harmonic oscillator parameter ¯hΩ, in particular for the heavier nuclei, in the current calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Note however that even at N3LO we anticipate the chiral truncation uncertainties will be larger than the indicated variations with respect to the harmonic oscillator parameter ¯hΩ due to the rather large value of the expansion parameter Q in the scattering calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Correlated truncation uncertainties For functional quantities y(x) we employ a correlated approach that includes information at nearby values of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This approach is better for observables such as a differential cross section, which we know does not vary wildly from values at nearby angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' It also starts from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (27) and treats the coefficients cn(x) as independent draws from an underlying Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This Gaussian process encodes information about the correlation length ℓ, and the qualities of the underlying distribution can be learned from the order-by-order results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This training is followed up by testing procedures which seek to confirm the Gaussian process has been appropriately fit to the available results, and if not, to diagnose potential issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' From a well-fit Gaussian process we can then extract truncation uncertainties for the functional quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For more details and applications, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [36, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In the following examples, we examine proton scattering for 4He, 12C, and 16O at various projectile energies and compare to the available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In each case, we show the convergence with respect to chiral order 1 One commonly applies a Similarity Renormalization Group (SRG) transformation to the NN potential in order to improve the conver- gence of the many-body calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, this leads to induced three-nucleon forces that are non-negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' omitting those would lead to a strong dependence on the SRG parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We therefore choose to not employ such a transformation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the binding energies we use an exponential extrapolation to the complete basis, with associated uncertainties, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [71] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Radii converge rather slowly in a harmonic oscillator basis, and they do not necessarily converge monotonically with increasing Nmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' furthermore, in the scattering calculations we use densities obtained at fixed values of the harmonic oscillator parameters Nmax and ¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We therefore simply give in Table I our results for the point-proton radii of 12C and 16O at Nmax = 10, averaged over the range 16 ≤ ¯hΩ ≤ 28 MeV (the same range as is used for the scattering calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The numerical uncertainty estimates for the radii listed in Table I correspond to the spread over this ¯hΩ interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' this is a systematic uncertainty due to the Gaussian fall-off of harmonic oscillator basis functions, and is therefore strongly correlated for the different chiral orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, the trend of a significant increase in the radii going from LO to NLO, followed by a smaller increase going from NLO to N2LO, is robust, and correlates with the decrease in binding energies going from LO to NLO to N2LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Note that we did not include any chiral EFT corrections to the R2 operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' and the experimental point-proton radii are extracted from the charge radius measured in electron scattering experiments, using standard proton and neutron finite-size corrections, relativistic corrections, and meson-exchange corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 9 and the resulting decrease in the size of the chiral truncation uncertainties, as well as discuss any associated physics insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To avoid concerns about the expansion parameter increasing at larger angles, we mostly restrict our analysis to forward angles where we expect the expansion parameter to be independent of the scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For proton scattering on 4He, we see good agreement with experiment for the differential cross sections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 4) at lower projectile energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Below 100 MeV, most data points fall within the 2σ uncertainty band, and at 100 MeV a majority of the data points are within the 1σ band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At the highest energy of 200 MeV, the chosen interaction seems unable to reproduce the experimental data, though this is not uncommon for scattering from 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The analyzing powers for proton scattering on 4He (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 5) is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the lower energies of 65 and 71 MeV, the experimental data shows a near zero value, regardless of scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In the scattering of a spin-1/2 particle from a spin-0 nucleus, this indicates that there is no spin-orbit force at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This behavior is only reproduced by the LO result, for which the chiral NN interaction only contains the one-pion exchange and contact terms, which do not produce a spin-orbit force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At NLO the two-pion exchange diagrams are responsible for reproducing the NN p-waves and thus provide a spin-orbit force that leads to a non-zero value for the analyzing power in NA scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At N2LO there are no new terms in the two-nucleon sector, and thus Ay does not change its shape at that chiral order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Therefore, one needs to conclude that in this case other physics which goes beyond the leading order NA effective interaction may be needed to describe the analyzing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the higher energy of 200 MeV, all of the experimental data points are within the 2σ uncertainty band, though there is a slight offset in the shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In all cases, the analyzing power is more difficult to reproduce using this interaction, though other interactions have done better [39, 41] For proton scattering from 12C, the differential cross sections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 6) are reliably reproduced by the central value of the N2LO calculations up to 100 MeV laboratory kinetic energy, and systematically over-predict at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As the projectile energy increases, the expansion parameter increases and as a result uncertainty bands become larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This is most noticeable at 160 MeV: the experimental data is within the 1σ band, but the size of that band, as well as the 2σ band, are so large that they are not practically useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The gray bars in the cross section panels for N2LO indicate the momentum transfer up to where we expect the expansion parameter to be dominated by the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' momentum pNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Once the momentum transfer exceeds the value given by the bar, the uncertainty is dominated by the momentum transfer q, and is thus underrepresented by the method we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Note that the vertical bar is at the same scattering angle θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=', but different momentum transfer q, as function of the projectile energy since pNA is a function of the projectile energy as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Looking at the lower energies, the increasing agreement with experiment in the first peak and minimum as higher orders in the chiral NN interaction are included gives the correct trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Minima in the differential cross section correlate with the size of the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' It is well well known [31], and also evident from Table I, that the nuclear binding energy calculated with the LO of the chiral NN interaction is way too large and correspondingly the radius much too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Only when going to NLO and N2LO the binding energy as well as the radius move into the vicinity of their experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This finding from structure calculations is corroborated by the calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 6, where with increasing chiral order the calculated first diffraction minimum moves towards smaller momentum transfers indicating a larger nuclear size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The analyzing powers for proton scattering on 12C are at 65 MeV also almost zero for small momentum transfers and rise at q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='2 fm−1 to its maximum value of +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This is captured by the NLO calculation where spin- contributions occur in the NN interaction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For 65 MeV, the experimental data is mostly within the 2σ band until approximately θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' = 60◦, where we expect the expansion parameter to being increasing and the uncertainty bands to thus be underestimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For 122 MeV, the very forward direction is inside the 1σ band, but the overall shape of the experimental data is not well captured by this interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For proton scattering from 16O, the differential cross sections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 8) are similar to the 12C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Namely, the lower energies do reasonably well at describing the data within the 2σ bands, but as the projectile energy increases the uncertainty bands increase to unhelpful sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At the lowest energy of 65 MeV, we see a better and better reproduction of the first minimum in the differential cross section as the chiral order increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Again, this first minimum is known to be related to the size of the nucleus, so this is an important feature to reproduce from both a structure, see Table I, and reaction perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The analyzing powers for proton scattering on 16O (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 9) are again similar to the 12C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At lower energies (65 and 100 MeV), we again see a good reproduction to within 1σ or 2σ of the forward direction data, but beyond θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' = 60◦, the experimental data is outside the uncertainty bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At the higher energy of 135 MeV, many of the experimental data are within the uncertainty bands but for a nucleus of this size, the expansion parameter has already increased such that the resulting uncertainty bands are unhelpfully large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As stated toward the beginning of the section we omit three-nucleon forces for consistency with the leading-order optical potential which only treats two active nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Those three-nucleon forces already appear at N2LO in the chiral expansion, however, including them consistently in the structure as well as reaction calculation requires going beyond the leading-order optical potential and is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the sake of investigating truncation errors in the chiral NN force, one may carry out inconsistent calculation in the sense that the structure part of the 10 calculation is kept fixed at N2LO, and in the reaction part higher orders in the NN force are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Proceeding in this fashion is sensible, since the scattering calculation is more sensitive to the NN force compared to the structure calculation, provided this structure calculation gives a reasonable description of the ground state one-body density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To show how the chiral truncation error develops when higher chiral orders in the NN interaction are introduced, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 10 proton scattering from 16O at 100 MeV projectile energy, where the higher chiral orders are only employed in the scattering part through the corresponding Wolfenstein amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In both, the differential cross section as well as the analyzing power the two most right panels depicting the inconsistent calculation show that the uncertainty bands become smaller when higher chiral orders in the NN interaction are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, these uncertainty bands are not necessarily realistic due to missing higher-body effects, which include higher orders in the chiral force as well as higher orders in the multiple scattering expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Therefore, we can not draw firm conclusions from the fact that data are outside the uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nevertheless, it is obvious that the decrease in the uncertainties in the chiral truncation is rather slow due to the large expansion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Furthermore, the medians of the calculations shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 8 and 9 do not change when higher chiral orders are considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 10, which further indicates that the smaller error bands of the higher order chiral truncations may be artificial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Analysis of Posteriors Even while restricting our analysis to a region where we expect the expansion parameter to be constant, we can still observe effects on the uncertainty bands if the expansion parameter is large, as noted in many of the results at larger projectile energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In fact, this behavior will place limits on the size of nucleus that can be considered with this approach, since pNA as defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (29) will continue to increase as A increases, yielding Q > 1 eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' While this situation is not ideal, we nonetheless find support for it in our analysis after examining the posteriors for Q, in accordance with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [36, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 11, we calculated posteriors for the differential cross sections in proton scattering from 4He, 12C, and 16O at the energies previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' From these, we can extract a single best guess for the value of Q based on the order-by-order calculations and compare that to the expectation for Q based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For 16O, the largest nucleus considered, we see generally good agreement between the expected value of Q and the best guess value from the posteriors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='11c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, as the nucleus decreases in size and as the laboratory energy decreases, some differences begin to emerge between the two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 11b for 12C, the comparisons are roughly similar to the 16O case, but for the 4He analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 11a), the differences are more pronounced, especially for the lower laboratory energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A similar analysis of neutron scattering on 12C did not show any significant differences between the two values [43], which implies 4He may be the outlier in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This analysis may imply scattering from 4He with projectiles at lower energies could be analyzed with a smaller expansion parameter Q, though the higher energy results still favor the larger expansion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' As the smallest nucleus considered here, it may also point to the few-body character of 4He, which has not historically been well captured in an optical potential approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' OUTLOOK Procedures that quantify the theoretical uncertainties associated with the underlying chiral EFT NN interaction are by now well established for the NN and nucleon-deuteron systems as well as nuclear structure calculations, while the systematic study of chiral truncation uncertainty is not as widely used in ab initio effective interaction employed to describe the scattering of protons or neutrons from nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Contributing factors for this relatively slow development include that when considering a multiple scattering approach to deriving this effective NA interaction in an ab initio fashion only recent progress in calculating the leading-order term in the multiple scattering approach has allowed to treat the NN interaction on the same footing in the structure and reaction part [41] by considering the spin of the struck target nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Though calculations showed that the latter does not contribute significantly to observables when considering scattering from nuclei with a 0+ ground state, one nevertheless needs a consistent ab initio implementation of the leading-order term of the effective NA interaction in order to study the theoretical uncertainties imprinted on NA observables by the chiral EFT NN interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this work we carry out a systematic study of chiral truncation uncertainties of the EKM chiral interaction on the ab initio effective NA interaction calculated in leading order of the spectator expansion for 4He, 12C, and 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We find that this interaction allows for a good description of experiment at energies around 100 MeV projectile kinetic energy and slightly lower, provided we focus on regions of momentum transfer where the analysis of the EFT truncation uncertainty is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' When considering the lower energy of 65 MeV, the agreement with data starts to deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This is an indication that errors other than the truncation error in the chiral interaction should come into play, specifically errors that result from the spectator expansion itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Theoretical consideration of the next-to-leading- 11 order term in the spectator expansion are described in some detail in this work in order to lay out necessary theoretical and computational developments for this nontrivial endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At at the next-to-leading order three-nucleon forces will naturally enter the effective interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' At present this step has only been attempted in approximative fashions, namely by approximating the next-to-leading order in the propagator expansion via a nuclear mean field force [54] or by introducing an effective, density dependent NN potential in the scattering part of the calculation [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Since we are not considering next-to-leading order terms in the spectator expansion, we restrict our analysis to N2LO in the chiral interaction and only consider two-nucleon forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this case the choice of the EKM interaction with a semi- local coordinate space regulator of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 fm is advantageous [38], since this specific interaction gives a slightly better description of the ground state energies in the upper p-shell compared to other more recent chiral EFT interactions when using two-nucleon interactions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In our study the chiral truncation errors at energies larger than 100 MeV increase considerably and the agreement with experiment deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The increase in the chiral truncation error can simply be traced back to the expansion parameter in our approach is getting too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The deterioration of the agreement with experiment when going to higher energies is more difficult to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' One conclusion may be that the specific EKM chiral interaction employed here in using the leading-order in the spectator expansion is not well suited to describe proton-nucleus scattering observables for 4He, 12C, and 16O at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the chiral NN interaction from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [73] this is not the case as shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Therefore one will have to investigate what features of a chiral NN interaction are most relevant for a description of NA scattering observables for light nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' To put this in perspective, let us reconsider the basic ideas of the spectator expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' By design, the leading-order term should be dominant at energies 150 MeV projectile kinetic energy and higher, since the reaction time of the projectile with nucleons inside the nucleus is short, and thus an ‘impulse approximation’ is in general very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, we do not want to consider here projectile energies larger than 400 MeV, where a relativistic treatment e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' via the Dirac equation may be preferred [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus at energies around 200 MeV the leading order term by design should give a reasonably good description of NA scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This has been the case in the microscopic calculations of the 1990s (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [9–14]) and a set of recent calculations with specific chiral NN interactions [40, 41, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Attempts to go beyond the leading order by incorporating 3NFs in a density dependent fashion into the many-body propagator [46] indicate that effects at 200 MeV are only visible at higher momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In a similar fashion, investigations going beyond the leading order term in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [54] indicate that those effects become important at around 100 MeV and at higher momentum transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Thus, if the 3NFs inherent in the chiral expansion are needed to influence calculations with chiral NN forces in the leading order of the spectator expansion at higher energies, then a new look at the interplay between NN and 3NFs in the leading-order spectator expansion must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' ACKNOWLEDGMENTS R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' and Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' gratefully acknowledge fruitful discussions with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Furnstahl and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Phillips about quan- tifying truncation errors in EFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' acknowledges useful discussions with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nogga about the LENPIC chiral NN interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' This work was performed in part under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Department of Energy under contract Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' DE-FG02- 93ER40756, DE-SC0018223 and DE-SC0023495, and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' NSF (PHY-1913728).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The numerical computations benefited from computing resources provided by the Louisiana Optical Network Initiative and HPC resources provided by LSU, together with resources of the National Energy Research Scientific Computing Center, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 56, 2080 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 12 [12] Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Elster, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Cheon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Redish, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Love, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 42, 652 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Entem and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nogga, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Potter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Roth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Skibi´nski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Topolnicki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Vary, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Wita�la (LENPIC Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 93, 044002 (2016), arXiv:1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='07218 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [72] One commonly applies a Similarity Renormalization Group (SRG) transformation to the NN potential in order to improve the convergence of the many-body calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, this leads to induced three-nucleon forces that are non-negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' omitting those would lead to a strong dependence on the SRG parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We therefore choose to not employ such a transformation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' For the binding energies we use an exponential extrapolation to the complete basis, with associated uncertainties, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [71] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Radii converge rather slowly in a harmonic oscillator basis, and they do not necessarily converge monotonically with increasing Nmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' furthermore, in the scattering calculations we use densities obtained at fixed values of the harmonic oscillator parameters Nmax and ¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' We therefore simply give in Table I our results for the point- proton radii of 12C and 16O at Nmax = 10, averaged over the range 16 ≤ ¯hΩ ≤ 28 MeV (the same range as is used for the scattering calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The numerical uncertainty estimates for the radii listed in Table I correspond to the spread over this ¯hΩ interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' this is a systematic uncertainty due to the Gaussian fall-off of harmonic oscillator basis functions, and is therefore strongly correlated for the different chiral orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' However, the trend of a significant increase in the radii going from LO to NLO, followed by a smaller increase going from NLO to N2LO, is robust, and correlates with the decrease in binding energies going from LO to NLO to N2LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Note that we did not include any chiral EFT corrections to the R2 operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' and the experimental point-proton radii are extracted from the charge radius measured in electron scattering experiments, using standard proton and neutron finite-size corrections, relativistic corrections, and meson-exchange corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [73] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ekstr¨om, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Baardsen, C.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 27, 459 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [85] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ieiri, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Sakaguchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nakamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Sakamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hirata, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nakano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Noro, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ikegami, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A 257, 253 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [86] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Sakaguchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Nakamura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hatanaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Goto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Noro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ohtani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Sakamoto, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kobayashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' B 89, 40 (1979).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kelly, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Bertozzi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Buti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Finn, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hersman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hyde-Wright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kovash, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Murdock, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Norum, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Pugh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Rad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Bacher, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Emery, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Foster, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Miller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Berman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Love, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Carr, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Petrovich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 39, 1222 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 14 [89] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kelly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Finn, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Bertozzi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Buti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hersman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hyde-Wright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Hynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Kovash, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Murdock, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ulmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Bacher, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Emery, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Foster, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Miller, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Berman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' C 41, 2504 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' TABLES 4He 12C 16O Binding energy (MeV) LO 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='45(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='01) 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') NLO 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='53(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='01)(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=')(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=')(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') N2LO 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='01)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=')(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=')(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=') expt 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='30 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='16 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='62 Point-proton radius (fm) LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='85(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='17) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='2) NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='40(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='02)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='08) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='04(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='16)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='09) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='16)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='10) N2LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='42(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='02)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='02) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='12(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='15)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='03) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='15)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='03) expt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='58 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Ground state binding energies (top) and point-proton RMS radii (bottom) of 4He, 12C, and 16O with LO, NLO, and N2LO LENPIC SCS NN potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Both our estimated numerical uncertainties (first set of uncertainties) and chiral truncation uncertainty estimates (second set of uncertainties, not evaluated for LO) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The angular distribution of the differential cross section divided by the Rutherford cross section (upper panel) and the analyzing power (Ay) for elastic proton scattering from 8He at 71 MeV laboratory kinetic energy as function of the momentum transfer q and the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' angle calculated with the LENPIC SCS chiral interaction [19] with a cutoff R = 1 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The calculations are based on nonlocal densities using ¯hΩ = 14 MeV at Nmax = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The solid (red) line stands for using the Watson optical potential while the black (dashed) line represents the KMT prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Oc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [deg] 20 40 60 80 100 120 103 102 101 8He 100 71 MeV 10-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 KMT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 Watson 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 q [fm-1]16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The expansion parameter Q, defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' (30) where Λb = 600 MeV, as a function of the center-of-mass angle θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' for a range of lab projectile energies Elab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' In this case of nucleon-nucleus (NA) elastic scattering, the transition between when the expansion parameter is dominated by the center-of-mass momentum and the momentum transfer can easily be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='6 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='4 A = 4, Eiab = 65 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='2 A = 4,Eiab = 71 MeV A = 4,Elab = 100 MeV A = 4, Eiab = 200 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 20 40 60 80 100 120 140 160 0 180 Oc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [deg]17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Reaction cross section for proton scattering on (a) 4He at 65 MeV and (b) 16O at 100 MeV, both at N2LO as a function of Nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The error bars show a 68% credible interval (CI) from using a pointwise error estimation with the LO, NLO, and N2LO results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The shaded regions show variations with respect to the harmonic oscillator parameter ¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The values of the expansion parameters used were Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='47 for4He at 65 MeV and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='69 for 16O at 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Note the different scales in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 160 4He(p,p)4He, 65 MeV (a) 155 150 145 (mb) 140 135 6 130 125 hΩ = 16 - 28 MeV 120 68% CI 115 8 10 12 14 16 18 20 Nmax600 160(p,p)160, 100 MeV (b) 500 400 (mb) 300 200 100 hΩ = 16 - 28 MeV 68% CI 0 2 4 6 8 10 Nmax18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Differential cross section divided by Rutherford for proton scattering on 4He at (first row) 65 MeV, (second row) 71 MeV, (third row) 100 MeV, and (fourth row) 200 MeV for LO (left column), NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Black dots are experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [76] (65 MeV), [77] (71 MeV), [78] (100 MeV), and [79] (200 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [deg] Oc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [deg] Oc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [deg] 10 20 30 40 50 60 10 20 30 40 50 60 10 20 30 40 50 60 500 N2LO 400 LO NLO 65 MeV /Ruth 300 200 100 0 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='4 q [fm- 11 q [fm-1] q [fm-21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Analyzing power for proton scattering on 12C at (first row) 65 MeV and (second row) 122 MeV for LO (left column), NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Black dots are experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [85] (65 MeV) and [84] (122 MeV).' 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 N2LO LO NLO 122 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content='6 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Analyzing power for proton scattering on 16O at (first row) 65 MeV, (second row) 100 MeV, and (third row) 135 MeV for LO (left column), NLO (middle column), and N2LO (right column) with corresponding 1σ (darker bands) and 2σ (lighter bands) error bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Black dots are experimental data from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [86] (65 MeV), [87] (100 MeV), and [88] (135 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Figure taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' Differential cross section divided by the Rutherford cross section (top) and analyzing power (bottom) for proton scattering from 16O at 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfEAn3/content/2301.04293v1.pdf'} +page_content=' The first 3 columns are the same as the second rows of Figs.' metadata={'source': 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+PACS 03.65.Ud – Entanglement and quantum non-locality +PACS 04.62.+v – Quantum fields in curved spacetime +Abstract –We study the genuine tripartite entanglement (GTE), one-tangle and two-tangle of +W state of fermionic fields in the background of a Schwarzschild black hole. We find that, with +the increase of the Hawking temperature, the GTE of W state first decreases and then tends to +zero, while the GTE of GHZ state first decreases and then freezes. We also find that the Hawking +effect can completely destroy the two-tangle of W state, whilie one-tangle first decreases and then +appears freezing phenomenon with the growth of the Hawking temperature. These results are +helpful to guide us to select appropriate quantum states and quantum resources to deal with +relativistic quantum information tasks. +Introduction. – +Quantum entanglement arises from +the tensorproduct structure of the Hilbert space and the +superposition principle. +Quantum entanglement is one +of the most valuable resource of quantum tasks, such as +quantum teleportation [1], dense coding [2] and quantum +key distribution [3]. One of the most famous multipartite +entangled states is the W state. +Up to now, many ex- +periments can generate W states. By tracing out a single +particle from the W state, the remaining bipartite state +is still entangled. Therefore, the W state exhibits a high +persistency of quantum entanglement against particle loss +[4]. W state also has the advantage that bipartite entan- +glement of its particles persists even if the third particle +suffers from decoherence [5,6]. Because of these merits, W +state has been widely used in quantum information [7]. +General relativity predicts black holes, and the gravi- +tational waves are detected by the Virgo and LIGO de- +tectors from a binary black hole merger system that in- +directly proves the existence of black holes in our unverse +[8]. +On the other hand, the releasing of the images of +M87* [9–14] and Sgr A* [15] directly proved the existence +of black holes. Black holes are basal objects in gravity +of Einstein which are totally composed of several con- +(a)Email: smwu@lnnu.edu.cn +(b)Email: xwfan0825@163.com +(c)Email: huangxiaoli1982@foxmail.com (corresponding author) +(d)Email: hszeng@hunnu.edu.cn (corresponding author) +served quantities, such as angular momentum, mass, and +charge. +The interior and exterior of the event horizon +of the black holes are separated. The Hawking radiation +comes from the particle-antiparticle pairs of autonomous +creation by quantum fluctuations in vicinal event horizon +[16–18]. +Nowadays, there are a growing number of re- +searchers who pay attention to the study of black holes. +Specifically, the Schwarzschild solution of the four space- +time dimensions may represent the simplest black hole. +Quantum information takes a significant role in the study +of the information loss problem and thermodynamics of +black holes [19–22]. Therefore, it is important that the in- +fluence of the relativistic effect on quantum maneuverabil- +ity is studied in curved spacetime. Bipartite entanglement +has been studied extensively in Schwarzschild spacetime +[23–28], while tripartite entanglement of W state has not +been studied in Schwarzschild spacetime. This is the main +motivation of our study. +Various types of tripartite entanglement are important +and interesting for Alice, Bob and Charlie. Bipartite en- +tanglement includes entanglement between two particles +and entanglement between one particle and the remaining +two particles as one party. Tripartite entanglement exists +in the whole tripartite system, which cannot be simplified +by any combination of various bipartite entanglement [29]. +Recently, quantum entanglement has come a long way in +quantum information theory. The concepts of one-tangle +p-1 + +F. Author et al. +and two-tangle have been proposed. The one-tangle de- +scribes the bipartite entanglement between one particle +and the remaining two particles, and the two-tangle de- +scribes the bipartite entanglement in any reduced bipartite +systems [30,31]. In terms of one-tangle and two-tangle, we +can define the measure of tripartite entanglement, which +is called the residual-tangle or residual entanglement [30]. +We define the smallest residual entanglement as genuine +tripartite entanglement (GTE). GTE is an important type +of quantum entanglement which provides significant ad- +vantages in quantum tasks. GTE is a crucial resources +for measurement-based quantum computing [32] and high- +precision metrology [33], and has an important function in +quantum phase transitions [34,35]. +In this paper, we study GTE, one-tangle and two-tangle +of W state for free fermionic modes in the background of +eternal Schwarzschild black hole. We assume that Alice, +Bob and Charlie initially share a tripartite pure state in +an asymptotically flat region. Then Alice still stays sta- +tionary at an asymptotically flat region, while Bob and +Charlie hover near the event horizon of the black hole. Al- +ice observes a vacuum state, which would be detected as a +thermal state from Bob and Charlie’s point of view. The +Hawking temperature T of the thermal bath rests with +the surface gravity κ of the black hole from a viewpoint +of general relativity. We will study the influence of the +Hawking effect on GTE, one-tangle and two-tangle of W +state in Schwarzschild spacetime, by making a comparison +with the GHZ state in Schwarzschild spacetime. +The structure of the paper is as follows. In the next +section, we briefly introduce the measures of one-tangle, +two-tangle and GTE. In the third section, we describe +the quantization of Dirac fields in a Schwarzschild black +hole. +In the fourth section, we study the influence of +the Hawking effect on GTE, one-tangle and two-tangle +in Schwarzschild spacetime. The last section is devoted to +a brief conclusion. +Measures of one-tangle, two-tangle and genuine +tripartite entanglement. – +Quantifying entanglement +in tripartite systems is generally complicated. A method +to determine the existence of tripartite correlation in an +entangled state is to explore the entanglement distribution +in the tripartite system. Different from classical correla- +tions, quantum entanglement is monogamous, which is not +freely shared in multiple subsystems of a quantum system +[36]. Therefore, we can use the residual entanglement as a +way for measuring nonclassical correlations of the tripar- +tite systems. The basis for analyzing residual entangle- +ment is negativity. Negativity has been used extensively, +which quantifies the entanglement in a state as the degree +of seeing whether the system is still entangled. A system is +entangled when its density matrix of the partial transpose +has negative eigenvalues. The bipartite entanglement of +a tripartite system has two types: entanglement between +two subsystems, and entanglement between one subsys- +tem and the remaining two subsystems as one party. The +bipartite entanglement between one subsystem and the re- +maining two subsystems is called one-tangle, +Nα(βγ) = ∥ρTα +αβγ∥ − 1. +(1) +The bipartite entanglement between two subsystems is +called two-tangle, +Nαβ = ∥ρTα +αβ∥ − 1. +(2) +Here Tα is partial transpose of ραβγ and ραβ relative to +observer α. Note that ∥A∥ − 1 is actually equal to the +two times of the sum of absolute values of the negative +eigenvalues of the operator A [30, 31]. Thus, one-tangle +and two-tangle also can be expressed as +Nα(βγ) = 2 +n +� +i=1 +|λ(−) +α(βγ)|i, +(3) +Nαβ = 2 +n +� +j=1 +|λ(−) +αβ |j. +(4) +For three parties, the Coffman-Kundu-Wootters in- +equality describes the monogamy constraint +N 2 +α(βγ) ≥ N 2 +αβ + N 2 +αγ. +(5) +The right hand side of inequality (5) represents the sum +of the square of the two two-tangles between subsystem α +and the every one of remaining subsystems. The other side +quantifies the square of the one-tangle between subsystem +α and the remaining subsystems. +We choose the minimum of each non-negative difference +between the two sides of inequality in a subsystem, which +is called minimally residual tripartite entanglement +E(α|β|γ) = min +(α,β,γ)[N 2 +α(βγ) − N 2 +αβ − N 2 +αγ], +(6) +where (α, β, γ) shows all the permutations of the three +mode indices. +In the tripartite quantum system, a sig- +nificance of the minimally residual entanglement denotes +the genuine tripartite entanglement (GTE) shared by the +three subsystems [30,31]. +Quantization of Dirac fields in a Schwarzschild +black hole . – +The metric in Schwarzschild spacetime +can be expressed as +ds2 += +−(1 − 2M +r )dt2 + (1 − 2M +r )−1dr2 ++r2(dθ2 + sin2 θdϕ2), +(7) +where M is the mass of the black hole, and r is radial +coordinates. We set G, c, ¯h and kB as unity in this paper. +The Dirac equation [37] in a general spacetime is written +as +[γaeaµ(∂µ + Γµ)]Φ = 0, +(8) +p-2 + +Title +where γa are the Dirac matrices, the four-vectors eaµ is +the inverse of the tetrad eaµ defined by gµν = ηabeaµebν +with ηab = diag(−1, 1, 1, 1), and Γµ = +1 +8[γa, γb]eaνebν;µ +are the spin connection coefficients. +Specifically, the Dirac equation in Schwarzschild space- +time can be expressed as [38] +− +γ0 +� +1 − 2M +r +∂Φ +∂t + γ1 +� +1 − 2M +r [ ∂ +∂r + 1 +r +(9) ++ +M +2r(r − 2M)]Φ + γ2 +r ( ∂ +∂θ + cot θ +2 +)Φ + +γ3 +r sin θ +∂Φ +∂ϕ = 0, +where γi (i = 0, 1, 2, 3) are the Dirac matrices. By solving +Eq.(9), we obtain the positive (fermion) frequency outgo- +ing solutions outside and inside regions of the event hori- +zon [38–41] +Φ+ +k,out ∼ φ(r)e−iωu, +(10) +Φ+ +k,in ∼ φ(r)eiωu, +(11) +where φ(r) represents four-component Dirac spinor, ω is +a monochromatic frequency, k is the wave vector, ω = |k| +in the massless Dirac field and u = t−r∗ with the tortoise +coordinate r∗ = r + 2M ln r−2M +2M . Therefore, we expand +Dirac field Φ through Eqs.(10) and (11) as +Φ += +� +dk[ˆain +k Φ+ +k,in + ˆbin† +−kΦ− +−k,in +(12) ++ +ˆaout +k Φ+ +k,out + ˆbout† +−k Φ− +−k,out], +where ˆain +k and ˆbin† +−k are the annihilation operator of fermion +and the creation operator of antifermion for the interior +of the event horizon, respectively, and ˆaout +k +and ˆbout† +−k +are +the annihilation operator of fermion and creation operator +of antifermion for the exterior of the event horizon in the +Schwarzschild black hole, respectively. +According to Domour and Ruffini’s suggestions [42], one +provides a complete basis for the positive energy mode +(Kruskal mode) by the analytic extension of Eqs.(10) and +(11) +Ψ+ +k,out = e−2πMωΦ− +−k,in + e2πMωΦ+ +k,out, +(13) +Ψ+ +k,in = e−2πMωΦ− +−k,out + e2πMωΦ+ +k,in. +(14) +Therefore, we also use the Kruskal modes to expand the +Dirac field [39] +Φ += +� +dk[2 cosh(4πMω)]− 1 +2 [ˆcin +k Ψ+ +k,in +(15) ++ +ˆdin† +−kΨ− +−k,in + ˆcout +k +Ψ+ +k,out + ˆdout† +−k Ψ− +−k,out], +where ˆcσ +k and ˆdσ† +k +with σ = (in, out) are the annihila- +tion operators of fermion and creation operators of an- +tifermion acting on the Kruskal vacuum. +Eqs.(12) and +(15) are shown that Dirac field can be decomposed by +Schwarzschild and Kruskal modes, respectively, which lead +to the Bogoliubov transformations between Schwarzschild +and Kruskal operators +ˆcout +k += +1 +√ +e−8πMω + 1 +ˆaout +k +− +1 +√ +e8πMω + 1 +ˆbin† +−k, +(16) +ˆcout† +k += +1 +√ +e−8πMω + 1 +ˆaout† +k +− +1 +√ +e8πMω + 1 +ˆbin +−k. +(17) +According to Bogoliubov transformations, the expres- +sions of the Kruskal vacuum and excited states in the +Schwarzschild black hole are written as +|0⟩K += +1 +� +e− ω +T + 1 +|0⟩out|0⟩in + +1 +� +e +ω +T + 1 +|1⟩out|1⟩in, +|1⟩K += +|1⟩out|0⟩in, +(18) +where T = +1 +8πM is the Hawking temperature, |n⟩out and +|n⟩in are the number states for the fermion in the exterior +region and the antifermion in the interior region of the +event horizon of the black hole. +When an outside observer travels through the Kruskal +vacuum, Bob’s detector records the number of particles, +which can be expressed as +NF =K ⟨0|ˆaout† +k +ˆaout +k |0⟩K = +1 +e +ω +T + 1. +(19) +The equation represents that the observer detects a ther- +mal Fermi-Dirac distribution of the particles in the exte- +rior of the event horizon of the black hole. +The influence of the Hawking effect on GTE, +one-tangle and two-tangle in the Schwarzschild +black hole. – +We assume that Alice, Bob and Charlie +initially stay stationary at an asymptotically flat region +and share a W state +|W⟩ = +1 +√ +3[|0A0B1C⟩ + |0A1B0C⟩ + |1A0B0C⟩], +(20) +where the subscripts A, B and C denote the qubits shared +by Alice, Bob and Charlie, respectively. Subsequently, we +consider Alice still stays stationary at an asymptotically +flat region, while Bob and Charlie hover near the event +horizon of the black hole. According to Eqs.(18) and (20), +the wave function of W state can be rewritten as +| ¯W⟩ += +1 +√ +3 +[µ|0A0B0 ¯ +B1C0 ¯ +C⟩ + µ|0A1B0 ¯ +B0C0 ¯ +C⟩ ++ +ν|0A1B0 ¯ +B1C1 ¯ +C⟩ + ν|0A1B1 ¯ +B1C0 ¯ +C⟩ ++ +µ2|1A0B0 ¯ +B0C0 ¯ +C⟩ + µν|1A0B0 ¯ +B1C1 ¯ +C⟩ ++ +µν|1A1B1 ¯ +B0C0 ¯ +C⟩ + ν2|1A1B1 ¯ +B1C1 ¯ +C⟩], (21) +where µ = +1 +√ +e− ω +T +1 and ν = +1 +√ +e +ω +T +1. +Since Bob and +Charlie cannot access the modes inside event horizon of +the black hole, we should trace over the inaccessible ¯B +p-3 + +F. Author et al. +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T +EA B C +W +GHZ +Fig. 1: The GTE of W and GHZ states as a function of the +Hawking temperature T . +and ¯C modes. Therefore, by tracing over the inaccessible +modes, we obtain the density matrix +ρABC = 1 +3 + + + + + + + + + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +µ2 +µ2 +0 +µ3 +0 +0 +0 +0 +µ2 +µ2 +0 +µ3 +0 +0 +0 +0 +0 +0 +2ν2 +0 +µν2 +µν2 +0 +0 +µ3 +µ3 +0 +µ4 +0 +0 +0 +0 +0 +0 +µν2 +0 +µ2ν2 +0 +0 +0 +0 +0 +µν2 +0 +0 +µ2ν2 +0 +0 +0 +0 +0 +0 +0 +0 +ν4 + + + + + + + + + + + + +. (22) +According +to +Eq.(6), +the +GTE +of +W +state +in +Schwarzschild spacetime can be expressed as +EW +(A|B|C) += +1 +576{8µ4 − 8µ2 + 2 +√ +2[35µ4 + 28µ4(2µ2 − 1) ++ +µ4(8µ4 − 8µ2 + 1)] +1 +2 }2 − 1 +18{ +√ +2[5 ++ +4(2µ2 − 1) + (8µ4 − 8µ2 + 1)] +1 +2 − 2}2. +(23) +From Eq.(23) we can see that the GTE of W state +depends on the Hawking temperature T , which means +that the Hawking radiation will affect the GTE in the +Schwarzschild black hole. On the other hand, the GTE of +GHZ state in curved spacetime reads EGHZ +(A|B|C) = 1 +4[µ2 − +µ2ν2 + µ +� +ν4µ2 + µ2]2 [38]. +In Fig.1, we plot the GTE of W and GHZ states +as a function of the Hawking temperature T in the +Schwarzschild black hole. +We find that the GTE of W +state first decreases and then tends to zero with the in- +crease of the Hawking temperature T , while GTE of GHZ +state first decreases and then freezes with the increase of +the Hawking temperature T [38]. We also find that the +GTE of W state is smaller than that of GHZ state. This +implies that the GTE of GHZ state is more effective for +resisting the Hawking effect and is more suitable for pro- +cessing relativistic quantum information. +We also study bipartite entanglement NA(BC), NB(AC), +NC(AB), +NAB, +NAC +and NBC +for the W state in +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T +Ent +NA (BC) +NB (AC) +Fig. 2: The one-tangles of W state as a function of the Hawking +temperature T . +Schwarzschild spacetime. +Firstly, we consider the one- +tangles NA(BC), NB(AC) and NC(AB). Taking the trans- +pose of ρABC with respect mode A, we get +ρTA +ABC = 1 +3 + + + + + + + + + + + + +0 +0 +0 +0 +0 +µ3 +µ3 +0 +0 +µ2 +µ2 +0 +0 +0 +0 +µν2 +0 +µ2 +µ2 +0 +0 +0 +0 +µν2 +0 +0 +0 +2ν2 +0 +0 +0 +0 +0 +0 +0 +0 +µ4 +0 +0 +0 +µ3 +0 +0 +0 +0 +µ2ν2 +0 +0 +µ3 +0 +0 +0 +0 +0 +µ2ν2 +0 +0 +µν2 +µν2 +0 +0 +0 +0 +ν4 + + + + + + + + + + + + +, +which has the negative eigenvalue +1 +48[−8µ4 + 8µ2 − +2 +√ +2 +� +35µ4 + 28µ4(2µ2 − 1) + µ4(8µ4 − 8µ2 + 1)]. +Thus +the one-tangle NA(BC) is +NA(BC) += +1 +24{8µ4 − 8µ2 + 2 +√ +2[35µ4 + 28µ4 +(2µ2 − 1) + µ4(8µ4 − 8µ2 + 1)] +1 +2 }. (24) +Similarly, taking the transpose with respect the mode B, +ρTB +ABC = 1 +3 + + + + + + + + + + + + +0 +0 +0 +µ2 +0 +0 +µ3 +0 +0 +µ2 +0 +0 +µ3 +0 +0 +µν2 +0 +0 +µ2 +0 +0 +0 +0 +0 +µ2 +0 +0 +2ν2 +0 +0 +µν2 +0 +0 +µ3 +0 +0 +µ4 +0 +0 +0 +0 +0 +0 +0 +0 +µ2ν2 +0 +0 +µ3 +0 +0 +µν2 +0 +0 +µ2ν2 +0 +0 +µν2 +0 +0 +0 +0 +0 +ν4 + + + + + + + + + + + + +. +Due to the complexity of the expression of NB(AC), we do +not write it out here. Since Bob and Charlie are symmet- +ric, we obtain NB(AC) = NC(AB). +In Fig.2, we plot the one-tangles of W state as a func- +tion of the Hawking temperature T . We can see that one- +tangles of W state first decrease and then appear freezing +phenomenon with the increase of the Hawking temper- +ature T . +We also find that one-tangle of GHZ state is +bigger than one-tangle of W state in curved spacetime, +which means that one-tangle of GHZ state can effectively +p-4 + +Title +resist Hawking effect [38]. This indicates that one-tangle +of GHZ state is more suitable for processing relativistic +quantum information. +It has been shown that [43, 44], +however, the coherence of W state is always bigger than +the coherence of GHZ state in Schwarzschild spacetime, +meaning that the coherence of W state is more suitable +for processing relativistic quantum information than the +GHZ state. Therefore, we should choose suitable quan- +tum resources as required to process relativistic quantum +information. +Next, we consider the two-tangles NAB, NAC and NBC. +Taking trace over the mode C from ρABC, one gets +ρAB = 1 +3 + + + + +µ2 +0 +0 +0 +0 +1 + ν2 +µ +0 +0 +µ +µ2 +0 +0 +0 +0 +ν2 + + + + . +(25) +The transpose with respect mode A is +ρTA +AB = 1 +3 + + + + +µ2 +0 +0 +µ +0 +1 + ν2 +0 +0 +0 +0 +µ2 +0 +µ +0 +0 +ν2 + + + + . +(26) +According to Eqs.(4) and (26), the two-tangle NAB can +be expressed as +NAB = 1 +6[ +√ +2 +� +4(2µ2 − 1) + (8µ4 − 8µ2 + 1) + 5 − 2]. (27) +Since +Bob +and +Charlie +are +symmetric, +we +have +NAB=NAC. +Similarly, we can get +ρBC = 1 +3 + + + + +µ4 +0 +0 +0 +0 +µ2(1 + ν2) +µ2 +0 +0 +µ2 +µ2(1 + ν2) +0 +0 +0 +0 +ν2(2 + ν2) + + + + , (28) +and its transpose +ρTB +BC = 1 +3 + + + + +µ4 +0 +0 +µ2 +0 +µ2(1 + ν2) +0 +0 +0 +0 +µ2(1 + ν2) +0 +µ2 +0 +0 +ν2(2 + ν2) + + + + . (29) +Employing Eqs.(4) and (29), the two-tangle NBC can be +written as +NBC += +1 +12(−12 − 8µ4 + 16µ2 +(30) ++ +2 +√ +2 +� +18 + 40µ4 − 48µ2). +In Fig.3, we plot the two-tangles of W state as a function +of the Hawking temperature T . It is shown that the two- +tangle NAB between Alice and Bob first decreases and +then freezes with the increase of the Hawking temperature +T . However, the two-tangle NBC between Bob and Charlie +first decreases and then suffers from sudden death with +2 +4 +6 +8 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +T +Ent +NAB +NBC +Fig. 3: The NAB and NBC of W state as a function of the +Hawking temperature T . +the growth of the Hawking temperature T . This means +that the Hawking effect completely destroys the two-tangle +NBC of W state. These results are in contrast with the +two-tangles of the tripartite GHZ state, which are zero in +curved spacetime [38]. +Finally, +we +compare +fermionic +entanglement +with +bosonic entanglement of tripartite states in Schwarzschild +spacetime. We initially assume that Alice, Bob and Char- +lie stay stationary at an asymptotically flat region and +share GHZ and W states of bosonic field. According to +Bogoliubov transformations, the Kruskal vacuum and ex- +cited states of bosonic field in Schwarzschild spacetime can +be expressed as +|0⟩B +K += +� +1 − e− ω +T +∞ +� +n=0 +e− nω +2T |n⟩B +out|n⟩B +in, +(31) +|1⟩B +K += +(1 − e− ω +T ) +∞ +� +n=0 +e− nω +2T √ +n + 1|n + 1⟩B +out|n⟩B +in, +where |n⟩B +out and |n⟩B +in are the number states for the boson +in the exterior region and the antiboson in the interior +region of the event horizon of the black hole, respectively +[24]. Hereafter, we omit the mark B for simplicity unless it +causes confusion. Because the tripartite entanglement of +bosonic field is very complex in Schwarzschild spacetime, +we consider a simpler model: Charlie hovers near the event +horizon of the black hole, while Alice and Bob still stay +stationary at an asymptotically flat region. According to +Eq.(31), the wave functions of GHZ and W states can be +rewritten as +|GHZ⟩B +ABC += +1 +√ +2α +∞ +� +n=0 +γn(|00n⟩|n⟩in +(32) ++ +√n + 1 +α +|11n + 1⟩|n⟩in), +|W⟩B +ABC += +1 +√ +3α +∞ +� +n=0 +γn( +√n + 1 +α +|00n + 1⟩ +(33) ++|01n⟩ + |10n⟩) +� +|n⟩in, +p-5 + +F. Author et al. +where α = +1 +√ +1−e− ω +T and γ = +1 +√ +e +ω +T . Since Charlie cannot +access the modes inside the event horizon of the black hole, +we should trace over the inaccessible mode and obtain the +density operators +ρB +GHZ += +1 +2α2 +∞ +� +n=0 +γ2n{|00n⟩⟨00n| +(34) ++ +√n + 1 +α +[|00n⟩⟨11n + 1| + |11n + 1⟩⟨00n|] ++n + 1 +α2 |11n + 1⟩⟨11n + 1|}, +ρB +W += +1 +3α2 +∞ +� +n=0 +γ2n{n + 1 +α2 |00n + 1⟩⟨00n + 1| +(35) ++|01n⟩⟨01n| + |10n⟩⟨10n| + +√n + 1 +α +[|00n + 1⟩ +⟨01n| + |01n⟩⟨00n + 1| + |00n + 1⟩⟨10n| + +|10n⟩⟨00n + 1|] + |01n⟩⟨10n| + |10n⟩⟨01n|}. +Employing Eqs.(3), (4) and (6), we can obtain the GTE +and bipartite entanglement N B +C(AB) of the GHZ and W +states of bosonic field in Schwarzschild spacetime. Since +the expressions are very complex, we do not write them. +In Fig.4, we plot the GTE, N B,GHZ +C(AB) and N B,W +C(AB) of GHZ +and W states of bosonic field as a function of the Hawking +temperature T . From Fig.4 (a), we can see that the GTEs +of GHZ and W states of bosonic field vanish in the infinite +Hawking temperature T , while the GTE of GHZ state of +fermionic field always survives in curved spacetime [38]. +This means that the GTE of GHZ state of fermionic field +may be more suitable for relativistic quantum information +tasks. By the numerical calculation, we find +lim +T →∞ N B,GHZ +C(AB) = 0, +lim +T →∞ N B,W +C(AB) = 0. +(36) +It means that N B,GHZ +C(AB) and N B,W +C(AB) of GHZ and W states +of bosonic field vanish in the infinite Hawking temperature +T . However, one-tangles of GHZ and W states of fermionic +field always survive when only Charlie hovers near the +event horizon of the black hole [38]. The disparity between +the Dirac and scalar fields is caused by the differences +between Bose-Einstein and Fermi-Dirac statistics. This is +because that Fermi-Dirac distribution protects tripartite +entanglement of fermionic field. In addition, one-tangles +N B,GHZ +A(BC) +(N B,GHZ +B(AC) +) and N B,W +A(BC) (N B,W +B(AC) ) of GHZ and +W states of bosonic field can survive for any T [45]. This +conclusion is consistent with the fermionic field [38]. +Conclusions . – +The effect of the Hawking effect +on the genuine tripartite entanglement (GTE), one-tangle +and two-tangle of W state in Schwarzschild spacetime have +been investigated. We assume that Alice, Bob and Charlie +initially share a W state at an asymptotically flat region. +Then Alice still stays stationary at an asymptotically flat +region, while Bob and Charlie hover near the event horizon +GHZ +W +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +T +EBA B C +( +a) +NC (AB) +B,��� +NC (AB) +B,W +2 +4 +6 +8 +1� +0.0 +0� � +0.4 +�� � +� + +  + +Ent +(b) +Fig. 4: The GTE, N B,GHZ +C(AB) +and N B,W +C(AB) of GHZ and W states +of bosonic field as a function of the Hawking temperature T . +of the black hole. We have found that, with the increase +of the Hawking temperature, the GTE of W state first de- +creases and then approaches zero, while GTE of GHZ state +first decreases and then appears freezing phenomenon [38]. +This implies that the GTE of GHZ state is more effective +for resisting Hawking effect. +We have also found that, with the growth of the Hawk- +ing temperature, the two-tangle between Alice and Bob +(Charlie) of W state first decreases and then freezes, while +two-tangle between Bob and Charlie first reduces and then +suffers from sudden death. This is different from the case +of GHZ state, whose two-tangles are zero in Schwarzschild +spacetime [38]. +We have shown that the one-tangle of +W state first decreases and then appears freezing phe- +nomenon with the increase of the Hawking temperature, +which is always smaller than the one-tangle of GHZ state +in curved spacetime. This is different from the behavior +of quantum coherence, where the coherence of W state is +bigger than that of GHZ state in Schwarzschild spacetime +[43,44]. Therefore, for different quantum states, we should +choose suitable quantum resources to process relativistic +quantum information. +Finally, +we +compare +bosonic +entanglement +with +p-6 + +Title +fermionic entanglement of tripartite states when only +Charlie hovers near the event horizon of the Schwarzschild +black hole. We find that the GTEs of GHZ and W states +of bosonic field reduce to zero with the growth of the +Hawking temperature, while the GTE of GHZ state of +fermionic field can survive for any Hawking temperature. +We also find that not all the one-tangles of GHZ and W +states of bosonic field can survive in the infinite Hawking +temperature limit, while all one-tangles of GHZ and W +states of fermionic field always survive in Schwarzschild +spacetime. +This is because that Fermi-Dirac distribu- +tion protects tripartite entanglement of fermionic field in +Schwarzschild spacetime. It means that tripartite entan- +glement of fermionic field is more suitable for processing +relativistic quantum information. +∗ ∗ ∗ +This work is supported by the National Natural Science +Foundation of China (Grant Nos. 12205133, 1217050862, +11275064, 11975064 and 12075050 ), LJKQZ20222315 and +2021BSL013. +REFERENCES +[1] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. +Peres, and W. K. Wootters, Phys. 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A 83, 012111 +(2011). +p-7 + diff --git a/X9AzT4oBgHgl3EQfmf32/content/tmp_files/load_file.txt b/X9AzT4oBgHgl3EQfmf32/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b92bd24b07448df7b7ec5a4d54eb0b90608e378c --- /dev/null +++ b/X9AzT4oBgHgl3EQfmf32/content/tmp_files/load_file.txt @@ -0,0 +1,519 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf,len=518 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='01566v1 [quant-ph] 4 Jan 2023 epl draft Genuine tripartite entanglement of W state subject to Hawking effect of a Schwarzschild black hole Shu-Min Wu1(a), Xiao-Wei Fan1(b), Xiao-Li Huang1(c), Hao-Sheng Zeng2(d) 1 Department of Physics, Liaoning Normal University, Dalian 116029, China 2 Department of Physics, Hunan Normal University, Changsha 410081, China PACS 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='Dy – Quantum aspects of black holes, evaporation, thermodynamics PACS 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='Ud – Entanglement and quantum non-locality PACS 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='+v – Quantum fields in curved spacetime Abstract –We study the genuine tripartite entanglement (GTE), one-tangle and two-tangle of W state of fermionic fields in the background of a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We find that, with the increase of the Hawking temperature, the GTE of W state first decreases and then tends to zero, while the GTE of GHZ state first decreases and then freezes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We also find that the Hawking effect can completely destroy the two-tangle of W state, whilie one-tangle first decreases and then appears freezing phenomenon with the growth of the Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' These results are helpful to guide us to select appropriate quantum states and quantum resources to deal with relativistic quantum information tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' – Quantum entanglement arises from the tensorproduct structure of the Hilbert space and the superposition principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Quantum entanglement is one of the most valuable resource of quantum tasks, such as quantum teleportation [1], dense coding [2] and quantum key distribution [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' One of the most famous multipartite entangled states is the W state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Up to now, many ex- periments can generate W states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' By tracing out a single particle from the W state, the remaining bipartite state is still entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, the W state exhibits a high persistency of quantum entanglement against particle loss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' W state also has the advantage that bipartite entan- glement of its particles persists even if the third particle suffers from decoherence [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Because of these merits, W state has been widely used in quantum information [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' General relativity predicts black holes, and the gravi- tational waves are detected by the Virgo and LIGO de- tectors from a binary black hole merger system that in- directly proves the existence of black holes in our unverse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' On the other hand, the releasing of the images of M87* [9–14] and Sgr A* [15] directly proved the existence of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Black holes are basal objects in gravity of Einstein which are totally composed of several con- (a)Email: smwu@lnnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='cn (b)Email: xwfan0825@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='com (c)Email: huangxiaoli1982@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='com (corresponding author) (d)Email: hszeng@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='cn (corresponding author) served quantities, such as angular momentum, mass, and charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The interior and exterior of the event horizon of the black holes are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The Hawking radiation comes from the particle-antiparticle pairs of autonomous creation by quantum fluctuations in vicinal event horizon [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Nowadays, there are a growing number of re- searchers who pay attention to the study of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Specifically, the Schwarzschild solution of the four space- time dimensions may represent the simplest black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Quantum information takes a significant role in the study of the information loss problem and thermodynamics of black holes [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, it is important that the in- fluence of the relativistic effect on quantum maneuverabil- ity is studied in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Bipartite entanglement has been studied extensively in Schwarzschild spacetime [23–28], while tripartite entanglement of W state has not been studied in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This is the main motivation of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Various types of tripartite entanglement are important and interesting for Alice, Bob and Charlie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Bipartite en- tanglement includes entanglement between two particles and entanglement between one particle and the remaining two particles as one party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Tripartite entanglement exists in the whole tripartite system, which cannot be simplified by any combination of various bipartite entanglement [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Recently, quantum entanglement has come a long way in quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The concepts of one-tangle p-1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Author et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' and two-tangle have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The one-tangle de- scribes the bipartite entanglement between one particle and the remaining two particles, and the two-tangle de- scribes the bipartite entanglement in any reduced bipartite systems [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In terms of one-tangle and two-tangle, we can define the measure of tripartite entanglement, which is called the residual-tangle or residual entanglement [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We define the smallest residual entanglement as genuine tripartite entanglement (GTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' GTE is an important type of quantum entanglement which provides significant ad- vantages in quantum tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' GTE is a crucial resources for measurement-based quantum computing [32] and high- precision metrology [33], and has an important function in quantum phase transitions [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In this paper, we study GTE, one-tangle and two-tangle of W state for free fermionic modes in the background of eternal Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We assume that Alice, Bob and Charlie initially share a tripartite pure state in an asymptotically flat region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Then Alice still stays sta- tionary at an asymptotically flat region, while Bob and Charlie hover near the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Al- ice observes a vacuum state, which would be detected as a thermal state from Bob and Charlie’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The Hawking temperature T of the thermal bath rests with the surface gravity κ of the black hole from a viewpoint of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We will study the influence of the Hawking effect on GTE, one-tangle and two-tangle of W state in Schwarzschild spacetime, by making a comparison with the GHZ state in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In the next section, we briefly introduce the measures of one-tangle, two-tangle and GTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In the third section, we describe the quantization of Dirac fields in a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In the fourth section, we study the influence of the Hawking effect on GTE, one-tangle and two-tangle in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The last section is devoted to a brief conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Measures of one-tangle, two-tangle and genuine tripartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' – Quantifying entanglement in tripartite systems is generally complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' A method to determine the existence of tripartite correlation in an entangled state is to explore the entanglement distribution in the tripartite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Different from classical correla- tions, quantum entanglement is monogamous, which is not freely shared in multiple subsystems of a quantum system [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, we can use the residual entanglement as a way for measuring nonclassical correlations of the tripar- tite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The basis for analyzing residual entangle- ment is negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Negativity has been used extensively, which quantifies the entanglement in a state as the degree of seeing whether the system is still entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' A system is entangled when its density matrix of the partial transpose has negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The bipartite entanglement of a tripartite system has two types: entanglement between two subsystems, and entanglement between one subsys- tem and the remaining two subsystems as one party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The bipartite entanglement between one subsystem and the re- maining two subsystems is called one-tangle, Nα(βγ) = ∥ρTα αβγ∥ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (1) The bipartite entanglement between two subsystems is called two-tangle, Nαβ = ∥ρTα αβ∥ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (2) Here Tα is partial transpose of ραβγ and ραβ relative to observer α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Note that ∥A∥ − 1 is actually equal to the two times of the sum of absolute values of the negative eigenvalues of the operator A [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Thus, one-tangle and two-tangle also can be expressed as Nα(βγ) = 2 n � i=1 |λ(−) α(βγ)|i, (3) Nαβ = 2 n � j=1 |λ(−) αβ |j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (4) For three parties, the Coffman-Kundu-Wootters in- equality describes the monogamy constraint N 2 α(βγ) ≥ N 2 αβ + N 2 αγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (5) The right hand side of inequality (5) represents the sum of the square of the two two-tangles between subsystem α and the every one of remaining subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The other side quantifies the square of the one-tangle between subsystem α and the remaining subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We choose the minimum of each non-negative difference between the two sides of inequality in a subsystem, which is called minimally residual tripartite entanglement E(α|β|γ) = min (α,β,γ)[N 2 α(βγ) − N 2 αβ − N 2 αγ], (6) where (α, β, γ) shows all the permutations of the three mode indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In the tripartite quantum system, a sig- nificance of the minimally residual entanglement denotes the genuine tripartite entanglement (GTE) shared by the three subsystems [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Quantization of Dirac fields in a Schwarzschild black hole .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' – The metric in Schwarzschild spacetime can be expressed as ds2 = −(1 − 2M r )dt2 + (1 − 2M r )−1dr2 +r2(dθ2 + sin2 θdϕ2), (7) where M is the mass of the black hole, and r is radial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We set G, c, ¯h and kB as unity in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The Dirac equation [37] in a general spacetime is written as [γaeaµ(∂µ + Γµ)]Φ = 0, (8) p-2 Title where γa are the Dirac matrices, the four-vectors eaµ is the inverse of the tetrad eaµ defined by gµν = ηabeaµebν with ηab = diag(−1, 1, 1, 1), and Γµ = 1 8[γa, γb]eaνebν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='µ are the spin connection coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Specifically, the Dirac equation in Schwarzschild space- time can be expressed as [38] − γ0 � 1 − 2M r ∂Φ ∂t + γ1 � 1 − 2M r [ ∂ ∂r + 1 r (9) + M 2r(r − 2M)]Φ + γ2 r ( ∂ ∂θ + cot θ 2 )Φ + γ3 r sin θ ∂Φ ∂ϕ = 0, where γi (i = 0, 1, 2, 3) are the Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (9), we obtain the positive (fermion) frequency outgo- ing solutions outside and inside regions of the event hori- zon [38–41] Φ+ k,out ∼ φ(r)e−iωu, (10) Φ+ k,in ∼ φ(r)eiωu, (11) where φ(r) represents four-component Dirac spinor, ω is a monochromatic frequency, k is the wave vector, ω = |k| in the massless Dirac field and u = t−r∗ with the tortoise coordinate r∗ = r + 2M ln r−2M 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, we expand Dirac field Φ through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (10) and (11) as Φ = � dk[ˆain k Φ+ k,in + ˆbin† −kΦ− −k,in (12) + ˆaout k Φ+ k,out + ˆbout† −k Φ− −k,out], where ˆain k and ˆbin† −k are the annihilation operator of fermion and the creation operator of antifermion for the interior of the event horizon, respectively, and ˆaout k and ˆbout† −k are the annihilation operator of fermion and creation operator of antifermion for the exterior of the event horizon in the Schwarzschild black hole, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' According to Domour and Ruffini’s suggestions [42], one provides a complete basis for the positive energy mode (Kruskal mode) by the analytic extension of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (10) and (11) Ψ+ k,out = e−2πMωΦ− −k,in + e2πMωΦ+ k,out, (13) Ψ+ k,in = e−2πMωΦ− −k,out + e2πMωΦ+ k,in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (14) Therefore, we also use the Kruskal modes to expand the Dirac field [39] Φ = � dk[2 cosh(4πMω)]− 1 2 [ˆcin k Ψ+ k,in (15) + ˆdin† −kΨ− −k,in + ˆcout k Ψ+ k,out + ˆdout† −k Ψ− −k,out], where ˆcσ k and ˆdσ† k with σ = (in, out) are the annihila- tion operators of fermion and creation operators of an- tifermion acting on the Kruskal vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (12) and (15) are shown that Dirac field can be decomposed by Schwarzschild and Kruskal modes, respectively, which lead to the Bogoliubov transformations between Schwarzschild and Kruskal operators ˆcout k = 1 √ e−8πMω + 1 ˆaout k − 1 √ e8πMω + 1 ˆbin† −k, (16) ˆcout† k = 1 √ e−8πMω + 1 ˆaout† k − 1 √ e8πMω + 1 ˆbin −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (17) According to Bogoliubov transformations, the expres- sions of the Kruskal vacuum and excited states in the Schwarzschild black hole are written as |0⟩K = 1 � e− ω T + 1 |0⟩out|0⟩in + 1 � e ω T + 1 |1⟩out|1⟩in, |1⟩K = |1⟩out|0⟩in, (18) where T = 1 8πM is the Hawking temperature, |n⟩out and |n⟩in are the number states for the fermion in the exterior region and the antifermion in the interior region of the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' When an outside observer travels through the Kruskal vacuum, Bob’s detector records the number of particles, which can be expressed as NF =K ⟨0|ˆaout† k ˆaout k |0⟩K = 1 e ω T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (19) The equation represents that the observer detects a ther- mal Fermi-Dirac distribution of the particles in the exte- rior of the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The influence of the Hawking effect on GTE, one-tangle and two-tangle in the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' – We assume that Alice, Bob and Charlie initially stay stationary at an asymptotically flat region and share a W state |W⟩ = 1 √ 3[|0A0B1C⟩ + |0A1B0C⟩ + |1A0B0C⟩], (20) where the subscripts A, B and C denote the qubits shared by Alice, Bob and Charlie, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Subsequently, we consider Alice still stays stationary at an asymptotically flat region, while Bob and Charlie hover near the event horizon of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (18) and (20), the wave function of W state can be rewritten as | ¯W⟩ = 1 √ 3 [µ|0A0B0 ¯ B1C0 ¯ C⟩ + µ|0A1B0 ¯ B0C0 ¯ C⟩ + ν|0A1B0 ¯ B1C1 ¯ C⟩ + ν|0A1B1 ¯ B1C0 ¯ C⟩ + µ2|1A0B0 ¯ B0C0 ¯ C⟩ + µν|1A0B0 ¯ B1C1 ¯ C⟩ + µν|1A1B1 ¯ B0C0 ¯ C⟩ + ν2|1A1B1 ¯ B1C1 ¯ C⟩], (21) where µ = 1 √ e− ω T +1 and ν = 1 √ e ω T +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Since Bob and Charlie cannot access the modes inside event horizon of the black hole, we should trace over the inaccessible ¯B p-3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Author et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 T E\uf000A B C\uf006 W GHZ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 1: The GTE of W and GHZ states as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' and ¯C modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, by tracing over the inaccessible modes, we obtain the density matrix ρABC = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 0 0 0 0 0 µ2 µ2 0 µ3 0 0 0 0 µ2 µ2 0 µ3 0 0 0 0 0 0 2ν2 0 µν2 µν2 0 0 µ3 µ3 0 µ4 0 0 0 0 0 0 µν2 0 µ2ν2 0 0 0 0 0 µν2 0 0 µ2ν2 0 0 0 0 0 0 0 0 ν4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (22) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (6), the GTE of W state in Schwarzschild spacetime can be expressed as EW (A|B|C) = 1 576{8µ4 − 8µ2 + 2 √ 2[35µ4 + 28µ4(2µ2 − 1) + µ4(8µ4 − 8µ2 + 1)] 1 2 }2 − 1 18{ √ 2[5 + 4(2µ2 − 1) + (8µ4 − 8µ2 + 1)] 1 2 − 2}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (23) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (23) we can see that the GTE of W state depends on the Hawking temperature T , which means that the Hawking radiation will affect the GTE in the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' On the other hand, the GTE of GHZ state in curved spacetime reads EGHZ (A|B|C) = 1 4[µ2 − µ2ν2 + µ � ν4µ2 + µ2]2 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='1, we plot the GTE of W and GHZ states as a function of the Hawking temperature T in the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We find that the GTE of W state first decreases and then tends to zero with the in- crease of the Hawking temperature T , while GTE of GHZ state first decreases and then freezes with the increase of the Hawking temperature T [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We also find that the GTE of W state is smaller than that of GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This implies that the GTE of GHZ state is more effective for resisting the Hawking effect and is more suitable for pro- cessing relativistic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We also study bipartite entanglement NA(BC), NB(AC), NC(AB), NAB, NAC and NBC for the W state in 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 T Ent NA (BC) NB (AC) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 2: The one-tangles of W state as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Firstly, we consider the one- tangles NA(BC), NB(AC) and NC(AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Taking the trans- pose of ρABC with respect mode A, we get ρTA ABC = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 0 µ3 µ3 0 0 µ2 µ2 0 0 0 0 µν2 0 µ2 µ2 0 0 0 0 µν2 0 0 0 2ν2 0 0 0 0 0 0 0 0 µ4 0 0 0 µ3 0 0 0 0 µ2ν2 0 0 µ3 0 0 0 0 0 µ2ν2 0 0 µν2 µν2 0 0 0 0 ν4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , which has the negative eigenvalue 1 48[−8µ4 + 8µ2 − 2 √ 2 � 35µ4 + 28µ4(2µ2 − 1) + µ4(8µ4 − 8µ2 + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Thus the one-tangle NA(BC) is NA(BC) = 1 24{8µ4 − 8µ2 + 2 √ 2[35µ4 + 28µ4 (2µ2 − 1) + µ4(8µ4 − 8µ2 + 1)] 1 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (24) Similarly, taking the transpose with respect the mode B, ρTB ABC = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 µ2 0 0 µ3 0 0 µ2 0 0 µ3 0 0 µν2 0 0 µ2 0 0 0 0 0 µ2 0 0 2ν2 0 0 µν2 0 0 µ3 0 0 µ4 0 0 0 0 0 0 0 0 µ2ν2 0 0 µ3 0 0 µν2 0 0 µ2ν2 0 0 µν2 0 0 0 0 0 ν4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Due to the complexity of the expression of NB(AC), we do not write it out here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Since Bob and Charlie are symmet- ric, we obtain NB(AC) = NC(AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='2, we plot the one-tangles of W state as a func- tion of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We can see that one- tangles of W state first decrease and then appear freezing phenomenon with the increase of the Hawking temper- ature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We also find that one-tangle of GHZ state is bigger than one-tangle of W state in curved spacetime, which means that one-tangle of GHZ state can effectively p-4 Title resist Hawking effect [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This indicates that one-tangle of GHZ state is more suitable for processing relativistic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' It has been shown that [43, 44], however, the coherence of W state is always bigger than the coherence of GHZ state in Schwarzschild spacetime, meaning that the coherence of W state is more suitable for processing relativistic quantum information than the GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, we should choose suitable quan- tum resources as required to process relativistic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Next, we consider the two-tangles NAB, NAC and NBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Taking trace over the mode C from ρABC, one gets ρAB = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed µ2 0 0 0 0 1 + ν2 µ 0 0 µ µ2 0 0 0 0 ν2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (25) The transpose with respect mode A is ρTA AB = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed µ2 0 0 µ 0 1 + ν2 0 0 0 0 µ2 0 µ 0 0 ν2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (26) According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (4) and (26), the two-tangle NAB can be expressed as NAB = 1 6[ √ 2 � 4(2µ2 − 1) + (8µ4 − 8µ2 + 1) + 5 − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (27) Since Bob and Charlie are symmetric, we have NAB=NAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Similarly, we can get ρBC = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed µ4 0 0 0 0 µ2(1 + ν2) µ2 0 0 µ2 µ2(1 + ν2) 0 0 0 0 ν2(2 + ν2) \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (28) and its transpose ρTB BC = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed µ4 0 0 µ2 0 µ2(1 + ν2) 0 0 0 0 µ2(1 + ν2) 0 µ2 0 0 ν2(2 + ν2) \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (29) Employing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (4) and (29), the two-tangle NBC can be written as NBC = 1 12(−12 − 8µ4 + 16µ2 (30) + 2 √ 2 � 18 + 40µ4 − 48µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='3, we plot the two-tangles of W state as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' It is shown that the two- tangle NAB between Alice and Bob first decreases and then freezes with the increase of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' However, the two-tangle NBC between Bob and Charlie first decreases and then suffers from sudden death with 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 T Ent NAB NBC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 3: The NAB and NBC of W state as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' the growth of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This means that the Hawking effect completely destroys the two-tangle NBC of W state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' These results are in contrast with the two-tangles of the tripartite GHZ state, which are zero in curved spacetime [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Finally, we compare fermionic entanglement with bosonic entanglement of tripartite states in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We initially assume that Alice, Bob and Char- lie stay stationary at an asymptotically flat region and share GHZ and W states of bosonic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' According to Bogoliubov transformations, the Kruskal vacuum and ex- cited states of bosonic field in Schwarzschild spacetime can be expressed as |0⟩B K = � 1 − e− ω T ∞ � n=0 e− nω 2T |n⟩B out|n⟩B in, (31) |1⟩B K = (1 − e− ω T ) ∞ � n=0 e− nω 2T √ n + 1|n + 1⟩B out|n⟩B in, where |n⟩B out and |n⟩B in are the number states for the boson in the exterior region and the antiboson in the interior region of the event horizon of the black hole, respectively [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Hereafter, we omit the mark B for simplicity unless it causes confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Because the tripartite entanglement of bosonic field is very complex in Schwarzschild spacetime, we consider a simpler model: Charlie hovers near the event horizon of the black hole, while Alice and Bob still stay stationary at an asymptotically flat region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (31), the wave functions of GHZ and W states can be rewritten as |GHZ⟩B ABC = 1 √ 2α ∞ � n=0 γn(|00n⟩|n⟩in (32) + √n + 1 α |11n + 1⟩|n⟩in), |W⟩B ABC = 1 √ 3α ∞ � n=0 γn( √n + 1 α |00n + 1⟩ (33) +|01n⟩ + |10n⟩) � |n⟩in, p-5 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Author et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' where α = 1 √ 1−e− ω T and γ = 1 √ e ω T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Since Charlie cannot access the modes inside the event horizon of the black hole,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' we should trace over the inaccessible mode and obtain the density operators ρB GHZ = 1 2α2 ∞ � n=0 γ2n{|00n⟩⟨00n| (34) + √n + 1 α [|00n⟩⟨11n + 1| + |11n + 1⟩⟨00n|] +n + 1 α2 |11n + 1⟩⟨11n + 1|},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' ρB W = 1 3α2 ∞ � n=0 γ2n{n + 1 α2 |00n + 1⟩⟨00n + 1| (35) +|01n⟩⟨01n| + |10n⟩⟨10n| + √n + 1 α [|00n + 1⟩ ⟨01n| + |01n⟩⟨00n + 1| + |00n + 1⟩⟨10n| + |10n⟩⟨00n + 1|] + |01n⟩⟨10n| + |10n⟩⟨01n|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Employing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (3), (4) and (6), we can obtain the GTE and bipartite entanglement N B C(AB) of the GHZ and W states of bosonic field in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Since the expressions are very complex, we do not write them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4, we plot the GTE, N B,GHZ C(AB) and N B,W C(AB) of GHZ and W states of bosonic field as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 (a), we can see that the GTEs of GHZ and W states of bosonic field vanish in the infinite Hawking temperature T , while the GTE of GHZ state of fermionic field always survives in curved spacetime [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This means that the GTE of GHZ state of fermionic field may be more suitable for relativistic quantum information tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' By the numerical calculation, we find lim T →∞ N B,GHZ C(AB) = 0, lim T →∞ N B,W C(AB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' (36) It means that N B,GHZ C(AB) and N B,W C(AB) of GHZ and W states of bosonic field vanish in the infinite Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' However, one-tangles of GHZ and W states of fermionic field always survive when only Charlie hovers near the event horizon of the black hole [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' The disparity between the Dirac and scalar fields is caused by the differences between Bose-Einstein and Fermi-Dirac statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This is because that Fermi-Dirac distribution protects tripartite entanglement of fermionic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' In addition, one-tangles N B,GHZ A(BC) (N B,GHZ B(AC) ) and N B,W A(BC) (N B,W B(AC) ) of GHZ and W states of bosonic field can survive for any T [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This conclusion is consistent with the fermionic field [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' – The effect of the Hawking effect on the genuine tripartite entanglement (GTE), one-tangle and two-tangle of W state in Schwarzschild spacetime have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We assume that Alice, Bob and Charlie initially share a W state at an asymptotically flat region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Then Alice still stays stationary at an asymptotically flat region, while Bob and Charlie hover near the event horizon GHZ W 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 T EB\uf000A B C\uf006 ( a) NC (AB) B,��� NC (AB) B,W 2 4 6 8 1� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='0 0� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content='4 �� � � \x0e \x0f Ent (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 4: The GTE, N B,GHZ C(AB) and N B,W C(AB) of GHZ and W states of bosonic field as a function of the Hawking temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We have found that, with the increase of the Hawking temperature, the GTE of W state first de- creases and then approaches zero, while GTE of GHZ state first decreases and then appears freezing phenomenon [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This implies that the GTE of GHZ state is more effective for resisting Hawking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We have also found that, with the growth of the Hawk- ing temperature, the two-tangle between Alice and Bob (Charlie) of W state first decreases and then freezes, while two-tangle between Bob and Charlie first reduces and then suffers from sudden death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This is different from the case of GHZ state, whose two-tangles are zero in Schwarzschild spacetime [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We have shown that the one-tangle of W state first decreases and then appears freezing phe- nomenon with the increase of the Hawking temperature, which is always smaller than the one-tangle of GHZ state in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This is different from the behavior of quantum coherence, where the coherence of W state is bigger than that of GHZ state in Schwarzschild spacetime [43,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Therefore, for different quantum states, we should choose suitable quantum resources to process relativistic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' Finally, we compare bosonic entanglement with p-6 Title fermionic entanglement of tripartite states when only Charlie hovers near the event horizon of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We find that the GTEs of GHZ and W states of bosonic field reduce to zero with the growth of the Hawking temperature, while the GTE of GHZ state of fermionic field can survive for any Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' We also find that not all the one-tangles of GHZ and W states of bosonic field can survive in the infinite Hawking temperature limit, while all one-tangles of GHZ and W states of fermionic field always survive in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' This is because that Fermi-Dirac distribu- tion protects tripartite entanglement of fermionic field in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' It means that tripartite entan- glement of fermionic field is more suitable for processing relativistic quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' ∗ ∗ ∗ This work is supported by the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' 12205133, 1217050862, 11275064, 11975064 and 12075050 ), LJKQZ20222315 and 2021BSL013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQfmf32/content/2301.01566v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': 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School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi +464-8602, Japan +∗E-mail: sakemi.haruka@astrophysics.jp +Received ⟨reception date⟩; Accepted ⟨acception date⟩ +Abstract +Microquasar SS 433 located at the geometric center of radio nebula W50 is a suitable source +for investigating the physical process of how galactic jets affect the surrounding interstellar +medium (ISM). Previous studies have searched for evidence of the interaction between the +SS 433 jet and ISM, such as neutral hydrogen gas and molecular clouds; however, it is still +unclear which ISM interacts with the jet. We looked for new molecular clouds that possibly +interact at the terminal of the SS 433 eastern jet using the Nobeyama 45-m telescope and the +Atacama Submillimeter Telescope Experiment (ASTE). We identified two molecular clouds, +comprising many small clumps, in the velocity range of 30.1–36.5 km s−1 for the first time. +These clouds have complex velocity structures, and one of them has a density gradient toward +SS 433. Although it is difficult to conclude the relation between the molecular clouds and the +SS 433/W50 system, there is a possibility that the eastern structure of W50 constructed by +the SS 433 jet swept up tiny molecular clumps drifting in the surroundings and formed the +molecular clouds that we identified in this study. +Key words: ISM: individual (W50)1 — ISM: jets and outflows2 — ISM: molecules3 — stars: individual +1 +arXiv:2301.13333v1 [astro-ph.GA] 30 Jan 2023 + +(SS 433)4 +1 Introduction +Galactic X-ray binary (GXB) jets inject matter and energy into their surroundings and thus +change the physical and chemical states of the interstellar medium (ISM). The study of the +interaction between the jets and surrounding ISM is thus essential to understanding the evolu- +tion of our galaxy. One of the essential contributions of the jets is the formation of molecular +clouds. Asahina et al. (2014) and Asahina et al. (2017) carried out a magnetohydrodynamical +simulation to investigate the effect of jets on surrounding neutral hydrogen (HI). In their sim- +ulation, jets compressed gas and triggered a cooling instability, and ultimately, the jets formed +molecular clouds after ∼106 yr. Other studies suggested that GXB jets drive interstellar tur- +bulence, produce high-energy cosmic rays, and sometimes stimulate star formation (Cooper et +al. 2020; Heinz et al. 2008; Mirabel et al. 2015). Recent observational studies confirmed the +interaction between galactic microquasar jets and the surrounding ISM. Tetarenko et al. (2018) +studied molecular clouds located on the jet axis of GRS 1915+105 using observation datasets of +the Atacama Large Millimeter/sub-millimeter Array (ALMA) and found that the jet drives a +shock at the impact site of the molecular cloud. Additionally, Tetarenko et al. (2020) observed +density and temperature tracers from molecular clouds around GRS 1758-258 and 1E 1740.7- +2942 and investigated the jet feedback to the surroundings. Nevertheless, there still have been +few observations of the interaction between jets and the ISM. It is thus important to increase +the number of sample observations to clarify what happens in the region of interaction between +jets and the ISM. +The system of the microquasar SS 433 and surrounding radio nebula W50 is a suitable +target for investigating the effects of GXB jets on their surroundings. W50 is a large radio +nebula located at (αJ2000, δJ2000) = (19h 12m 19.92s, +4d 55m 01.2s) (figure 1). It extends over +an area of 2◦ × 1◦ in the sky (Geldzahler et al. 1980). The microquasar SS 433 is located at +the geometric center of W50. SS 433 ejects precessing jets in the east–west direction with a jet +velocity of 0.26c, an opening angle of 20◦ and a precession period of 162 days (Abell & Margon +1979; Hjellming and Johnston 1981; Margon 1984; Margon & Anderson 1989; Davydov, Esipov, +& Cherepashchuk 2008). The elongated structures of W50, called “ears”, are believed to be +the result of an interaction between the jets and the nebula (Downes et al. 1981a; Downes et al. +2 + +1981b; Downes et al. 1986; Dubner et al. 1998). The western side of W50 is near the Galactic +plane, and the length of the western ear along the jet axis is shorter than that of the eastern +ear (19 versus 82 pc assuming a distance of 5.5 kpc). The distance of the SS 433/W50 system +is still under discussion but believed to be in the range of 3.0 to 5.5 kpc. +Many studies of the ISM around SS 433/W50 have tried to search the evidence of the +jet-ISM interaction and determine the kinematic distance of the system. The first observation +of HI gas was reported by Dubner et al. (1998) with the Green Bank Telescope. They identified +an HI cavity at the position of W50 for a velocity of vLSR = 42 km s−1. Additionally, a large +HI clump was detected at the southeast of the eastern ear, and it has been suggested that this +HI clump collided with the eastern ear and changed the morphology. A distance to the SS +433/W50 system of 3.0 kpc was determined from these results. Sakemi et al. (2021) compared +the spatial distributions of W50 and the surrounding HI using GALFA-HI survey datasets taken +with the 305-m Arecibo Radio Telescope (Peek et al. 2011) and confirmed that the HI cavity +identified by Dubner et al. (1998) has a clear spatial correlation with W50 in the velocity range +from 33 to 55 km s−1. Meanwhile, Lockman et al. (2007) observed the absorption line of HI in +the direction of SS 433 and concluded that the HI cloud at a velocity of 75 km s−1 is related to +the SS 433/W50 system. The velocity is clearly different from that suggested by Dubner et al. +(1998), and the kinematic distance is 5.5 kpc. The distance is consistent to the value derived +by comparing a deep-integrated radio image of the SS 433 jet with the kinematic jet model +based on the speed and the precession period of the jet (Blundell & Bowler 2004). +Not only the HI gas but also molecular clouds have been identified around the SS +433/W50 system. We summarize the previous studies in table 1 and show the positions of +the clouds in figure 1. Yamamoto et al. (2008) suggested that the molecular clouds in the +velocity range from 42 to 56 km s−1 are related to the SS 433 jet, which is distributed around +the jet axis (S1–S6, N1–N4). The eastern and western clouds of SS 433 are in the velocity range +of 42 to 45 km s−1 and 49 to 56 km s−1, respectively. These velocities correspond to kinematic +distances of 3.0 and 3.5 kpc for the flat rotation curve (Brand & Blitz 1993). The eastern clouds +(S1–S6) are distributed in a region far from the eastern edge of W50; i.e., at approximately +0.4◦–1.6◦ (outside area of figure 1). The western-side clouds (N1–N4) are closer to SS 433, and +Liu et al. (2020) observed N1, N2, and N3 with higher spatial resolution and concluded that a +part of them is located within W50. Recently, Yamamoto et al. (2022) reported the observa- +tion of N4. The molecular cloud has an asymmetric spectrum for CO emission and an extreme +temperature gradient. They thus suggested that the cloud is interacting with the SS 433 jet. +Meanwhile, Su et al. (2018) found molecular clouds in another velocity range of 73–84 km s−1, +3 + +Chimney +Eastern +Edge +Eastern Ear +Beam sizes +SS433 +Precession +Cone +Jet Axis +N1 +N2 +N3 +N4 +G39.315-1.155 +Western Ear +Fig. 1. Radio continuum image of W50 taken with the Karl G. Jansky Very Large Array (JVLA) at 1.602 GHz (Sakemi et al. 2021). Yellow contours show the +positions of the newly identified molecular clouds in the velocity range 30.1–42.3 km s−1. The contour levels are the rms × [4, 8]. The yellow circle and +magenta ellipse in the bottom-left corner show the beam sizes of the CO emission and continuum, respectively. Purple and hot pink ellipses show the +positions of the molecular clouds identified by previous studies in the velocity range of 49–56 km s−1 and 73–77 km s−1, respectively. Note that the clouds +S1–S6 and G40.331–4.302 are outside the field of view. +which are located near the western ear (G39.315–1.155) and a straight extension of the eastern +ear about 0.9◦ (G40.331–4.302, outside area of figure 1). They suggested that the molecular +clouds located at a distance of 4.9 kpc and interacted with W50 approximately 105 years ago. +The molecular clouds identified in these previous studies have good spatial correlation with +the SS 433/W50 system, and some have spectral features (e.g., spectral broadening and wings) +suggesting interaction with the jet. It remains unclear at which velocities and distances the +association is real, and there is no consensus in explaining this situation. +In the present paper, we focus on the eastern edge of W50. Since there is a terminal +region of the SS 433 jet, it is a suitable target to investigate the jet-ISM interaction. +We +made observations with the Nobeyama 45-m telescope for 12CO(J=1–0) and 13CO(J=1–0) and +with the Atacama Submillimeter Telescope Experiment (ASTE) for 12CO(J=3–2) to identify +molecular clouds for the first time. We investigate the features of the molecular clouds that +we identified and provide the fundamental physical parameters based on the observed CO +line emissions, such as density and temperature. The remainder of the paper is structured as +follows. Section 2 describes the observations and the data reduction. Section 3 presents the +observational results. Section 4 presents a discussion of the results. Section 5 provides our +conclusions. +4 + +Table 1. The molecular clouds around the SS 433/W50 system +identified by previous studies∗ +Name +VLSR (km s−1) +Distance (kpc) +References +S1 +42.9 +3.0 +1 +S2 +45.4 +3.0 +1 +S3 +44.1 +3.0 +1 +S4 +43.2 +3.0 +1 +S5 +44.8 +3.0 +1 +S6 +42.1 +3.0 +1 +N1 +55.8 +3.5 +1,2,3 +N2 +53.7 +3.5 +1,2,3 +N3 +53.0 +3.5 +1,2,3 +N4 +49.4 +3.5 +1,3 +G39.315–1.155 +73 +4.9 +4 +G40.331–4.302 +74 +4.9 +4 +∗ Column 1: Name of each molecular cloud. Column 2: Peak velocity. +Column 3: Kinematic distance referred to in each paper. Note that the +values of N1 to N4 refer to reference 1. Column 4: References +(1)Yamamoto et al. (2008), (2)Liu et al. (2020), (3)Yamamoto et al. +(2022), (4)Su et al. (2018). +2 Observations +2.1 12CO(J=1–0) and 13CO(J=1–0) +The +12CO(J=1–0) and +13CO(J=1–0) data were obtained with the 45-m telescope of the +Nobeyama Radio Observatory. We scanned an area of 30.1 × 32.6 arcmin2 in on-the-fly map- +ping mode (Sawada et al. 2008). We scanned in the right ascension and declination directions +to suppress the scanning effect. The four-beam receiver FOREST (Minamidani et al. 2016) and +the autocorrelation spectrometer SAM45 (Kuno et al. 2011) were used. Typical noise temper- +atures of the system including the atmosphere were between 300 and 400 K at 115 GHz. The +bandwidth and resolution were 62.5 MHz and 30.52 kHz, respectively. The pointing accuracy +was checked every 2 hours by observing R Aquilae (αJ2000, δJ2000) = (19h 06m 22.25s, +8d 13m +48.0s) with the frontend H40 and confirmed to be better than 3 arcsec. All observations were +conducted in equatorial coordinates. We used a chopper wheel to obtain the antenna temper- +ature T ∗ +a (Kutner & Ulich 1981). We observed W51 as a standard source to fix gain variations +between the eight output signals on December 6 to 20, 2019, February 7 to March 6, 2020, and +December 1 to 4, 2020. The spectral intensity was calibrated and converted to the TMB scale by +applying a main beam efficiency ηMB corresponding to each polarization, frequency, and observ- +5 + +ing season provided by the observatory. The final angular resolutions were 20.3 and 20.5 arcsec +at 115 and 110 GHz, respectively. The spatial and velocity grids had sizes of 8.5 arcsec and 0.2 +km s−1, respectively. The velocity coverage was from -45 to 113 km s−1. The root-mean-square +(rms) noise level in TMB was 0.60 and 0.26 K at 115 and 110 GHz, respectively. +2.2 12CO(J=3–2) +The 12CO(J=3–2) data were obtained with the ASTE on August 9 to 11, 2019 (Ezawa et al. +2004). We adopted the on-the-fly mode, and the mapping area had dimensions of 11 × 11 +arcmin2 centered at (αJ2000, δJ2000) = (19h 16m 47.5243s, +4d 51m 31.709s). The frontend was +a 2SB SIS mixer receiver called DASH 345. The typical system temperature was 263 K in +the single side band. We used WHSF as the backend in F-FX mode. The band width and +resolution were 64 MHz and 31.25 kHz, respectively. The pointing accuracy was checked every +3–4 hours by observing R Aquilae. We convolved the intensity scale into TMB by assuming the +W44 peak to be TMB = 35.5 K (Wang et al. 1994). The final angular resolution, grid size, and +velocity resolution were 28.2 arcsec, 11 arcsec, and 0.2 km s−1, respectively. The rms noise level +was 0.09 K at 345 GHz. +3 Results +We report the molecular cloud distribution observed with the Nobeyama 45-m telescope and +ASTE at the eastern edge region of W50. Additionally, we explain the velocity features and +the line-intensity ratios of the clouds. +3.1 Spatial distributions +We show the distribution of the 12CO(J=1–0) line emission in the velocity range of 30.1–42.3 +km s−1 observed with the Nobeyama 45-m telescope in figure 1 with yellow contours. They +distribute around a bright filamentary structure of W50 at αJ2000 = 19h 16m 00.0s. In this +region, we found a pointed structure to the north in the radio continuum image, called the +chimney (see figure 1, Dubner et al. 1998; Farnes et al. 2017; Broderick et al. 2018). +We +identified a molecular cloud near the chimney. Additionally, there was another cloud of similar +size at the eastern edge of the ear. Figure 2 is a magnified view of the target region. The color is +the integrated intensity of 12CO(J=1–0) in the velocity range 30.1–42.3 km s−1. These velocities +correspond well to those of HI gas, suggesting a relation with W50 (Dubner et al. 1998; Sakemi +et al. 2021). We refer to the newly identified prominent clouds as the chimney cloud and edge +6 + +[K km s-1] +Chimney cloud +Edge cloud +Fig. 2. Velocity-integrated intensity maps of 12CO(J=1–0). The velocity range is 30.1–42.3 km s−1. Magenta contours show the radio continuum observed +with the JVLA at 1.602 GHz. The contour levels are the rms × [2, 4, 8, 16]. The beam sizes of CO emission and continuum observations are shown in the +bottom-left corner of each panel by a black circle and magenta ellipse, respectively. +� � +� +� +� +� +� � +� +� � +� +� +� +� +� � +� +[K km s-1] +(a) 12CO( =1-0) 30.1-36.5 km s-1 +J +[K km s-1] +(b) 13CO( =1-0) 30.1-36.5 km s-1 +J +(c) 12CO( =3-2) 30.1-36.5 km s-1 +J +[K km s-1] +Fig. 3. Velocity-integrated intensity maps of 12CO(J=1–0) (left), 13CO(J=1–0) (middle), and 12CO(J=3–2) emissions (right). The velocity range is +30.1–36.5 km s−1. Magenta contours show the radio continuum observed with the JVLA at 1.602 GHz. The beam sizes of CO emission and continuum +observations are shown in the bottom-left corner of each panel by a white circle and magenta ellipse, respectively. The numbers in the middle panel show +the position where the physical parameters are derived in section 4. See table 2 and 3 and figure 10. +cloud. Besides these, faint and clumpy clouds are distributed around the filamentary structure. +Hereafter, we focus on the chimney and edge clouds. Panel (a) of figure 3 is a velocity- +integrated intensity map of 12CO(J=1–0) in the velocity range 30.1–36.5 km s−1. The chimney +cloud has two peaks in the southern and northern parts. The edge cloud is patchy, and it is +brightest in the northwest region. Panel (b) of figure 3 is a velocity-integrated intensity map +of 13CO(J=1–0) in the same velocity range. The intensity distribution traces the high-density +regions of observed molecular clouds. Although the distribution is similar to that of 12CO(J=1– +7 + +[K] +Fig. 4. Peak TMB distribution in 12CO(J=1–0). The velocity range is 30.1–36.5 km s−1. Cyan contours show the radio continuum observed with the JVLA +at 1.602 GHz. The beam sizes of CO emission and continuum observations are shown in the bottom-left corner of each panel by a white circle and cyan +ellipse, respectively. +0), the integrated intensity map of 13CO(J=1–0) shows structures that are more clumpy. We +thus expect that both the chimney and edge clouds comprise many separated clumps. Panel (c) +of figure 3 is a velocity-integrated intensity map of 12CO(J=3–2). Note that we only observed +the edge cloud. We detected a difference from the trend in 12CO(J=1–0) in that the intensity +was higher westward rather than eastward in the edge cloud. This cloud thus has different +physical properties on eastern and western sides. +We derived the peak TMB distribution in 12CO(J=1–0) (figure 4). The northern part +of the chimney cloud has a flat temperature distribution. Similar to the distribution of the +velocity-integrated intensity, there is a temperature peak at the southern part. In the case of +the edge cloud, the temperature peak is located eastward of the northern part. This position +does not correspond to the peak area of the velocity-integrated intensity. +3.2 Velocity structures +We next explain the velocity structures of the molecular clouds in the eastern region of W50. +Figure 5 is a position–velocity diagram of the chimney cloud. Note that we inclined the boxes +to align the integration axis with the direction of the SS 433 jet axis and thus assess the +effect of the jet activity. We divided the chimney cloud into eight regions and investigated the +8 + +0 +-100 +100 +200 +300 +-200 +-300 +1 +2 +3 +4 +5 +6 +7 +8 +19h16m50s +40s +30s +20s +J2000 Right Ascension +J2000 Declination +5 06’ +∘ +03’ +00’ +4 57’ +∘ +25.0 +20.0 +15.0 +10.0 +5.0 +0.0 +[K km s-1] +Offset [arcsec] +-300 +-200 +-100 +0 +100 +200 +300 +30 32 34 36 +LSR [km s-1] +V +1 +2 +3 +4 +5 +6 +7 +8 +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +Fig. 5. Position–velocity diagrams of the 12CO(J=1–0) emission of the chimney cloud. The intensity is integrated along the short axes of each square. +Offset (arcsec) +-250 +-200 +-100 +100 +200 +0 +-150 +-50 +50 +150 +250 +0 +-100 +100 +200 +-200 +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +15.0 +10.0 +5.0 +0.0 +[K km s-1] +J2000 Declination +4 57’ +∘ +54’ +51’ +48’ +19h17m10s +00s +16m50s +40s +J2000 Right Ascension +30s +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +30 32 34 36 +LSR [km s-1] +V +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +0 +2 +4 +[K degree] +6 +8 +25.0 +20.0 +Fig. 6. Similar to figure 5 but for the edge cloud. +velocity structures in the northeast-to-southwest direction. The most outstanding feature is +seen in region 4, where the chimney cloud seems to touch the chimney of W50. In this region, +there is a curved structure of the position–velocity diagram. The center velocity seems to shift +approximately 1 km s−1 along the curve. A similar trend is seen for the edge cloud (figure 6). +We divided the edge cloud into seven regions. The curved structures are in regions 4 and 5. +Especially in region 5, the curved structure is above an offset of 0 arcsec, where the edge cloud +also touches the eastern edge of W50 in the sky plane. Additionally, the velocity width is thick +(∼ 4 km s−1) at an offset less than 0 arcsec in region 5. +Figure 7 presents spectra for different parts of the chimney cloud. At the northern peak, +the spectrum tends to have a wide velocity width and seems to have a spectral wing toward +lower velocity. Between the northern and southern peaks, spectral wings form toward higher +velocity. Spectral wings often imply the existence of an external force generated by surrounding +sources. However, these wings may originate from multiple clouds in the line of sight and at +slightly different velocities. Figure 8 presents spectral plots of the edge cloud. At the peak +positions in the northwest region, we clearly see a wide spectrum not only for 12CO(J=1-0) +9 + +25 +5°06' +20 +Declination (J2000) +03' +15 +kms-1 +10 +.00 +5 +4°57' +19h16m50s +405 +305 +20s +RightAscension (J2000)4°57' +25 +20 +Declination(2000) +54* +15 +kms-1 +51' +10 +48' +19h17m105 +oos +16m50s +405 +305 +Right Ascension (J2000)0 +4 +8 +30 +36 +LSR [km s-1] +V +MB [K] +T +15.0 +10.0 +5.0 +0.0 +[K km s-1] +19h16m48s +42s +36s +J2000 Right Ascension +30s +J2000 Declination +5 04’ +∘ +02’ +00’ +4 58’ +∘ +25.0 +20.0 +Fig. 7. (Left) Integrated intensity map of the 12CO(J=1–0) emission of the chimney cloud. The velocity range is 30.1–36.5 km s−1. (Right) 12CO(J=1–0) +(black) and three times 13CO(J=1–0) (red) spectra for the regions shown in the grids in the left panel. +0 +4 +30 +36 +LSR [km s-1] +V +MB [K] +T +15.0 +10.0 +5.0 +0.0 +[K km s-1] +19h17m00s 16m54s +48s +J2000 Right Ascension +42s +J2000 Declination +4 54’ +∘ +52’ +50’ +48’ +20.0 +25.0 +8 +Fig. 8. Similar to figure 7 but for the edge cloud. +10 + +25 +5°04' +20 +Declination (j2000) +02'- +15 +- +kms +10 Y +00' +5 +4°58' +0 +19h16m485425 +365 +305 +RightAscension(j2000)MMMt25 +4°54' +20 +Declination (J2000) +52' +15 +kms-1 +10 +50° +5 +48' +0 +1gh17m00516m545 +485 +425 +RightAscension(j2000)(a) 13CO( =1-0) / 12CO( =1-0) +J +J +(b) 12CO( =3-2) / 12CO( =1-0) +J +J +Fig. 9. Intensity ratios of 13CO(J=1–0) and 12CO(J=1–0) (a) and 12CO(J=3–2) and 12CO(J=1–0) (b). Only the regions where the intensities of the CO +emission are higher than three times the rms value are plotted. Black contours show the values of [0.1, 0.15, 0.2, 0.25]. Magenta contours show the radio +continuum observed with the JVLA at 1.602 GHz. The resolutions of intensity ratio maps and continuum are given at the bottom-left corner of each panel by +black circle and magenta ellipse, respectively. +but also for 13CO(J=1-0) with a mean velocity width of 4.26 km s−1, which is a result similar +to that in figure 6. In the northeast region of the edge cloud, spectral wings form toward high +velocity. +3.3 Intensity ratios +Panel (a) of figure 9 shows the intensity ratio of 13CO(J=1–0)/12CO(J=1–0), which traces high- +density areas of molecular clouds. Most parts of the chimney and edge clouds have values below +0.3. In the edge cloud, we see again the patchy structure, and we analyze the physical properties +of each clump in section 4. +Panel (b) of figure 9 shows the intensity ratio of 12CO(J=3– +2)/12CO(J=1–0) of the edge cloud. There is a gradient from west to east, which suggests a +difference in temperature and/or density in the western and eastern parts. +4 Discussion +In this section, we consider the relation between the molecular clouds that we identified and +their surroundings. We first derive the physical parameters of the chimney and edge clouds. +We then consider the interaction between these clouds and the SS 433/W50 system. We also +11 + +mention the relation between these clouds and a giant molecular filament in front of the SS +433/W50 system. We finally discuss the high-velocity clouds in terms of the relation with the +clouds identified by Su et al. (2018). +4.1 Physical properties of the molecular clouds +We first estimated the column densities of each region shown in panel (b) of figure 3 adopting two +methods; i.e., assuming the X-factor and local thermodynamic equilibrium (LTE). The X-factor +converts the 12CO(J=1–0) integrated intensity W(12CO(J =1–0)) to the column density N(H2), +and we adopt the value of N(H2)/W(12CO(J =1–0)) = 2×1020 cm−2 (K km s−1)−1 with ± 30% +uncertainty (Bolatto, Wolfire, & Leroy 2013). Alternatively, we used the following procedures +to derive the column density N(H2) by assuming LTE (Wilson, Rohlfs, & H¨uttemeister 2009). +Assuming the 12CO(J=1–0) line is optically thick, the excitation temperature Tex is derived +from the 12CO peak intensity TMB: +Tex = 5.5 +� +ln +� +1 + +5.5 +TMB + 0.82 +��−1 +[K]. +(1) +The optical depth τ13(v) is calculated for the 13CO(J=1–0) brightness temperature of each +velocity channel TMB(v) as +τ13(v) = −ln +� +��1 − TMB(v) +5.3 +� +� +� +1 +exp +� +5.3 +Tex +� +− 1 +− 0.16 +� +� +� +−1� +��. +(2) +Using the velocity resolution ∆v = 0.2 km s−1, the column density of 13CO, N(13CO), is +calculated as +N(13CO) = 2.4 × 1014 × +� +v +Texτ13(v)∆v +1 − exp +� +− 5.3 +Tex +� [cm−2]. +(3) +We finally adopt the conversion factor from N(13CO) to N(H2) of 7.7 × 105 (Frerking, Langer, +& Wilson 1982; Pineda et al. 2010; Wilson & Rood 1994). We present the results in table 2. +The column densities derived by assuming the X-factor are slightly higher than those derived +using the LTE; however, we can explain these differences by errors in the observations and the +abundance ratios [13CO]/[H2] = 1–3.5 × 10−6 and the X-factor = 1.8–2.0 × 1020 cm−2 (K km +s−1)−1 depending on the surrounding environment. We calculated the masses of the chimney +and edge clouds using N(H2) derived by assuming the X-factor, +Mmol = ¯µmH +� +[N(H2) × Ω × D2], +(4) +where ¯µ = 2.8, mH = 1.67 × 10−24 g, Ω = 1.7 × 10−9 sr, and D pc are the mean molecular +weight, the mass of the atomic hydrogen, the solid angle subtended by the grid spacing, and +12 + +Table 2. Column densities∗ +Region +N1−0 +X +N13,1−0 +LTE +chimney 1 +3.18 +2.28 +chimney 2 +4.14 +2.44 +chimney 3 +5.27 +2.70 +chimney 4 +4.72 +2.14 +edge 1 +2.66 +1.90 +edge 2 +3.68 +2.97 +edge 3 +2.82 +1.40 +edge 4 +2.03 +1.72 +edge 5 +2.08 +1.64 +∗ Column 1: Region name given in +panel (b) of figure 3. Column 2: +Column density of H2 derived from +12CO(J=1–0) in 1021 cm−2 +assuming the X-factor. Column 3: +Column density of H2 derived from +13CO(J=1–0) in 1021 cm−2 +assuming LTE. +Table 3. Results of RADEX calculation.∗ +Region +R13/12 +1−0 +R12 +3−2/1−0 +τ(13CO) +n(H2) +Tkin +edge 1 +0.154±0.028 +0.190±0.020 +1.87+0.17 +−0.12 +320+120 +−100 +8.6+1.6 +−1.2 +edge 2 +0.195±0.020 +0.201±0.014 +0.72+0.04 +−0.04 +880+220 +−180 +7.7+0.8 +−0.6 +edge 3 +0.123±0.028 +0.137±0.018 +1.64+0.13 +−0.10 +270+110 +−90 +8.3+2.2 +−1.3 +edge 4 +0.204±0.040 +0.228±0.030 +0.56+0.07 +−0.06 +800+400 +−280 +8.2+1.8 +−1.2 +edge 5 +0.166±0.040 +0.244±0.033 +0.93+0.18 +−0.16 +740+660 +−340 +9.5+2.8 +−1.8 +∗ Column 1: Region name given in figures 3 and 10. Column 2: Intensity ratio +of 13CO(J=1–0)/12CO(J=1–0). Column 3: Intensity ratio of +12CO(J=3–2)/12CO(J=1–0). Column 4: Optical depth of 13CO. Column 5: +Number density of H2 in cm−3. Column 6: Kinematic temperature in K. +the distance to the molecular clouds, respectively. If we assume the distance of D = 5500 pc, +the masses of the chimney and edge clouds are 3800 and 2300 M⊙, respectively. Also, the +masses of these clouds are 1100 and 700 M⊙ assuming the distance of D = 3000 pc. +Additionally, we carried out the RADEX calculation to estimate the temperature and +density of the edge cloud using non-local thermodynamic equilibrium radiative transfer code +(van der Tak et al. 2007). We input parameter sets of Tk and n(H2) for intervals of 100.1 K in +the range of 1 to 100 K and intervals of 100.05 cm−3 in the range of 10 to 104 cm−3, and we +calculated the intensities 12CO(J=1–0), 13CO(J=1–0) and 12CO(J=3–2). The line intensity +13 + +ratios of 13CO(J=1–0)/12CO(J=1–0) and 12CO(J=3–2)/12CO(J=1–0) were then calculated in +n(H2)–Tk space. We estimated N(12CO) as +N(12CO) = N(H2) × 10−4, +(5) +where N(H2) was derived by assuming the X-factor. +Additionally, we assumed N(12CO)/ +N(13CO)=62 (Milam et al. 2005). We analyzed the local peaks of 13CO(J=1–0) of the edge +cloud shown in the top-left panel of figure 10. The observed line intensity ratios of 13CO(J=1– +0)/12CO(J=1–0) and 12CO(J=3–2)/12CO(J=1–0), with one sigma error, are plotted in n(H2)– +Tk space with red and blue lines in five plots of figure 10 with the expected optical depth +τ(13CO). The parameters at the crossover points of the line intensity ratios correspond to the +physical values derived by the RADEX calculation and are listed in table 3. We note that the +densities of all regions are much lower than the critical density of 13CO(J=1–0); nevertheless, +the 13CO(J=1–0) emission is observed. This implies that the edge cloud comprises tiny clumps +that cannot be spatially resolved by our observations and are distributed with the low beam- +filling factor. In that case, the emission is smoothed out, and the densities are estimated to be +lower. Additionally, we find that the western side of the edge cloud has higher density. This +result is consistent with the trend of the intensity ratio 12CO(J=3–2)/12CO(J=1–0) shown in +panel (b) of figure 9. +4.2 Association with the SS 433/W50 system +We here consider the relation between the chimney and edge clouds and the SS 433/W50 +system. Our observations revealed that these clouds are close to the eastern region of W50 in +the plane of the sky. The clouds have clumpy spatial structures and complex velocity structures, +including a shift in the central velocity (figures 5 and 6), spectral wings, and wider velocity +widths (figures 7 and 8). Additionally, we identified that the density distribution of the edge +cloud is not flat (panel b of figure 9 and figure 10). Near the chimney and edge clouds, there +is no heating source that might explain their complexity, such as the HII region radiating UV +photons. Also, the gamma-ray emission regions of the SS 433 jet are far from these clouds, +above 0.6◦, and should have less influence. Therefore, the properties of these clouds imply the +interaction with the eastern ear of W50. +Unfortunately, we cannot conclude the interaction from only our observations. There +is a possibility that the clouds are mere foreground or background sources of W50. As an +example, Lin et al. (2020) identified a giant molecular filament in the Milky Way Imaging +Scroll Painting (MWISP) in front of the SS 433/W50 system in the velocity range from 27 to +14 + +40 km s−1, named GMF MWISP G041-01. We thus need to discuss the association between +the molecular clouds that we identified and this giant molecular filament. Lin et al. (2020) +suggested that GMF MWISP G041-01 comprises four components, three being filamentary +structures. Two such giant filaments possibly collided in the region around (αJ2000, δJ2000) = +(19h 10m 01.3525s, +7d 15m 03.330s) and (19h 10m 29.6389s, +6d 10m 32.713s), while they +might be just overlapping along the line of sight. If such a collision actually occurred, spectral +wings, such as those observed for the chimney and edge clouds, should be seen. The clouds that +we identified might then be part of GMF MWISP G041-01. However, the filament–filament +collision is located away from the clouds, by approximately 2.4 degrees, corresponding to 71 pc +assuming the distance of GMF MWISP G041-01 to be 1.7 kpc. We thus infer that the clouds +that we identified are independent of the system of GMF MWISP G041-01. +Hereafter, we assume that the chimney and edge clouds are related to W50 and consider +the formation mechanism of these molecular clouds. An instinctive formation scenario is that +the eastern ear of W50 compressed the surrounding HI gas, and a shock induced the molecular +cloud formation (Asahina et al. 2014; Asahina et al. 2017). However, this mechanism would take +a long time, approximately 106 yr, to form molecular clouds, and this time is in disagreement +with the age of W50 (a few 104 yr). We then consider the alternative scenario of sweeping +tiny molecular clumps by the surface of the eastern ear of W50, corresponding to the surface +of the jet cocoon. As mentioned above, the chimney and edge clouds seem to comprise more +small clumps. Additionally, we identified faint and clumpy clouds in the eastern region of W50 +in the velocity range of 30.1–42.3 km s−1 except for the chimney and edge clouds (figure 2). +This means that many tiny clumps are drifting at the position of the eastern ear of W50. The +origin of the tiny clumps remains unknown, but there is a possibility that they were formed by +the activity of the progenitor star of SS 433, such as by the compression of the surrounding HI +gas by the stellar wind. Here, we imagine that the faint clouds are distributed throughout the +region before the eastern ear goes through. The mean column density of H2 in the velocity range +of 30.1–42.3 km s−1 except for the chimney and edge clouds is 7.18 × 1020 cm−2. We regard +the eastern ear as a cylinder with a diameter of 0.3 deg and a height of 0.7 deg. Assuming a +distance of 3.0 and 5.5 kpc, the eastern ear can collect clumps to a mass of 7.2 × 103 M⊙ and +2.4 × 104 M⊙, respectively. If we reduce the velocity range and consider only the contribution +of the northern part of the eastern ear, the mass is comparable to that of the total mass of the +chimney and edge clouds. Additionally, this scenario explains the complexity of the velocity +structures of the clouds. Note that these clumps might influence the formation theory of the +eastern ear of W50 (Goodall, Alouani-Bibi, & Blundell 2011; Ohmura et al. 2021). To consider +15 + +how they affect, we require to carry out additional numerical simulations; however, it is out of +the focus of this paper. +Also, we discuss the distance of the SS 433/W50 system based on the relation with +the newly identified clouds. +Using the velocity of 33 km s−1, roughly the peak velocity of +the chimney and edge clouds, the kinematic distance is 2.2 kpc. This value is far from the +historically discussed one in the range of 3.0–5.5 kpc. Here, we suggest that the clouds are +unsuitable for deriving the distance based on the model for Galactic rotation. As mentioned +above, these clouds probably consist of many small clumps, and they are affected by the motion +of the SS 433 jet. Small clumps can be easily shifted the central velocity. Therefore, there is +a possibility that these clouds do not follow simply the rotation of the galactic plane. This +scenario can explain the reason why the velocities of these clouds are lower shifted; since the +eastern jet is approaching, the clouds have been pushed toward us. +We mention the differences between the clouds we identified and reported by Yamamoto +et al. (2022), N4 (see figure 1 and table 1). N4 is not similar to the typical molecular clouds in +the Galactic plane, and it shows clear intensity and velocity gradients and a velocity shift from +the systemic. In addition, the kinematic temperature of a part of N4 is as high as ∼ 50 K. The +authors suggested that some external force is required to explain these trends. Considering the +sources around N4, the SS 433 jet is the most plausible and is thought to be colliding with +the cloud. In fact, the X-ray jet and N4 overlap in the plane of the sky, and this scenario is +reasonable. On the other hand, although the edge cloud we identified seems to have a density +gradient, the absolute values are low compared to the typical Galactic clouds. Also, the cloud +has the typical kinematic temperature. The difference between N4 and the newly identified +clouds may come from the difference in impact given by the jet. N4 is closer to SS 433 and its +jet axis, and the active jet is interacting with N4. On the other hand, the clouds we identified +are far from SS 433 and the jet axis, and they are interacting with the surface of the eastern ear +of W50, corresponding to the surface of the jet cocoon. Additional observations such as shock +tracer should verify the difference of influence of the SS 433 jet on these molecular clouds. +4.3 High-velocity clouds in the eastern region of W50 +Finally, we mention molecular clouds in a higher velocity range. The left panel of figure 11 +is a velocity-integrated intensity map of 12CO(J=1–0) at a velocity of 77.7–83.7 km s−1. As +mentioned in section 1, Su et al. (2018) suggested that the clouds in this velocity range relate +to the SS 433/W50 system, and we also identified faint clumps with our observation in the +16 + +closer region of W50. Since they have completely different velocities, these high-velocity clouds +should be independent structures from the chimney and edge clouds. We show the spectrum of +the peak intensity position in right panel of figure 11. The center velocity and velocity width +are 79.4 km s−1 and 2.41 km s−1, respectively. +In fact, the clumps that we identified seem to be similar to the molecular clouds reported +by Su et al. (2018). However, they have a different feature (see figure 13 of their paper). Su et al. +(2018) suggested that the eastern molecular clouds of the SS 433/W50 system are approaching +us. In their model, the clouds closer to the SS 433/W50 system have a higher velocity owing +to their interaction with the jet. Although the clumps that we identified are closer to W50 +than their clouds, the central velocity is lower than their highest cloud velocity of 84 km s−1. +This means that the high-velocity clouds in the present study are not a member of the clouds +identified by Su et al. (2018), or they have slightly different kinematic properties owing to the +offset from the SS 433 jet axis. Note that the detected signal from these clumps is very weak, +and our observation is not suitable for discussing the details. We require to observe them deeper +to extract more robust conclusion. +5 Conclusion +We reported the observation of molecular clouds at the eastern edge of W50 with the Nobeyama +45-m telescope and the ASTE. We identified two clouds that possible interact with the SS +433/W50 system for the first time. One is in the northern region of the eastern ear of W50, +where there is a protruding structure called the chimney. The other is located at the eastern +edge of the ear. Both clouds comprise small clumps that might not be resolved sufficiently by our +observations. They have complex velocity structures with spectral wings and broadening, and +it is unclear whether these features are due to an interaction with W50 or merely overlapping +multiple components. The distribution of the intensity ratio and the results of RADEX analysis +reveal that the western side of the edge cloud has higher density. Although it is difficult to +conclude the relationship between the clouds that we identified and the SS 433/W50 system, +the clouds were possibly formed by the propagation of the eastern ear of W50 and the sweeping +of tiny clumps. Finally, we identified the high-velocity clouds at the eastern edge of W50 in +the same velocity range as the clouds reported by Su et al. (2018), while it is unclear whether +they belong to the same series. +17 + +Acknowledgements +We are grateful to Drs. +M. Kohno, K. Tsuge, Y. Yamane and Mr. +D. Tsutsumi for sup- +porting our observations. +We thank Dr. +H. Sano for helpful comments and discussion for +the interaction of the cloud with the GXB jet. We thank the anonymous referee for useful +comments and constructive suggestions. This work was supported by JSPS KAKENHI Grant +Numbers HS: 20J13339, 22K20386, MM: 19K03916, 20H01941, 22H01272, and KT: 18H05440, +20H01945, 22H00152. The Nobeyama 45-m radio telescope is operated by Nobeyama Radio +Observatory, a branch of National Astronomical Observatory of Japan. The ASTE telescope +is operated by National Astronomical Observatory of Japan (NAOJ). The National Radio +Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under +a cooperative agreement by Associated Universities, Inc. Data analysis was partly carried out +on the Multi-wavelength Data Analysis System operated by the Astronomy Data Center (ADC) +at the National Astronomical Observatory of Japan. 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L., Rohlfs K., H¨uttemeister S., 2009, tra..book. doi:10.1007/978-3-540-85122-6 +Yamamoto, H., Ito, S., Ishigami, S. et al. 2008, PASJ, 60, 715 +Yamamoto H., Okamoto R., Murata Y., Nakanishi H., Imai H., Kurahara K., 2022, PASJ, 74, 493. +doi:10.1093/pasj/psac012 +20 + +� +� +� +� +� +� +� +� +� +� +[K km s-1] +1 +3 +5 +2 +4 +Fig. 10. Integrated intensity map of the 12CO(J=1–0) emission of the chimney cloud in the velocity range of 30.1–36.5 km s−1 (top-left) and the results of +RADEX analysis conducted at the points shown on the map. Red and blue solid lines respectively represent the intensity ratios 13CO(J=1–0)/12CO(J=1–0) +and 12CO(J=3–2)/12CO(J=1–0) at each position. The dashed lines associated with each solid line show the 1σ of the ratio. The color represents the optical +depth τ(13CO). The parameters at the crossover points of the line intensity ratios correspond to the physical values derived by the RADEX calculation. +21 + +102 +2.00 + 1.75 +1.50 +1.25 +(O0 +Tkin +101 +0.75 +0.50 +0.25 +0.00 +100 +101 +102 +103 +104 +n(H2) [cm-3]102 +2.00 +1.75 +1.50 +1.25 +1.00 +Tkin +101 +0.75 +0.50 +0.25 +100 +0.00 +101 +102 +103 +104 +n(H2) [cm-3]102 +2.00 +F1.75 +1.50 +1.25 +(O) +X +101 +0.75 +0.50 +0.25 +100 +0.00 +101 +102 +103 +104 +n(H2) [cm-3]102 +2.00 +1.75 +1.50 +1.25 +(O0 +101 +0.75 +0.50 +0.25 +100 +0.00 +101 +102 +103 +104 +n(H2) [cm-3]102 +2.00 +F1.75 +1.50 +1.25 +(O) +X +101 +0.75 +0.50 +0.25 +100 +0.00 +101 +102 +103 +104 +n(H2) [cm-3][K km s-1] +MB [K] +T +0 +1 +2 +3 +70 +75 +80 +85 +LSR [km s-1] +V +4 +Fig. 11. (Left) Velocity-integrated intensity maps of 12CO(J=1–0). The velocity range is 77.7–83.7 km s−1. Black contours show the line of five times the rms +value. Magenta contours show the radio continuum observed with the JVLA at 1.602 GHz. The beam sizes of CO emission and continuum observations are +shown in the bottom-left corner of each panel by a white circle and magenta ellipse, respectively. (Right) 12CO(J=1–0) spectra of the peak position of the +high-velocity clouds shown in left panel. The red-dashed line shows the value of three times the rms. +22 + +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 \ No newline at end of file diff --git a/YdFQT4oBgHgl3EQfeDYW/content/tmp_files/load_file.txt b/YdFQT4oBgHgl3EQfeDYW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a522573b0c667b47076e0b35e9dc037f9732fba --- /dev/null +++ b/YdFQT4oBgHgl3EQfeDYW/content/tmp_files/load_file.txt @@ -0,0 +1,1264 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf,len=1263 +page_content='Molecular clouds at the eastern edge of radio nebula W50 Haruka SAKEMI1,∗, Mami MACHIDA2, Hiroaki YAMAMOTO3, and Kengo TACHIHARA3 1Graduate School of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, Kagoshima 890-0065, Japan 2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 3Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan ∗E-mail: sakemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='haruka@astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='jp Received ⟨reception date⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Accepted ⟨acception date⟩ Abstract Microquasar SS 433 located at the geometric center of radio nebula W50 is a suitable source for investigating the physical process of how galactic jets affect the surrounding interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Previous studies have searched for evidence of the interaction between the SS 433 jet and ISM, such as neutral hydrogen gas and molecular clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' however, it is still unclear which ISM interacts with the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We looked for new molecular clouds that possibly interact at the terminal of the SS 433 eastern jet using the Nobeyama 45-m telescope and the Atacama Submillimeter Telescope Experiment (ASTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We identified two molecular clouds, comprising many small clumps, in the velocity range of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1 for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' These clouds have complex velocity structures, and one of them has a density gradient toward SS 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Although it is difficult to conclude the relation between the molecular clouds and the SS 433/W50 system, there is a possibility that the eastern structure of W50 constructed by the SS 433 jet swept up tiny molecular clumps drifting in the surroundings and formed the molecular clouds that we identified in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Key words: ISM: individual (W50)1 — ISM: jets and outflows2 — ISM: molecules3 — stars: individual 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='13333v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='GA] 30 Jan 2023 (SS 433)4 1 Introduction Galactic X-ray binary (GXB) jets inject matter and energy into their surroundings and thus change the physical and chemical states of the interstellar medium (ISM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The study of the interaction between the jets and surrounding ISM is thus essential to understanding the evolu- tion of our galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' One of the essential contributions of the jets is the formation of molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Asahina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2014) and Asahina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2017) carried out a magnetohydrodynamical simulation to investigate the effect of jets on surrounding neutral hydrogen (HI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In their sim- ulation, jets compressed gas and triggered a cooling instability, and ultimately, the jets formed molecular clouds after ∼106 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Other studies suggested that GXB jets drive interstellar tur- bulence, produce high-energy cosmic rays, and sometimes stimulate star formation (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Heinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Mirabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Recent observational studies confirmed the interaction between galactic microquasar jets and the surrounding ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018) studied molecular clouds located on the jet axis of GRS 1915+105 using observation datasets of the Atacama Large Millimeter/sub-millimeter Array (ALMA) and found that the jet drives a shock at the impact site of the molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2020) observed density and temperature tracers from molecular clouds around GRS 1758-258 and 1E 1740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7- 2942 and investigated the jet feedback to the surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Nevertheless, there still have been few observations of the interaction between jets and the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' It is thus important to increase the number of sample observations to clarify what happens in the region of interaction between jets and the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The system of the microquasar SS 433 and surrounding radio nebula W50 is a suitable target for investigating the effects of GXB jets on their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' W50 is a large radio nebula located at (αJ2000, δJ2000) = (19h 12m 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='92s, +4d 55m 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2s) (figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' It extends over an area of 2◦ × 1◦ in the sky (Geldzahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The microquasar SS 433 is located at the geometric center of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' SS 433 ejects precessing jets in the east–west direction with a jet velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='26c, an opening angle of 20◦ and a precession period of 162 days (Abell & Margon 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Hjellming and Johnston 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Margon 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Margon & Anderson 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Davydov, Esipov, & Cherepashchuk 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The elongated structures of W50, called “ears”, are believed to be the result of an interaction between the jets and the nebula (Downes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1981a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Downes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2 1981b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Downes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The western side of W50 is near the Galactic plane, and the length of the western ear along the jet axis is shorter than that of the eastern ear (19 versus 82 pc assuming a distance of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The distance of the SS 433/W50 system is still under discussion but believed to be in the range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Many studies of the ISM around SS 433/W50 have tried to search the evidence of the jet-ISM interaction and determine the kinematic distance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The first observation of HI gas was reported by Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (1998) with the Green Bank Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' They identified an HI cavity at the position of W50 for a velocity of vLSR = 42 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, a large HI clump was detected at the southeast of the eastern ear, and it has been suggested that this HI clump collided with the eastern ear and changed the morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' A distance to the SS 433/W50 system of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 kpc was determined from these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Sakemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2021) compared the spatial distributions of W50 and the surrounding HI using GALFA-HI survey datasets taken with the 305-m Arecibo Radio Telescope (Peek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2011) and confirmed that the HI cavity identified by Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (1998) has a clear spatial correlation with W50 in the velocity range from 33 to 55 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Meanwhile, Lockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2007) observed the absorption line of HI in the direction of SS 433 and concluded that the HI cloud at a velocity of 75 km s−1 is related to the SS 433/W50 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity is clearly different from that suggested by Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (1998), and the kinematic distance is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The distance is consistent to the value derived by comparing a deep-integrated radio image of the SS 433 jet with the kinematic jet model based on the speed and the precession period of the jet (Blundell & Bowler 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Not only the HI gas but also molecular clouds have been identified around the SS 433/W50 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We summarize the previous studies in table 1 and show the positions of the clouds in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2008) suggested that the molecular clouds in the velocity range from 42 to 56 km s−1 are related to the SS 433 jet, which is distributed around the jet axis (S1–S6, N1–N4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The eastern and western clouds of SS 433 are in the velocity range of 42 to 45 km s−1 and 49 to 56 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' These velocities correspond to kinematic distances of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc for the flat rotation curve (Brand & Blitz 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The eastern clouds (S1–S6) are distributed in a region far from the eastern edge of W50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=', at approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4◦–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6◦ (outside area of figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The western-side clouds (N1–N4) are closer to SS 433, and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2020) observed N1, N2, and N3 with higher spatial resolution and concluded that a part of them is located within W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Recently, Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2022) reported the observa- tion of N4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The molecular cloud has an asymmetric spectrum for CO emission and an extreme temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' They thus suggested that the cloud is interacting with the SS 433 jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Meanwhile, Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018) found molecular clouds in another velocity range of 73–84 km s−1, 3 Chimney Eastern Edge Eastern Ear Beam sizes SS433 Precession Cone Jet Axis N1 N2 N3 N4 G39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='315-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='155 Western Ear Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Radio continuum image of W50 taken with the Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Jansky Very Large Array (JVLA) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz (Sakemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Yellow contours show the positions of the newly identified molecular clouds in the velocity range 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The contour levels are the rms × [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The yellow circle and magenta ellipse in the bottom-left corner show the beam sizes of the CO emission and continuum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Purple and hot pink ellipses show the positions of the molecular clouds identified by previous studies in the velocity range of 49–56 km s−1 and 73–77 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that the clouds S1–S6 and G40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='331–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='302 are outside the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' which are located near the western ear (G39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='315–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='155) and a straight extension of the eastern ear about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='9◦ (G40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='331–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='302, outside area of figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' They suggested that the molecular clouds located at a distance of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='9 kpc and interacted with W50 approximately 105 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The molecular clouds identified in these previous studies have good spatial correlation with the SS 433/W50 system, and some have spectral features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=', spectral broadening and wings) suggesting interaction with the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' It remains unclear at which velocities and distances the association is real, and there is no consensus in explaining this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In the present paper, we focus on the eastern edge of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Since there is a terminal region of the SS 433 jet, it is a suitable target to investigate the jet-ISM interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We made observations with the Nobeyama 45-m telescope for 12CO(J=1–0) and 13CO(J=1–0) and with the Atacama Submillimeter Telescope Experiment (ASTE) for 12CO(J=3–2) to identify molecular clouds for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We investigate the features of the molecular clouds that we identified and provide the fundamental physical parameters based on the observed CO line emissions, such as density and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The remainder of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Section 2 describes the observations and the data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Section 3 presents the observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Section 4 presents a discussion of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Section 5 provides our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The molecular clouds around the SS 433/W50 system identified by previous studies∗ Name VLSR (km s−1) Distance (kpc) References S1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 S2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 S3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 S4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 S5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 S6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 1 N1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 1,2,3 N2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 1,2,3 N3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 1,2,3 N4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 1,3 G39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='315–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='155 73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='9 4 G40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='331–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='302 74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='9 4 ∗ Column 1: Name of each molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 2: Peak velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 3: Kinematic distance referred to in each paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that the values of N1 to N4 refer to reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 4: References (1)Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2008), (2)Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2020), (3)Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2022), (4)Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2 Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 12CO(J=1–0) and 13CO(J=1–0) The 12CO(J=1–0) and 13CO(J=1–0) data were obtained with the 45-m telescope of the Nobeyama Radio Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We scanned an area of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6 arcmin2 in on-the-fly map- ping mode (Sawada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We scanned in the right ascension and declination directions to suppress the scanning effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The four-beam receiver FOREST (Minamidani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2016) and the autocorrelation spectrometer SAM45 (Kuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2011) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Typical noise temper- atures of the system including the atmosphere were between 300 and 400 K at 115 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The bandwidth and resolution were 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 MHz and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='52 kHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The pointing accuracy was checked every 2 hours by observing R Aquilae (αJ2000, δJ2000) = (19h 06m 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='25s, +8d 13m 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0s) with the frontend H40 and confirmed to be better than 3 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' All observations were conducted in equatorial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We used a chopper wheel to obtain the antenna temper- ature T ∗ a (Kutner & Ulich 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We observed W51 as a standard source to fix gain variations between the eight output signals on December 6 to 20, 2019, February 7 to March 6, 2020, and December 1 to 4, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The spectral intensity was calibrated and converted to the TMB scale by applying a main beam efficiency ηMB corresponding to each polarization, frequency, and observ- 5 ing season provided by the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The final angular resolutions were 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 arcsec at 115 and 110 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The spatial and velocity grids had sizes of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 arcsec and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity coverage was from -45 to 113 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The root-mean-square (rms) noise level in TMB was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='60 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='26 K at 115 and 110 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 12CO(J=3–2) The 12CO(J=3–2) data were obtained with the ASTE on August 9 to 11, 2019 (Ezawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We adopted the on-the-fly mode, and the mapping area had dimensions of 11 × 11 arcmin2 centered at (αJ2000, δJ2000) = (19h 16m 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5243s, +4d 51m 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='709s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The frontend was a 2SB SIS mixer receiver called DASH 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The typical system temperature was 263 K in the single side band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We used WHSF as the backend in F-FX mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The band width and resolution were 64 MHz and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='25 kHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The pointing accuracy was checked every 3–4 hours by observing R Aquilae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We convolved the intensity scale into TMB by assuming the W44 peak to be TMB = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 K (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The final angular resolution, grid size, and velocity resolution were 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 arcsec, 11 arcsec, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The rms noise level was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='09 K at 345 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 3 Results We report the molecular cloud distribution observed with the Nobeyama 45-m telescope and ASTE at the eastern edge region of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we explain the velocity features and the line-intensity ratios of the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 Spatial distributions We show the distribution of the 12CO(J=1–0) line emission in the velocity range of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1 observed with the Nobeyama 45-m telescope in figure 1 with yellow contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' They distribute around a bright filamentary structure of W50 at αJ2000 = 19h 16m 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In this region, we found a pointed structure to the north in the radio continuum image, called the chimney (see figure 1, Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Farnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Broderick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We identified a molecular cloud near the chimney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, there was another cloud of similar size at the eastern edge of the ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Figure 2 is a magnified view of the target region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The color is the integrated intensity of 12CO(J=1–0) in the velocity range 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' These velocities correspond well to those of HI gas, suggesting a relation with W50 (Dubner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Sakemi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We refer to the newly identified prominent clouds as the chimney cloud and edge 6 [K km s-1] Chimney cloud Edge cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Velocity-integrated intensity maps of 12CO(J=1–0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity range is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Magenta contours show the radio continuum observed with the JVLA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The contour levels are the rms × [2, 4, 8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The beam sizes of CO emission and continuum observations are shown in the bottom-left corner of each panel by a black circle and magenta ellipse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' � � � � � � � � � � � � � � � � � � [K km s-1] (a) 12CO( =1-0) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s-1 J [K km s-1] (b) 13CO( =1-0) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s-1 J (c) 12CO( =3-2) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1-36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s-1 J [K km s-1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Velocity-integrated intensity maps of 12CO(J=1–0) (left), 13CO(J=1–0) (middle), and 12CO(J=3–2) emissions (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity range is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Magenta contours show the radio continuum observed with the JVLA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The beam sizes of CO emission and continuum observations are shown in the bottom-left corner of each panel by a white circle and magenta ellipse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The numbers in the middle panel show the position where the physical parameters are derived in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' See table 2 and 3 and figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Besides these, faint and clumpy clouds are distributed around the filamentary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Hereafter, we focus on the chimney and edge clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Panel (a) of figure 3 is a velocity- integrated intensity map of 12CO(J=1–0) in the velocity range 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The chimney cloud has two peaks in the southern and northern parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The edge cloud is patchy, and it is brightest in the northwest region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Panel (b) of figure 3 is a velocity-integrated intensity map of 13CO(J=1–0) in the same velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The intensity distribution traces the high-density regions of observed molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Although the distribution is similar to that of 12CO(J=1– 7 [K] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Peak TMB distribution in 12CO(J=1–0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity range is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Cyan contours show the radio continuum observed with the JVLA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The beam sizes of CO emission and continuum observations are shown in the bottom-left corner of each panel by a white circle and cyan ellipse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 0), the integrated intensity map of 13CO(J=1–0) shows structures that are more clumpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thus expect that both the chimney and edge clouds comprise many separated clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Panel (c) of figure 3 is a velocity-integrated intensity map of 12CO(J=3–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that we only observed the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We detected a difference from the trend in 12CO(J=1–0) in that the intensity was higher westward rather than eastward in the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This cloud thus has different physical properties on eastern and western sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We derived the peak TMB distribution in 12CO(J=1–0) (figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The northern part of the chimney cloud has a flat temperature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Similar to the distribution of the velocity-integrated intensity, there is a temperature peak at the southern part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In the case of the edge cloud, the temperature peak is located eastward of the northern part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This position does not correspond to the peak area of the velocity-integrated intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 Velocity structures We next explain the velocity structures of the molecular clouds in the eastern region of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Figure 5 is a position–velocity diagram of the chimney cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that we inclined the boxes to align the integration axis with the direction of the SS 433 jet axis and thus assess the effect of the jet activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We divided the chimney cloud into eight regions and investigated the 8 0 100 100 200 300 200 300 1 2 3 4 5 6 7 8 19h16m50s 40s 30s 20s J2000 Right Ascension J2000 Declination 5 06’ ∘ 03’ 00’ 4 57’ ∘ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='[K km s-1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='Offset [arcsec] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='30 32 34 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='LSR [km s-1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Similar to figure 5 but for the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' velocity structures in the northeast-to-southwest direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The most outstanding feature is seen in region 4, where the chimney cloud seems to touch the chimney of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In this region, there is a curved structure of the position–velocity diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The center velocity seems to shift approximately 1 km s−1 along the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' A similar trend is seen for the edge cloud (figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We divided the edge cloud into seven regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The curved structures are in regions 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Especially in region 5, the curved structure is above an offset of 0 arcsec, where the edge cloud also touches the eastern edge of W50 in the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, the velocity width is thick (∼ 4 km s−1) at an offset less than 0 arcsec in region 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Figure 7 presents spectra for different parts of the chimney cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' At the northern peak, the spectrum tends to have a wide velocity width and seems to have a spectral wing toward lower velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Between the northern and southern peaks, spectral wings form toward higher velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Spectral wings often imply the existence of an external force generated by surrounding sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' However, these wings may originate from multiple clouds in the line of sight and at slightly different velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Figure 8 presents spectral plots of the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=" At the peak positions in the northwest region, we clearly see a wide spectrum not only for 12CO(J=1-0) 9 25 5°06' 20 Declination (J2000) 03' 15 kms-1 10 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content="00 5 4°57' 19h16m50s 405 305 20s RightAscension (J2000)4°57' 25 20 Declination(2000) 54* 15 kms-1 51' 10 48' 19h17m105 oos 16m50s 405 305 Right Ascension (J2000)0 4 8 30 36 LSR [km s-1] V MB [K] T 15." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 [K km s-1] 19h16m48s 42s 36s J2000 Right Ascension 30s J2000 Declination 5 04’ ∘ 02’ 00’ 4 58’ ∘ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (Left) Integrated intensity map of the 12CO(J=1–0) emission of the chimney cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity range is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (Right) 12CO(J=1–0) (black) and three times 13CO(J=1–0) (red) spectra for the regions shown in the grids in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 0 4 30 36 LSR [km s-1] V MB [K] T 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 [K km s-1] 19h17m00s 16m54s 48s J2000 Right Ascension 42s J2000 Declination 4 54’ ∘ 52’ 50’ 48’ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Similar to figure 7 but for the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=" 10 25 5°04' 20 Declination (j2000) 02'- 15 kms 10 Y 00' 5 4°58' 0 19h16m485425 365 305 RightAscension(j2000)MMMt25 4°54' 20 Declination (J2000) 52' 15 kms-1 10 50° 5 48' 0 1gh17m00516m545 485 425 RightAscension(j2000)(a) 13CO( =1-0) / 12CO( =1-0) J J (b) 12CO( =3-2) / 12CO( =1-0) J J Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Intensity ratios of 13CO(J=1–0) and 12CO(J=1–0) (a) and 12CO(J=3–2) and 12CO(J=1–0) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Only the regions where the intensities of the CO emission are higher than three times the rms value are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Black contours show the values of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Magenta contours show the radio continuum observed with the JVLA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The resolutions of intensity ratio maps and continuum are given at the bottom-left corner of each panel by black circle and magenta ellipse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' but also for 13CO(J=1-0) with a mean velocity width of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='26 km s−1, which is a result similar to that in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In the northeast region of the edge cloud, spectral wings form toward high velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 Intensity ratios Panel (a) of figure 9 shows the intensity ratio of 13CO(J=1–0)/12CO(J=1–0), which traces high- density areas of molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Most parts of the chimney and edge clouds have values below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In the edge cloud, we see again the patchy structure, and we analyze the physical properties of each clump in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Panel (b) of figure 9 shows the intensity ratio of 12CO(J=3– 2)/12CO(J=1–0) of the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' There is a gradient from west to east, which suggests a difference in temperature and/or density in the western and eastern parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4 Discussion In this section, we consider the relation between the molecular clouds that we identified and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We first derive the physical parameters of the chimney and edge clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We then consider the interaction between these clouds and the SS 433/W50 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We also 11 mention the relation between these clouds and a giant molecular filament in front of the SS 433/W50 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We finally discuss the high-velocity clouds in terms of the relation with the clouds identified by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 Physical properties of the molecular clouds We first estimated the column densities of each region shown in panel (b) of figure 3 adopting two methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=', assuming the X-factor and local thermodynamic equilibrium (LTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The X-factor converts the 12CO(J=1–0) integrated intensity W(12CO(J =1–0)) to the column density N(H2), and we adopt the value of N(H2)/W(12CO(J =1–0)) = 2×1020 cm−2 (K km s−1)−1 with ± 30% uncertainty (Bolatto, Wolfire, & Leroy 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Alternatively, we used the following procedures to derive the column density N(H2) by assuming LTE (Wilson, Rohlfs, & H¨uttemeister 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Assuming the 12CO(J=1–0) line is optically thick, the excitation temperature Tex is derived from the 12CO peak intensity TMB: Tex = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 � ln � 1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 TMB + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='82 ��−1 [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (1) The optical depth τ13(v) is calculated for the 13CO(J=1–0) brightness temperature of each velocity channel TMB(v) as τ13(v) = −ln � ��1 − TMB(v) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 � � � 1 exp � 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 Tex � − 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='16 � � � −1� ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2) Using the velocity resolution ∆v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 km s−1, the column density of 13CO, N(13CO), is calculated as N(13CO) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 × 1014 × � v Texτ13(v)∆v 1 − exp � − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 Tex � [cm−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (3) We finally adopt the conversion factor from N(13CO) to N(H2) of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 × 105 (Frerking, Langer, & Wilson 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Wilson & Rood 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We present the results in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The column densities derived by assuming the X-factor are slightly higher than those derived using the LTE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' however, we can explain these differences by errors in the observations and the abundance ratios [13CO]/[H2] = 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 × 10−6 and the X-factor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 × 1020 cm−2 (K km s−1)−1 depending on the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We calculated the masses of the chimney and edge clouds using N(H2) derived by assuming the X-factor, Mmol = ¯µmH � [N(H2) × Ω × D2], (4) where ¯µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8, mH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='67 × 10−24 g, Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 × 10−9 sr, and D pc are the mean molecular weight, the mass of the atomic hydrogen, the solid angle subtended by the grid spacing, and 12 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column densities∗ Region N1−0 X N13,1−0 LTE chimney 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='28 chimney 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='44 chimney 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='70 chimney 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='14 edge 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='90 edge 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='97 edge 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='40 edge 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='72 edge 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='64 ∗ Column 1: Region name given in panel (b) of figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 2: Column density of H2 derived from 12CO(J=1–0) in 1021 cm−2 assuming the X-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 3: Column density of H2 derived from 13CO(J=1–0) in 1021 cm−2 assuming LTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Results of RADEX calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='∗ Region R13/12 1−0 R12 3−2/1−0 τ(13CO) n(H2) Tkin edge 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='154±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='190±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='12 320+120 −100 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 edge 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='195±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='201±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='72+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='04 880+220 −180 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6 edge 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='123±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='137±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='64+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='10 270+110 −90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 edge 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='204±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='228±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='06 800+400 −280 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 edge 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='166±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='244±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='16 740+660 −340 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='8 ∗ Column 1: Region name given in figures 3 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 2: Intensity ratio of 13CO(J=1–0)/12CO(J=1–0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 3: Intensity ratio of 12CO(J=3–2)/12CO(J=1–0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 4: Optical depth of 13CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 5: Number density of H2 in cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Column 6: Kinematic temperature in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' the distance to the molecular clouds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' If we assume the distance of D = 5500 pc, the masses of the chimney and edge clouds are 3800 and 2300 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Also, the masses of these clouds are 1100 and 700 M⊙ assuming the distance of D = 3000 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we carried out the RADEX calculation to estimate the temperature and density of the edge cloud using non-local thermodynamic equilibrium radiative transfer code (van der Tak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We input parameter sets of Tk and n(H2) for intervals of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1 K in the range of 1 to 100 K and intervals of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='05 cm−3 in the range of 10 to 104 cm−3, and we calculated the intensities 12CO(J=1–0), 13CO(J=1–0) and 12CO(J=3–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The line intensity 13 ratios of 13CO(J=1–0)/12CO(J=1–0) and 12CO(J=3–2)/12CO(J=1–0) were then calculated in n(H2)–Tk space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We estimated N(12CO) as N(12CO) = N(H2) × 10−4, (5) where N(H2) was derived by assuming the X-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we assumed N(12CO)/ N(13CO)=62 (Milam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We analyzed the local peaks of 13CO(J=1–0) of the edge cloud shown in the top-left panel of figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The observed line intensity ratios of 13CO(J=1– 0)/12CO(J=1–0) and 12CO(J=3–2)/12CO(J=1–0), with one sigma error, are plotted in n(H2)– Tk space with red and blue lines in five plots of figure 10 with the expected optical depth τ(13CO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The parameters at the crossover points of the line intensity ratios correspond to the physical values derived by the RADEX calculation and are listed in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We note that the densities of all regions are much lower than the critical density of 13CO(J=1–0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' nevertheless, the 13CO(J=1–0) emission is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This implies that the edge cloud comprises tiny clumps that cannot be spatially resolved by our observations and are distributed with the low beam- filling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In that case, the emission is smoothed out, and the densities are estimated to be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we find that the western side of the edge cloud has higher density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This result is consistent with the trend of the intensity ratio 12CO(J=3–2)/12CO(J=1–0) shown in panel (b) of figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 Association with the SS 433/W50 system We here consider the relation between the chimney and edge clouds and the SS 433/W50 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Our observations revealed that these clouds are close to the eastern region of W50 in the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The clouds have clumpy spatial structures and complex velocity structures, including a shift in the central velocity (figures 5 and 6), spectral wings, and wider velocity widths (figures 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we identified that the density distribution of the edge cloud is not flat (panel b of figure 9 and figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Near the chimney and edge clouds, there is no heating source that might explain their complexity, such as the HII region radiating UV photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Also, the gamma-ray emission regions of the SS 433 jet are far from these clouds, above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6◦, and should have less influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Therefore, the properties of these clouds imply the interaction with the eastern ear of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Unfortunately, we cannot conclude the interaction from only our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' There is a possibility that the clouds are mere foreground or background sources of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' As an example, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2020) identified a giant molecular filament in the Milky Way Imaging Scroll Painting (MWISP) in front of the SS 433/W50 system in the velocity range from 27 to 14 40 km s−1, named GMF MWISP G041-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thus need to discuss the association between the molecular clouds that we identified and this giant molecular filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2020) suggested that GMF MWISP G041-01 comprises four components, three being filamentary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Two such giant filaments possibly collided in the region around (αJ2000, δJ2000) = (19h 10m 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3525s, +7d 15m 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='330s) and (19h 10m 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='6389s, +6d 10m 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='713s), while they might be just overlapping along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' If such a collision actually occurred, spectral wings, such as those observed for the chimney and edge clouds, should be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The clouds that we identified might then be part of GMF MWISP G041-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' However, the filament–filament collision is located away from the clouds, by approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 degrees, corresponding to 71 pc assuming the distance of GMF MWISP G041-01 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thus infer that the clouds that we identified are independent of the system of GMF MWISP G041-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Hereafter, we assume that the chimney and edge clouds are related to W50 and consider the formation mechanism of these molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' An instinctive formation scenario is that the eastern ear of W50 compressed the surrounding HI gas, and a shock induced the molecular cloud formation (Asahina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Asahina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' However, this mechanism would take a long time, approximately 106 yr, to form molecular clouds, and this time is in disagreement with the age of W50 (a few 104 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We then consider the alternative scenario of sweeping tiny molecular clumps by the surface of the eastern ear of W50, corresponding to the surface of the jet cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' As mentioned above, the chimney and edge clouds seem to comprise more small clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, we identified faint and clumpy clouds in the eastern region of W50 in the velocity range of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1 except for the chimney and edge clouds (figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This means that many tiny clumps are drifting at the position of the eastern ear of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The origin of the tiny clumps remains unknown, but there is a possibility that they were formed by the activity of the progenitor star of SS 433, such as by the compression of the surrounding HI gas by the stellar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Here, we imagine that the faint clouds are distributed throughout the region before the eastern ear goes through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The mean column density of H2 in the velocity range of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 km s−1 except for the chimney and edge clouds is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='18 × 1020 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We regard the eastern ear as a cylinder with a diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 deg and a height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Assuming a distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc, the eastern ear can collect clumps to a mass of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 × 103 M⊙ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 × 104 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' If we reduce the velocity range and consider only the contribution of the northern part of the eastern ear, the mass is comparable to that of the total mass of the chimney and edge clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additionally, this scenario explains the complexity of the velocity structures of the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that these clumps might influence the formation theory of the eastern ear of W50 (Goodall, Alouani-Bibi, & Blundell 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Ohmura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' To consider 15 how they affect, we require to carry out additional numerical simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' however, it is out of the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Also, we discuss the distance of the SS 433/W50 system based on the relation with the newly identified clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Using the velocity of 33 km s−1, roughly the peak velocity of the chimney and edge clouds, the kinematic distance is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This value is far from the historically discussed one in the range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Here, we suggest that the clouds are unsuitable for deriving the distance based on the model for Galactic rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' As mentioned above, these clouds probably consist of many small clumps, and they are affected by the motion of the SS 433 jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Small clumps can be easily shifted the central velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Therefore, there is a possibility that these clouds do not follow simply the rotation of the galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This scenario can explain the reason why the velocities of these clouds are lower shifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' since the eastern jet is approaching, the clouds have been pushed toward us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We mention the differences between the clouds we identified and reported by Yamamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2022), N4 (see figure 1 and table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' N4 is not similar to the typical molecular clouds in the Galactic plane, and it shows clear intensity and velocity gradients and a velocity shift from the systemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In addition, the kinematic temperature of a part of N4 is as high as ∼ 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The authors suggested that some external force is required to explain these trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Considering the sources around N4, the SS 433 jet is the most plausible and is thought to be colliding with the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In fact, the X-ray jet and N4 overlap in the plane of the sky, and this scenario is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' On the other hand, although the edge cloud we identified seems to have a density gradient, the absolute values are low compared to the typical Galactic clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Also, the cloud has the typical kinematic temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The difference between N4 and the newly identified clouds may come from the difference in impact given by the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' N4 is closer to SS 433 and its jet axis, and the active jet is interacting with N4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' On the other hand, the clouds we identified are far from SS 433 and the jet axis, and they are interacting with the surface of the eastern ear of W50, corresponding to the surface of the jet cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Additional observations such as shock tracer should verify the difference of influence of the SS 433 jet on these molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='3 High-velocity clouds in the eastern region of W50 Finally, we mention molecular clouds in a higher velocity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The left panel of figure 11 is a velocity-integrated intensity map of 12CO(J=1–0) at a velocity of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' As mentioned in section 1, Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018) suggested that the clouds in this velocity range relate to the SS 433/W50 system, and we also identified faint clumps with our observation in the 16 closer region of W50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Since they have completely different velocities, these high-velocity clouds should be independent structures from the chimney and edge clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We show the spectrum of the peak intensity position in right panel of figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The center velocity and velocity width are 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='4 km s−1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='41 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In fact, the clumps that we identified seem to be similar to the molecular clouds reported by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' However, they have a different feature (see figure 13 of their paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018) suggested that the eastern molecular clouds of the SS 433/W50 system are approaching us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' In their model, the clouds closer to the SS 433/W50 system have a higher velocity owing to their interaction with the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Although the clumps that we identified are closer to W50 than their clouds, the central velocity is lower than their highest cloud velocity of 84 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This means that the high-velocity clouds in the present study are not a member of the clouds identified by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018), or they have slightly different kinematic properties owing to the offset from the SS 433 jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Note that the detected signal from these clumps is very weak, and our observation is not suitable for discussing the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We require to observe them deeper to extract more robust conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 5 Conclusion We reported the observation of molecular clouds at the eastern edge of W50 with the Nobeyama 45-m telescope and the ASTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We identified two clouds that possible interact with the SS 433/W50 system for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' One is in the northern region of the eastern ear of W50, where there is a protruding structure called the chimney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The other is located at the eastern edge of the ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Both clouds comprise small clumps that might not be resolved sufficiently by our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' They have complex velocity structures with spectral wings and broadening, and it is unclear whether these features are due to an interaction with W50 or merely overlapping multiple components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The distribution of the intensity ratio and the results of RADEX analysis reveal that the western side of the edge cloud has higher density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Although it is difficult to conclude the relationship between the clouds that we identified and the SS 433/W50 system, the clouds were possibly formed by the propagation of the eastern ear of W50 and the sweeping of tiny clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Finally, we identified the high-velocity clouds at the eastern edge of W50 in the same velocity range as the clouds reported by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (2018), while it is unclear whether they belong to the same series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 17 Acknowledgements We are grateful to Drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Kohno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Tsuge, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Yamane and Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Tsutsumi for sup- porting our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Sano for helpful comments and discussion for the interaction of the cloud with the GXB jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thank the anonymous referee for useful comments and constructive suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This work was supported by JSPS KAKENHI Grant Numbers HS: 20J13339, 22K20386, MM: 19K03916, 20H01941, 22H01272, and KT: 18H05440, 20H01945, 22H00152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The Nobeyama 45-m radio telescope is operated by Nobeyama Radio Observatory, a branch of National Astronomical Observatory of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The ASTE telescope is operated by National Astronomical Observatory of Japan (NAOJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under a cooperative agreement by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Data analysis was partly carried out on the Multi-wavelength Data Analysis System operated by the Astronomy Data Center (ADC) at the National Astronomical Observatory of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This work was supported by the Japan Foundation for Promotion of Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' SAOImageDS9 development was made possible by funding from the Chandra X-ray Science Center (CXC), the High Energy Astrophysics Science Archive Center (HEASARC), and the JWST Mission office at the Space Telescope Science Institute (Joye & Mandel 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' This research used Astropy,1 a community-developed core Python package for Astronomy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' We thank Edanz (https://jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=', 2022, PASJ, 74, 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1093/pasj/psac012 20 � � � � � � � � � � [K km s-1] 1 3 5 2 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Integrated intensity map of the 12CO(J=1–0) emission of the chimney cloud in the velocity range of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 km s−1 (top-left) and the results of RADEX analysis conducted at the points shown on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Red and blue solid lines respectively represent the intensity ratios 13CO(J=1–0)/12CO(J=1–0) and 12CO(J=3–2)/12CO(J=1–0) at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The dashed lines associated with each solid line show the 1σ of the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The color represents the optical depth τ(13CO).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='25 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='00 101 102 103 104 n(H2) [cm-3][K km s-1] MB [K] T 0 1 2 3 70 75 80 85 LSR [km s-1] V 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (Left) Velocity-integrated intensity maps of 12CO(J=1–0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The velocity range is 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='7 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Black contours show the line of five times the rms value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' Magenta contours show the radio continuum observed with the JVLA at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='602 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The beam sizes of CO emission and continuum observations are shown in the bottom-left corner of each panel by a white circle and magenta ellipse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' (Right) 12CO(J=1–0) spectra of the peak position of the high-velocity clouds shown in left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' The red-dashed line shows the value of three times the rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content=' 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFQT4oBgHgl3EQfeDYW/content/2301.13333v1.pdf'} +page_content='5 3.' metadata={'source': 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E-mail: ceren.ocal@gmail.com; ORCID: 0000-0002-0652- +7386 +2 Yaşar University, Department of Management Information Systems, Universite Cad. 35, +35100, Bornova, Izmir, Turkey. E-mail: murat.komesli@yasar.edu.tr; ORCID: 0000-0002- +8240-5540 +3 Ege University, Department of Computer Engineering, Universite Cad. 9, 35100, Bornova, +Izmir, Turkey. E-mail: murat.osman.unalir@ege.edu.tr; ORCID: 0000-0003-4531-0566 +*Corresponding Author: Prof. Dr. Murat Komesli, Yaşar University, Department of +Management Information Systems, Universite Cad. 35, 35100, Bornova, Izmir, Turkey. E- +mail: murat.komesli@yasar.edu.tr; Tel (Office): +902325707817; Mobile: +905055718202 + + + + + + + + + + + + + + + + + + + + + +2 + +Semantic Web Enabled Geographic Question Answering Framework: GeoTR + +Abstract +With the considerable growth of linked data, researchers have focused on how to increase the +availability of semantic web technologies to provide practical usages for real-life systems. +Question answering systems are an example of real-life systems that communicate directly with +end-users, understand user intention and generate answers. End users do not care about the +structural query language or the vocabulary of the knowledge base where the point of a problem +arises. In this study, a question-answering framework that converts Turkish natural language +input into SPARQL queries in the geographical domain is proposed. Additionally, a novel +Turkish ontology, which covers a 10th-grade geography lesson named “Spatial Synthesis: +Turkey”, has been developed to be used as a linked data provider. Moreover, a gap in the +literature on Turkish question answering systems, which utilizes linked data in the geographical +domain, is addressed. A hybrid system architecture that combines natural language processing +techniques with linked data technologies to generate answers is also proposed. Further related +research areas are suggested. + +Key Words: question answering systems, linked data, ontology development, natural +language processing + + +1. Introduction +As the importance of technology and the amount of stored data increases, fast and easy access +to information for end-users has become a never-ending challenge. One of these challenges is +that, as natural language is the most common way for human beings to communicate as end- +users, user intention in the form of natural language must be transformed into a structural format +and represented with machine-understandable notation. Tokenizing natural language input, +analysis of each token, converting it into a machine-understandable representation, and +generating answers or outcomes based on user intention are the major elements focused on by +researchers in this area. Natural language processing (NLP), one of the research fields within +artificial intelligence, can be used to solve some of these aforementioned issues. +Processes that perform the tasks of tokenizing natural language input and carrying out linguistic +analysis on each token and their components are the main phases of NLP. However, despite the +deep analysis outcomes of NLP, there are some weaknesses in enriching input semantically. +Therefore, semantic technologies are required to analyse the outcomes of NLP and to focus on +supporting techniques and methodologies to achieve an upper-level understanding of +expressions (Berners-Lee et al. 2001). The technology behind semantic analysis consists of +ontologies that are composed of relationships and taxonomies to represent a specific +conceptualization in a specific domain. Ontologies play an essential role in sharing and +exchanging knowledge between different platforms of different domains. An ontology provides +flexibility and interoperability opportunities, which has resulted in a significant increase in the +number of studies of the ontology concept, OWL or RDF representation, and ontology query +languages like RDQL and SPARQL. +Advances in NLP techniques have also developed on account of improvements in semantic +technologies. Recently, a new discussion has arisen about how to understand natural language +input and generate appropriate answers with hybrid solutions, using a combination of NLP +techniques and semantic technologies. Researchers have attempted to improve NLP outcomes + +3 + +by applying semantic enrichment. Another important discussion concerns how NLP contributes +to ontology learning, ontology querying, and multilingual ontology mapping. Hybrid solutions +involve a reciprocal relationship between NLP and semantic technologies (R. Q. Guo et al. +2009). +Understanding a sentence involves two main phases: morphological analysis and semantic +analysis (Y. Guo et al. 2011). Morphological analysis focuses on determining the structure of +words, the relationships between words, POS (part of speech) tags (noun, verb, adjective, etc.), +and their positions in the sentence (subject, object, predicate, etc.). Named entity recognition +(NER) is one of the NLP techniques for semantic enrichment. Entities in an expression are +tagged by using predefined categories, such as person, organization, date, location, number, etc. +in the NER model (Collobert et al. 2011). The generated outcome is combined with +conceptualization and linked data defined in ontologies. The hybrid approach provides an +opportunity to use ontologies as data sources for question answering systems that have a natural +language input format. A knowledge requirement denoted in natural language is transformed +into ontology query language. Without dealing with additional technical details relating to the +ontology, such as query language, vocabulary or hierarchical structures, even end-users can +utilize the knowledge presented in ontologies (Bernstein et al. 2005). This approach bridges the +gap between end-users and semantic technologies and also provides practical implications for +ontologies. +This study firstly outlines the Question Answering Framework (QAF) over linked data, presents +a detailed comparison between them, and proposes a Turkish geographical question answering +framework that combines NLP techniques and semantic web technologies over linked data. The +current section introduces the problem, concepts, motivation, and discussions. The following +section includes background information and related work. Section 3 defines the natural +language-aware ontology development process. Following this, section 4 describes the +proposed framework and developed geographical ontology by dividing it into 3 subsections: +question pre-processing; query formulation; and experimental study. Conclusions and future +work are set out in Section 5. The contributions of this study to the literature are also discussed +in the final section. +2. Background Information and Related Work +12 question answering systems or frameworks over linked data with cutting edge approaches +are examined to give an overview of the literature. +Habernal and Konopík introduced a Semantic Web Search Using Natural Language (SWSNL) +system. SWSNL employs ontologies in order both to store data and domain structure, and to +return answers for users’ search queries in natural language. The pipeline described for the +system holds the phases for pre-processing, semantic analysis, semantic interpretation, and +executing SPARQL to generate answers. The authors claim there are significant differences +between their approach and other similar systems in that users can formulate natural language +queries in more than one sentence, and stochastic semantic analysis model pre-processing is +handled without requiring any syntactic parser. SWSNL has been tested for three domains in +different languages: accommodation in English, public transportation in Czech, and a subset of +the English ATIS dataset. Serving different domains and languages addressed the portability +feature of SWSNL in their study (Habernal and Konopik 2013). +Xser is a question answering system that generates answers over DBpedia by transforming +natural language questions into linked data query representation. Unstructured natural language +input is converted into structured SPARQL queries to be executed on the DBpedia endpoint. +The main motivation is to understand user intention accurately and to map user intention and + +4 + +ontology conceptualization, which form the model for concepts and their interrelationships in +an ontology (Xu et al. 2014). +Intui3 is proposed as a multilingual question answering platform over linked data that applies +both syntactic and semantic analysis of natural language input while formulating RDF triples +to represent user queries. Dima addresses the problems in traditional keyword-based searches +– that results are given only as a ranked list of documents rather than a precise answer - and +proposes a solution by offering a new alternative search paradigm. The Intui3 search paradigm +is based on a predicate and entity index and descriptions of all entities and predicates are +sourced from DBpedia (Dima 2014). +In the study of (He et al. 2014), a question answering system over linked data, namely +CASIA@V2, is described. A Markov Logic Network, a joint learning model, is used to detect +and classify phrases and perform mapping of phrases with semantic entities (Richardson and +Domingos 2006). In the first step, the input question is decomposed to detect phrases, followed +by mapping the selected candidate phrases and concepts in DBpedia. Decomposed phrases that +contribute to user intention are grouped to generate triples which are actually the components +of SPARQL. +(Park et al. 2015) propose ISOFT as a SPARQL template generator for natural language +questions that combines two notable fields: knowledge-based question answering and +information retrieval-based question answering. Semantic similarity is positioned as the focus +of this study. The natural language input query is parsed into constituent phrases to be matched +with SPARQL templates by imposing semantic similarity. Phrases interpreted as a contributor +to answer generation are extracted to be mapped with uniform resource identifiers (URIs) that +represent concepts in the ontology. + +The study of (Paredes-Valverde et al. 2015) proposes an ontology-based information retrieval +system called ONLI (Ontology-based Natural Language Interface). The structure and context +of a natural language question are represented with an ontology model in DBpedia to address +user intention. Questions are classified regarding their context. The probability of returning +accurate answers is improved by classifying the NL questions to reduce search space. The ONLI +pipeline has 3 main modules: question processing, knowledge bases, and answer searching and +building. The question processing module deals with the pre-processing of natural language +input syntactically to further improve the results of semantic analysis. Identifying semantically +similar terms or finding synonyms enables semantic enrichment of the input. An ontological +model, namely the Question Model, is used to organize answer searching and building. Applied +search methods vary in different domains. ONLI provides possible answers with relevance +ranking. +QAnswer is a question answering framework developed by (Ruseti et al. 2015) to convert +natural language input questions into a SPARQL query to be executed on a DBpedia endpoint. +The first task in their pipeline is to detect the DBpedia entities and the type of dependency +relationships between them. The outcome of the dependency-parsing phase is a directed graph +with vertices and edges, where vertices represent annotated tokens and edges hold collapsed +dependencies. After addressing the dependencies, entity type detection is performed. After +deciding on the various types and dependencies, candidate matches are generated. In other +words, multiple different directed candidate graphs that represent possible question patterns are +obtained. Finally, a scoring algorithm is applied to identify the graph with the highest score to +generate SPARQL. + +5 + +Baudis and Sedivy propose a question answering system pipeline called YodaQA that generates +answers sourced from DBpedia and Freebase. Fundamental processes included in the pipeline +can be listed as question analysis, answer production, answer analysis, answer merging and +scoring, and successive refining. They formulate each question by naming three dimensions, +“Clues”, “Focus” and “LAT (Lexical Answer Type)”. Clues represent keywords or expressions +that might play essential roles in user intention. Focus is the target object to be queried. LAT is +derived from focus and stands for answer type description. Semantic endpoints are searched for +each clue in the sentence. Returned answers are analysed to determine LAT. Answers are +classified with regard to their features. A scoring algorithm is applied for further processing +with successive refining. The answer with the highest score is selected as the outcome of the +pipeline (Baudiš and Šedivý 2015). +Mazzeo and Zaniolo put forward a question answering framework named CANaLI (Context- +Aware controlled Natural Language Interface) to return results for controlled natural language +(CNL) questions. CNL refers to restricting the grammar to make the language more easily +machine-interpretable and creating a formal structure to the input questions while still +protecting the natural format of the language. CANaLI suggests auto-generated question +patterns to end-users typing questions, thereby correcting them semantically, syntactically, and +grammatically in accordance with the concepts in the underlying knowledge base. DBpedia, +MusicBrainz, DrugBank, Diseasome, and SIDER are the domain knowledge bases. +Encyclopedic, music, and medicine are employed as data sources (Mazzeo and Zaniolo 2016). + +AMAL (Ask Me in Any Language) is a solution that allows users to ask questions in French +that are then automatically converted into SPARQL queries to find answers found in DBpedia +(Radoev et al. 2017). AMAL has four main phases in generating an answer: question +classification, entity extraction, property identification, and SPARQL query building. For now, +it only focuses on simple questions that hold one single entity and a single property. However, +the authors define AMAL as a still-evolving system that has future capacity for complex +questions which hold more than one entity-property match. +(Diefenbach et al. 2018) proposes a question answering component entitled WDAqua-core0 +that finds answers to natural language queries and keyword queries over DBpedia and Wikidata. +DBpedia provides language support only for English, whereas Wikidata extends the range to +French, German and Italian. WDAqua-core0 serves the research community through the +Qanary Ecosystem (Diefenbach et al. 2017b), which is a framework to integrate question- +answering components for reusability. The authors apply a combinatorial approach based on +the semantic items represented in linked data sources. Two triple patterns, SELECT and ASK, +are handled by WDAqua-core0. Only using the COUNT operator and not being able to handle +questions containing comparative and superlative expressions are expressed as limitations of +their study. + +A method called SPARQLtoUser is described in another study by (Diefenbach et al. 2017a). It +focuses on generating a user understandable version of SPARQL by converting user intention +to a representation of a SPARQL query. The noteworthy features of SPARQLtoUser are +claimed to be its multilingualism and portability. Users can generate answers to questions in +different languages from different knowledge bases. Instead of verbalizing the query, a generic +approach to building up a user understandable schematic representation of the query is proposed +in their study. + +The above-mentioned studies and this study are compared according to year, domain, language, +learning appliance, and employed methods in Table 1. + +6 + +Table 1. Comparison of this study (Geo-TR) and literature studies +System +Year +Domain +Language +Learning +Applied or +Not +Methods +SWSNL +2013 +Multidomain +(tested on +accommodation, +medicine, and +DBpedia) +Multilingual +(tested in Czech +and English) +Yes +POS Tagging, +Dependency +Analysis, Named +Entity +Recognition, +Lemmatizing +Xser +2014 +Multidomain +(tested on +Freebase and +DBpedia) +Multilingual +(tested in +English) +Yes +POS Tagging, +Dependency +Analysis, Named +Entity Recognition +Intui3 +2014 +Multidomain +(tested on +DBpedia) +English +No +POS Tagging, +Dependency +Analysis, Named +Entity Recognition, +Lemmatizing, +Chunking +CASIA@V2 +2014 +Multidomain +(tested on +DBpedia) +English +Yes +POS Tagging, +Dependency +Analysis, Lexicon +Model +ISOFT +2015 +Multidomain +English +No +Natural language +input query, +Imposing semantic +similarity, +Matching with +SPARQL templates +ONLI +2015 +Multidomain +(tested on +DBpedia) +Multilingual +(tested in +English) +No +POS Tagging, +Named Entity +Recognition, +Lemmatizing, +Synonym +Extension +QAnswer +2015 +Multidomain +(tested on +DBpedia) +English +No +Dependency +Analysis, Named +Entity Recognition, +Generate SPARQL +YodaQA +2015 +Multidomain +(tested on +Freebase and +DBpedia) +English +Yes +POS Tagging, +Dependency +Analysis, Named +Entity Recognition, +Lexicon Model +CANaLI +2016 +Multidomain +(tested on +DBpedia, +music, and +medicine) +English +No +Controlled Natural +Language +Grammar, +Ontology-Driven +Auto-Completion +Amal +2017 +Multidomain +(tested on +DBpedia) +French +Yes +Named Entity +Recognition, +Ontology-Driven +Property +Extraction, +Lexicon Model +WDAqua-core0 +2017 +Multidomain +Multilingual +(tested in +English, French, +No +Named Entity +Recognition, + +7 + + +3. Ontology Development (Natural Language Aware) +The term “ontology” comes from the combination of the ancient Greek words “ontos”, which +means “being”, and “logos”, which means “word”. In philosophy, ontology is defined as the +subject of existence by (Gaševic et al. 2006), and existence categorization for a specific domain +by (Sowa 2000). Ontologies are applied to represent knowledge by defining a set of concepts +and relationships in computer science. By modelling a specific domain with concepts, including +their properties, and relationships, an ontology represents a structural shared vocabulary among +multiple systems. +Underpinning the development of an ontology is the requirement that it allows for the sharing +of domain knowledge to provide semantic interoperability and facilitates the reusing of +ontological concepts in different platforms. The most critical factor during its development is +understanding the set of questions and related answers that an ontology must be able to answer +correctly. These are competency questions. (Uschold and Gruninger 1996; Grüninger and Fox +1995). User stories, possible use case scenarios, and normal and alternative processing steps +are helpful to determine such competency questions. Compared with other steps involved in +ontology development, dealing with competency questions is essential, not only to define the +context of the ontology but also to underpin the architecture behind it. Information contexts that +cover competency questions provide clues about how to encode semantic items in the ontology +and any application platform. Additionally, competency questions are dynamic. For long-term +sustainability purposes, they must be updated by extending the context of the ontology (Kendall +and McGuinness 2019). Competency questions help an ontology to be natural language-aware, +which fits well with question answering systems that take input in the form of natural language. +In this study, a natural language-aware geographical ontology was developed by applying the +steps defined in Ontology Development 101 (Noy and McGuinness 2001). Instead of using an +existing ontology encoded in a language other than Turkish and having to deal with the burden +of translation, an ontology (named GEO-TR) was developed in respect of the geography of +Turkey. All data and object properties, individuals, and class names are in Turkish. +Additionally, the development of a Turkish ontology in the geographical domain contributes to +the field and addresses a gap in the current literature. +The first step in ontology development 101 is determining the domain and scope by defining +competency questions. The questions: “Which type of domain will the ontology cover?”, “For +what the ontology will be used?” and “What types of questions will be answered by the +ontology?” should be answered during the first step of the development. While designing the +GEO-TR ontology, Chapter 6: “Spatial Synthesis of Turkey” from the 10th grade geography +(tested on +DBpedia and +Wikidata) +German and +Italian) +Ontology-Driven +Property Extraction +SPARQLtoUser +2018 +Multidomain +(tested in +English) +Multilingual +No +The extended +version of +WDAqua-core0 +with user +interaction +OUR STUDY +(Geo-TR) +2019 +Multidomain +(tested on +Geographic +domain) +Multilingual +(tested in +Turkish) +Yes +POS Tagging, +Dependency +Analysis, Named +Entity Recognition, +Ontology-Driven +Property, and +Instance Validation + +8 + +class from the national curriculum was chosen as the target scope. Competency questions were +derived from “Characteristics and distribution of landforms”, “Climate of Turkey”, “Features +of geospatial model of Turkey” and “Water resources of Turkey (rivers, seas, lakes)” which are +the subsections of this chapter. Sample competency questions from these subsections can be +listed as: “Lütfen, Türkiye'deki şehirleri listeler misiniz? (Please list the cities in Turkey?)”, +“İzmir’in komşularını gösterir misin? (Which provinces border Izmir?)”, “Akdeniz bölgesinde +bulunan dağları gösterir misin? (List the mountains in the Mediterranean region?)”, “Manisa +şehrinin çevresinde hangi şehirler konumlanır? (Which cities are located in the province of +Manisa?)”, “İzmir’ in en yüksek dağı hangisidir? (What is the highest mountain in Izmir?)”, +“Ege Bölgesi’ndeki nehirlerin uzunluklarını gösterir misin? (Can you tell the length of the rivers +in the Aegean region?), “Türkiye’ de en fazla yağış alan il hangisidir? (Which city has the most +rainfall in Turkey?)”. +Step 2 was the consideration of reusing of existing ontologies. This was rejected for language- +related reasons. Instead, the decision was taken to develop an ontology from scratch. Step 3 +concerned the determination of competency questions. This involved determining the main +conceptualization, scope, and hierarchies in the ontology to avoid overlap between concepts. +Concepts, corresponding properties, and candidate relations were then defined. +Important concepts and related terms in the GEO-TR ontology are set out in Table 2. +Table 2. Main concepts and related terms +Important +Concepts +Related Terms +Ada (Island) +konumlanir (locatedIn), nufus (population), +Bogaz (Strait) +konumlanir (locatedIn), uzunluk (length) +Bolge (Region) +konumlanir (locatedIn), konumVar (hasLocations), nufus (population), +yuzolcumu (surface area) +Dag (Mountain) +konumlanir (locatedIn), yukseklik (height) +Deniz (Sea) +konumlanir (locatedIn), derinlik (depth), tuzluluk (salinity) +Gol (Lake) +konumlanir (locatedIn), derinlik (depth) +Nehir (River) +konumlanir (locatedIn), uzunluk (length) +Ova (Plain) +konumlanir (locatedIn), yuzolcumu (surface area) +Sehir (City) +konumlanir (locatedIn), konumVar (hasLocations), nufus (population), +yuzolcumu (surface area), yukseklik (height), ortalamaYagis (average +rainfall), komsu (neighbourOf) +Ilce +(District) +(subclass of Sehir) +konumlanir (locatedIn), nufus (population), yuzolcumu (surface area) +Ulke (Country) +konumlanir (locatedIn), konumVar (hasLocations), nufus (population), +yuzolcumu (surface area), iklim (climate), baskent (capital) + + +After determining the main conceptualization and relationships, Step 4 involved defining +classes and applying a corresponding hierarchy. Within GEO-TR, important terms are +represented as classes. Each of these constitutes a subclass of the “Thing” class that represents +the root node in the ontology. Ada (Island), Bogaz (Strait), Bolge (Region), Dag (Mountain), +Deniz (Sea), Gol (Lake), Nehir (River), Ova (Plain), Sehir (City), Ilce (District) (and Ulke +(Country) were determined as such subclasses. The only defined hierarchy in the ontology is +between the subclasses Sehir (City) and Ilce (District). + +The class list and hierarchy are shown in Figures 1 and 2. + + + +9 + + + + +Figure 1. Class List in GEO-TR – OntoGraf view Protégé (Gennari et al. 2003). + + +Figure 2. The class hierarchy between Sehir (City) and Ilce (District) – (OntoGraf view +Protégé) + +Step 5 of the ontology development process centred around determining the properties of +classes. Related terms in the main conceptualization were selected as possible properties in the +ontology. There are two types of properties in an ontology, namely data and object properties. +Data properties imply a data holding element, whereas object properties hold object-oriented +information. For example; a mountain has a height property. A river has a length property. The +sea class’s property is salinity. The city class has the properties locatedIn, neighbourOf, etc. +The object and data properties of each class are illustrated in Figures 3 and 4. There is a +symmetric relationship between konumlanir (locatedIn) and konumVar (hasLocations), which +implies these two properties should also be defined inversely. + + + + + +Figure 3. Object properties in GEO-TR + +ODag +OThing + Sehir +OGol +Bogaz +OAda ++ +OUike +Bolge +OOva +Nehir +DenizO Ada +Bogaz +Bolge + Dag +Deniz +Gol + Nehir +.ova += Sehir +Ilce +oulke目Sehir +Sehir -- has subclass --? Ice +OllceObject property hierarchy: +日日口卤 +X +topobjectProperty +komsu +- konumlanir +konumVar10 + + + +Figure 4. Data properties in GEO-TR + +Step 6 related to defining the facets of properties that refer to value type, allowed values, +cardinality (number of allowed values), and other value features a property may have. Property +facets are defined as the domain and range of a property in an ontology. In ontology +development terminology, the range is defined for data properties and the domain is defined for +object properties. For instance, the data property nufus (population) should have an integer type +that declares the range for this property. Another example, the konumlanir (locatedIn) property +is valid between specific pairs like Sehir (City) – Ulke (Country) and Sehir (City) – Bolge +(Region) etc. to represent the domain of this property, which defines “a city is located in a +country” or “a city is located in a region”. An additional example can be given for the object +property komsu (neighbourOf). Classes Sehir (City), Ulke (Country), and Bolge (Region) are +defined as possible domains to apply the komsu (neighbourOf) property in GEO-TR. + +The final step in the process involved creating instances in the structural environment of the +ontology. A sample list of instances for the classes Sehir and Bolge in GEO-TR is shown in +Figures 5 and 6. + + +Figure 5. Sample list of instances of Sehir + +Data property hierarchy: topDataProperty 日 +风 +...topDataProperty +baskent +derinlik +iklim +koordinat +nufus +ortYagis +tuzluluk +uzunluk +yukseklik +yuzolcumuThing +Ankara +Sehir +O llce +O Ulke +JzI +Antalya +→Bolu +Sanllurfa +nquersi +Bursa +Van +→Manisa11 + + +Figure 6. Sample list of instances of Bolge + + +Names of all classes, data, and object properties and instances are in Turkish. Thus, GEO-TR +is coherent with semantic web-enabled geographic question-answering in Turkish, which is +the principal novel contribution of this study. + +4. Methodology +A geographical question answering framework over linked data is represented in this study for +given natural language sentences in Turkish. The main components and corresponding sub- +components in the system architecture are illustrated in Figure 7. + +Figure 7. System architecture +Three main processes are configured in the system architecture, with the following layer +naming: question pre-processing, query formulation, and query execution. Question pre- +processing is the step in which questions are analysed morphologically and NLP techniques +applied by morphologically disambiguating the POS tags of each token. In addition, named +entities are recognized and dependencies between each word extracted. Each component in the +question pre-processing layer acts as a pipeline, finally generating a pre-processed form of the +natural language input, which is prepared to further semantically enrich the sentence. The +proposed framework is designed and implemented to answer two types of questions, namely +informative and quantitative reasoning involved questions. The question pre-processing layer +is applied to both two types before deciding on the type of question. Next, the query formulation + +Thing +OBolge ++Marmara ++ Akdeniz ++ DoguAnadolu ++ +IcAnadolu +T +→Karadeniz +Gune ydoguAna dol ++Ege1)QuestionProcessing +2)QueryFormulation +3)QueryExecution +Morphological +AnswerType +Analyzer +Detection +StructuredQuery +NLP +SPARQL +Morphological +Output +Classification +Query +Disambiguator +(WEKA) +Geo-TR +Named Entity +Relation Extraction +Ontology +Recognizer +Dependency Analysis +Answer +ITU NLPPIPELINE12 + +layer accepts the processed natural language input that is generated by the question pre- +processing layer. The type of question is determined and further corresponding processing tasks +are applied. A structured query in the form of SPARQL (query language of the ontology) is the +outcome produced at this stage. The generated SPARQL query is executed on GEO-TR to +return the answer to the query. +4.1 Question Pre-processing +Understanding user intention requires a combination of syntactic and semantic analysis of +expressions. Eliminating tokens that do not have any contribution to achieve meaning; +understanding relationships between tokens to get the focus of the sentence; tagging named +entities; and extracting possible relations between these entities and the focus are the first steps +in converting user intent in an unstructured form to a structured query language in the proposed +study. +Turkish is a complex language that is agglutinative, morphologically different, and has free +constituent order. 2 types of suffixes contribute significantly to meaning: constructive and +inflectional suffixes. By adding constructive suffixes to a word, it is possible to form completely +different new words semantically or words with a similar context. Inflectional suffixes are used +for properly placing a word into a sentence (Erguvanli and Taylan 1984). Combinations that +have a different meaning or proper usage for a given word are generated by placing the suffixes +at the end of a word. The morphological features of Turkish make this language more +challenging for pre-processing, necessitating a customized solution. +The Turkish NLP pipeline (Eryiğit 2014), developed and served as SaaS (software as a service) +by the NLP research group of Istanbul Technical University (ITU), is used for the question pre- +processing layer in Figure 7. Processing components, namely a “Morphological Analyzer”, +“Morphological Disambiguator”, “Named Entity Recognizer” and “Dependency Parser” are +utilized in this study, and spotter methods are implemented to convert the input format +appropriately for each component. For a sample question input, each processing step is +described in the following subsections. +4.1.2 Morphological Analysis +The morphological analysis step includes two main processing layers, namely determination +and disambiguation of POS tags in a question input. The morphological analyzer component of +the ITU Turkish NLP pipeline is the first to apply at this stage. The method, which combines +the word lemmata lexicon with over 49321 entries and flag diacritics for Turkish to handle +exceptions regarding phonetic and morphological rules, is presented for further processing to +disambiguate the POS tags of each token. In the disambiguation layer, affixes are removed +recursively without having an additional lexicon (named affix stripping) to find the accurate +POS tags. Details of the method are represented in (Sahin et al. 2013). The sample question +output of the morphological analysis is shown in Figure 8. Disambiguated output in Figure 8. +represents the morphological structure of each word that is ready for further processing to +understand their role in the sentence to achieve user intention. +4.1.3. Named Entity Recognition +The pipeline further processes disambiguated output with the named entity recognition method +to extract location information by resolving the mentions. Several NER techniques are applied +for different types of applications. The ITU Turkish NLP tool uses the methodology of the +Conditional Random Fields (CRF) technique for statistical modeling (Lafferty et al. 2001) of +predefined entity categories, such as person, location, organization, money, number, etc. Details + +13 + +of their methodology are described in their study (Şeker and Eryiğit 2012). Sample named entity +recognizer output is illustrated in Figure 9. +“B-LOCATION” is the first location identifier token in which “B” stands for the beginning of +the expression. 3 types of prefixes exist in NER output format. The first token of a named +entity is tagged by using the prefix “B” and continues with other tokens (if possible) that are +location identifiers “I-LOCATION”. “I” stands for in the location expression. The last type of +output prefix is “O”, which represents out of any named entity tagged. + + +Figure 8. Morphological analyzer output + +Figure 9. NER output + + + +Ankara iline komsu olan illeri gosterir misin ? +Ankara+Noun+Prop+A3sg+Pnon+Nom +il+Noun+A3sg+P2sg+Dat il+Noun+A3sg+P3sg+Dat +komsu+Adj komsu+Noun+NAdj+A3sg+Pnon+Nom +ol+Verb+Pos^DB+Adj+PresPart +ol+Verb+Pos^DB+Adj+PresPart^DB+Noun+Zero+A3sg+Pnon+Nom +il+Noun+A3pl+P3pl+Nom il+Noun+A3pl+Pnon+Acc +il+Noun+A3pl+P3sg+Nom il+Noun+A3sg+P3pl+Nom +goster+Verb+Pos+Aor+A3sg goster+Verb+Pos^DB+Adj+AorPart +mi+Postp+Ques+Pres+A2sg +?+Punc ++ +Ankara Ankara+Noun+Prop+A3sg+Pnon+Nom +iline il+Noun+A3sg+P3sg+Dat +komsu komsu+Adj +olan ol+Verb+Pos^DB+Adj+PresPart +illeri il+Noun+A3pl+Pnon+Acc +gosterirgoster+Verb+Pos+Aor+A3sg +misin mi+Postp+Ques+Pres+A2sg +??+PuncAnkaraAnkara+Noun+Prop+A3sg+Pnon+Nom +ilineil+Noun+A3sg+P3sg+Dat +komsu komsu+Adj +olan ol+Verb+Pos^DB+Adj+PresPart +illeriil+Noun+A3pl+Pnon+Acc +gosterir goster+Verb+Pos+Aor+A3sg +misin mi+Postp+Ques+Pres+A2sg +??+Punc ++ +AnkaraAnkara+Noun+Prop+A3sg+Pnon+Nom B-LOCATION +iline il+Noun+A3sg+P3sg+Dat O +komsukomsu+AdjO +olan ol+Verb+Pos^DB+Adj+PresPart O +illeri il+Noun+A3pl+Pnon+Acc O +gosterirgoster+Verb+Pos+Aor+A3sgO +misin mi+Postp+Ques+Pres+A2sg O +? ?+Punc O14 + + 4.1.4 Dependency Analysis +Dependency analysis comprises tagging relationships between words to understand the roles of +each token in the sentence and determining its various components, such as an object, subject, +verb, or other modifiers. Generating a dependency graph composed of dependency nodes +(tokens) and relationships is the underlying methodology for most dependency analysis +algorithms (Nivre et al. 2010). The Conference on Computational Natural Language Learning +(CoNLL-X) (Buchholz and Marsi 2006) input format and tags of the Turkish Dependency +TreeBank (subject, object, modifier, classifier, possessor, etc.) are used in the dependency +analysis tool of the ITU Turkish NLP pipeline (Eryiğit 2014; Eryiğit et al. 2008). The tenth +CoNLL (CoNLL-X) promoted a shared training file format for multilingual dependency parsing +models that has a standardized structured, column-based form. The dependency analysis result +of the sample input sentence is demonstrated in Table 3. +Table 3. Dependency analysis output of the sample sentence +ID +FORM +LEMMA +CPOS +TAG +POSTAG +FEATS +HEAD +DEPREL +PHE +AD +PDEPREL +1 +Ankara +Ankara +Noun +Noun +Prop|A3sg|Pnon|No +m +_ +2 +_ +POSSESSOR +2 +iline +il +Noun +Noun +A3sg|P3sg|Dat +_ +4 +_ +MODIFIER +3 +komşu +komşu +Adj +Adj +_ +_ +4 +_ +MODIFIER +4 +olan +ol +Verb +Verb +Pos^DB|Adj|PresPar +t +_ +6 +_ +MODIFIER +5 +illeri +il +Noun +Noun +A3pl|Pnon|Acc +_ +6 +_ +OBJECT +6 +gösterir +göster +Verb +Verb +Pos|Aor|A3sg +_ +7 +_ +ARGUMENT +7 +misin +mi +Postp +Postp +Ques|Pres|A2sg +_ +0 +_ +PREDICATE +8 +? +? +Punc +Punc +_ +_ +7 +_ +PUNCTUATION + +Id is the counter to represent a token number. Form is the original token that is in the form of +an original word or punctuation symbol. Lemma is the stem of a word or the same as a form if +that token is a punctuation symbol. Cpostag stands for coarse-grained and Postag is the fine- +grained POS tag definition from a specific treebank. Feats is the set of syntactic and +morphological structure definitions, separated by “|” symbol or underscore if not available. +Head and Phead values are eliminated in CoNLL-X format (Nivre et al. 2007). Deprel +represents the related token and Pdeprel is the type of relationship, or in other words, type of +dependency. + +4.2 Query Formulation +The initial step of query formulation is answering type detection, entailing discovering the +mention and user intention that is critically helpful in deciding answer type. Answer type +classification is performed on 2 main types of questions that hold quantitative reasoning: +question type 1 (QT1), or not, question type 2 (QT2). A rule-based approach detects quantitative +reasoning required expressions such as “kaç tane/kaç” (how many), “ne kadar” (how many) or +“en (superlative expression in Turkish - Adverb)” and further checks for the bigram of the +words to detect “Adjective + Noun”, “Adverb + Adjective” or “Adverb + Noun” patterns. The +main flow for query formulation is shown in Figure 10. +For a given natural language input, if isQuantitative returns true, the question is determined as +QT2 and query components are classified to fulfil the items in SPARQL query patterns. Target +class, entity class, data property, object property, and function name are represented as +categorical variables in the training model and these parameters are all matched with semantic + +15 + +Yes +items in GEO-TR ontology. A multilayer perceptron, which is an artificial neural network, is +used to generate a learning model. The attributes and categories of the training model are shown +in Table 4. + + + + + + + + + + +Figure 10. Main Flow of Query Formulation + + +Table 4. Structure of training model +Attribute Name +Categories +target_class +{Sehir,Bolge,Ulke,Dag,Nehir,Gol,Ada,Ova,Deniz, +Ilce,null} +entity_class +{Sehir,Bolge,Ulke,Dag,Nehir,Gol,Ada,Ova, +Deniz, Ilce,null} +data_property +{yuzolcumu, populasyon, yukseklik, derinlik, +tuzluluk, ortYagis, sicaklik, enlemBoylam, +bitkiOrtusu, baskent, iklim, null} +object_property +{konumlanir,konumVar,komsu,null} +function_name +{count,min,max,sum,null} + +Considering the structure of the training model in Table 4, an instance sentence “Türkiye'nin +en derin denizi hangisidir? (Which sea is the deepest in Turkey?)” is modelled as target_class += Deniz (Sea), entity_class = Ulke (Country), data_property = derinlik (depth), object_property += konumlanir (located in) and function_name = max (maximum as aggregate function). A +SPARQL query is formulated by using classified components with corresponding query +patterns. 2 types of SPAQL patterns are designed for this study. The type of aggregate function +specifies the type of query pattern by using a sub query-based approach (Type 1) or not (Type +2). For the functions min and max, subquery formation is inevitable due to the nature of +SPARQL queries, whereas count and sum functions do not require it (Table 5). + + + +Start + +Use NLP output to formulate query +(Figure 12) + + +Answer requires +quantitative +reasoning or not +(Figure 11) +Use learning model to generate +query components + + +Formulate SPARQL query by +using classified components + + +No + +16 + +Table 5. Types of Query Pattern +Query Pattern: Type 1 +Query Pattern: Type 2 +SELECT ?y ?m +WHERE { ?y rdf:type +ontology_name_prefix:target-class . +?y property_prefix:data-property ?m . +{ SELECT (function_name(?var) as ?m) +WHERE { ?x rdf:type +ontology_name_prefix:entity-class . +?y rdf:type ontology_name_prefix:target-class +. +?y property_prefix:object-property ?x . +?y property_prefix:data-property ?var +FILTER(regex(str(?x),"named entity","i")) } +}} + +SELECT (function_name (?y) as ?total) +WHERE { ?x rdf:type ontology_name_prefix: +entity-class. +?y rdf:type ontology_name_prefix: target- +class. +?y property_prefix: data-property ?x +FILTER(regex(str(?x),"named entity","i")) } +If any quantitative expression is not held in the natural language input, NLP output is employed +in a manner based on ontology validation to formulate a SPARQL query. The algorithm +deciding the answer type is indicated in Figure 11. + +Figure 11. Answer Type Detection + +In the case of a sentence that does not hold any quantitative reasoning expression, NLP output +is combined with semantic web technologies to convert natural language input into a structured +query form. The algorithm designed for the conversion is mainly based on dependency analysis, + +No +Yes +No +Yes +No +Yes17 + +NER output, and ontology-based validation. In the Turkish language, the target intent of the +user is generally located in the object or subject of a sentence, or any other connected token +with them. Following that assumption, which is based on the rules of Turkish grammar, the +algorithm was designed to combine NLP output with ontology capabilities to improve accuracy +while understanding user intent. The algorithm is represented and described with examples +below. +Algorithm Algorithm to find the answer type of question in Turkish and generate +SPARQL query by using processed output by NLP techniques (Method name: generateSparql) +Require: sentence processed by NLP techniques +agenda: generate query by using NLP output +Ensure: final_query +1: nerEntities = pipeline.getNamedEntities(nerOutput) +2: if dependencyOutput.contains(“OBJECT”) +3: +answerType = objectTerm +4: +axiomType = pipeline.checkAxiomType(answerType) +5: +if axiomType == “CLASS” then +6: + +properties = pipeline.findProperties(answerType, nerEntities) +7: + +final_query = pipeline.formulate_query(properties, nerEntities, answerType) +8: end if +9: if axiomType == “DATA PROPERTY” then +10: + +relatedToken = pipeline.findRelatedToken(answerType, dependencyOutput) +11: + +axiomTypeRelated = pipeline.checkAxiomType(relatedToken) +12: + Go back to Step 5 call the method for the input axiomTypeRelated and +continue again +13: +end if +14: +if axiomType == “OBJECT PROPERTY” then +15: + +relatedClass = pipeline.findRelatedToken(answerType,dependencyOutput) +16: + +final_query = pipeline.formulate_query(answerType, nerEntities, relatedClass) +17: +end if +19: if dependencyOutput.contains(“SUBJECT”) then +20: +answerType = subjectTerm +21: +axiomType = pipeline.checkAxiomType(answerType) +22: +if axiomType == “CLASS” then +23: + +properties = pipeline.findProperties(answerType, nerEntities) +24: + + if properties == NONE then +25: +commonConnected = pipeline.findCommonConnected +(dependencyOutput, answerType) +26: + + +Go to Step 21 call the method for the input commonConnected and +continue again +27: + +end if +28: +end if +29: +else +30: + +final_query = pipeline.formulate_query(properties, nerEntities, answerType) +31: +if axiomType == “DATA PROPERTY” then +32: + +relatedToken = pipeline.findRelatedToken(answerType, dependencyOutput) +33: + +axiomTypeRelated = pipeline.checkAxiomType(relatedToken) +34: + Go back to Step 21 call the method for the input axiomTypeRelated and +continue again +35: +end if +36: + if axiomType == “INDIVIDUAL” then + +18 + +37: + connectedToken = pipeline.findConnectedToken (answerType, +dependencyOutput) +38: + +axiomTypeConnected = pipeline. checkAxiomType(connectedToken) +39: + +Go back to Step 21 call the method for the input axiomTypeConnected +and cont. again +40: +end if +41: +if axiomType == “OBJECT PROPERTY” then +42: + commonConnected = pipeline.findCommonConnected(dependencyOutput, +answerType) +43: + Go back to Step 21 call checkAxiomType for the input commonConnected and +continue +44: +end if +45: end if +The first processing step for the sample input question after applying NLP methods is deciding +on the type of dependency relationship the sentence holds. As illustrated in the dependency +analysis output of the sample sentence (Figure Table 3), token 5 (“il” (city)) is the object of +that sentence, and the axiom type of the object is checked to decide whether it is a class, a data +or object property, or an individual. City is a semantic item and represented as a class in the +ontology and determined as a target class for query formulation. For the condition that the object +of the sentence is a class (Algorithm – Step 5), the properties of that class with the named entity +(if it exists) in the sentence are found to generate a SPARQL query. “Ankara” is the named +entity for this sentence (Figure 9). The entity class for the individual “Ankara” is extracted +from the ontology as Sehir (entity class), which is the same class with the object token. Possible +relationships with “Ankara” and class Sehir are extracted from the ontology. The only +relationship is found to be an object property “komsu” (neighbourOf) for the sample case. The +generic query pattern for the SPARQL formulation is: +SELECT ?y +WHERE { ?x rdf:type ontology_name_prefix: entity-class. +?y rdf:type ontology_name_prefix: target-class. +?y property_prefix: nameOfproperty ?x +FILTER(regex(str(?x),"named entity","i")) }. +By using this pattern, the query is generated as follows: +SELECT ?y +WHERE { ?x rdf:type geo_turkce:Sehir . +?y rdf:type geo_turkce:Sehir . +?y ins:komsu ?x . +FILTER(regex(str(?x),"Ankara","i")) }. +Another sample question includes a subject phrase via a deep-thinking algorithm. For the +question “Ege Bölgesi’nin yüzölçümü ne kadardır? (How much is the total area of the Aegean +region?)”, the dependency analysis output is given in Table 6 and the NER result is: + +Ege ege+Noun+A3sg+Pnon+Nom B-LOCATION +Bölgesi'nin bölge+Noun+A3sg+P3sg+Gen I-LOCATION +yüzölçümü yüzölçüm+Noun+A3sg+P3sg+Nom O + +19 + +ne ne+Pron+Ques+A3sg+Pnon+Nom O +kadardır kadar+Postp+PCNom^DB+Noun+Zero+A3sg+Pnon+Nom^DB+Verb+Zero+Pres+A +3sg+Cop O +? ?+Punc O. +Table 6. Dependency Analysis Result of Sample Sentence +ID +FORM +LEMMA +CPOS +TAG +POST +AG +FEATS +DEP +REL +PDEPREL +1 +Ege +ege +Noun +Noun +A3sg|Pnon|Nom +2 +POSSESSOR +2 +Bölgesi’nin +bölge +Noun +Noun +A3sg|P3sg|Gen +3 +POSSESSOR +3 +yüzölçümü +yüzölçüm +Noun +Noun +A3sg|P3sg|Nom +5 +SUBJECT +4 +ne +ne +Pron +Pron +Ques|A3sg|Pnon|No +m +5 +ARGUMENT +5 +kadardır +kadar +Postp +Postp +PCNom^DB|Noun|Ze +ro|A3sg|Pnon|Nom^ +DB|Verb|Zero|Pres| +A3sg|Cop +0 +PREDICATE +6 +? +? +Punc +Punc +_ +5 +PUNCTUATION + +Token 3 is the subject of the sentence and axiom type determined as a data property from GEO- +TR. This is an indicator that quantitative analysis is required to handle user intent. Possible +classes that are assigned with the aforementioned data property are detected in the sentence +(Algorithm – Step 31). From the NER result, “Ege Bölgesi” (Aegean Region) is the named +entity, and the entity class is extracted as “Bölge (Region)” from the ontology. Additionally, +the dependency parsing result shows that token 2 is directly related to the subject token, and the +axiom type of token 2 is a class in GEO-TR. At that point, the algorithm moves back to Step +22 to check for possible properties from the ontology but, for that sample case, the answer type +is a property so searching for the commonly connected token with the subject expression is the +second thing to do (Step 25). The commonly connected token is already found because of the +fact that there is no other entity for that case and the algorithm formulates the query as: +SELECT ?variable +WHERE { ?x rdf:type geo_turkce:Bolge . +?x ins:yuzolcumu ?variable . +FILTER(regex(str(?x),"Ege","i")) } +Final sample input: “Ege Bölgesi'ndeki şehirlerin nüfuslarını gösterir misin ? (Can you show +me the populations of the cities in the Aegean Region?)”, is more complex and holds a +determinative group expression for possessive construction (“zincirleme isim tamlaması”). The +dependency analysis output is shown in Table 7 and the NER result of the sentence is as follows: + +Ege ege+Noun+A3sg+Pnon+Nom B-LOCATION +Bölgesi'ndeki bölge+Noun+A3sg+P3sg+Loc^DB+Adj+Rel I-LOCATION +şehirlerin şehir+Noun+A3pl+Pnon+Gen O +nüfuslarını nüfus+Noun+A3pl+P3sg+Acc O + +20 + +gösterir göster+Verb+Pos+Aor+A3sg O +misin mi+Postp+Ques+Pres+A2sg O +? ?+Punc O + +Table 7. Dependency Analysis Output of Final Sample Sentence +ID +FORM +LEMMA +CPOS +TAG +POST +AG +FEATS +DEP +REL +PDEPREL +1 +Ege +ege +Noun +Noun +A3sg|Pnon|Nom +2 +POSSESSOR +2 +Bölgesi’nd +eki +bölge +Noun +Noun +A3sg|P3sg|Loc^DB|A +dj|Rel +5 +MODIFIER +3 +şehirlerin +şehir +Noun +Noun +A3pl|Pnon|Gen +4 +POSSESSOR +4 +nüfuslarını +nüfus +Noun +Noun +A3pl|P3sg|Acc +5 +OBJECT +5 +gösterir +göster +Verb +Verb +Pos|Aor|A3sg +6 +ARGUMENT +6 +misin +mi +Postp +Postp +Ques|Pres|A2sg +0 +PREDICATE +7 +? +? +Punc +Punc +_ +6 +PUNCTUATION + +After determining token 4 (“nüfuslarını” (population)) as the object of the sample sentence, the +algorithm firstly checks the axiom type for the stemmed form of the token or synonyms (nüfus +(population)). The axiom type is determined as a data property and the algorithm continues with +Step 10 by using the findRelatedToken method to find the directly dependent token with the +token 3 object (“şehirlerin”). The dependency output is critical for specifically this type of +question in order to understand user intent to find the population of cities located in the Aegean +region rather than displaying the population of the Aegean region. Token 3 “şehir” (city) is +checked for the axiom type from the ontology and the result returns as the class that moves the +algorithm to Step 6 by assigning the answerType to the related token (“şehir”). Properties +defined between “Ege Bölgesi” and “şehir” are extracted from the ontology and only one object +property returns, namely “konumVar (hasLocation)”. The named entity, entity class, target +class, data, and object properties are assigned to formulate the query as follows: +SELECT ?variable +WHERE { ?x rdf:type geo_turkce:Sehir . +?y rdf:type geo_turkce:Bolge . +?y ins:konumVar ?x . +?x ins:populasyon ?variable . +FILTER(regex(str(?y),"Ege","i")) + + + +21 + + +Figure 12. Flowchart of Algorithm + + + + + + +Start +Output of +Outputof +No +dependency +No +dependency +Determinethat +final query cannot +analysis holds +analysis holds +be formulated +“"SUBJECT" +“OBJECT" relation +relation or not +or not +Yes +Yes +Axiom type of the +Axiom typeof +/ Axiom type of +Axiom type of the +No +No +No +No +current (subject) +the current +the current +current (object) +token is data +(subject) token +(object) token + token is data +propertyor not +is class or not +is class or not +propertyornot +Yes +Yes +Yes +Yes +Check if properties exist between +Find properties between +Find related token with the +Find related token with +named entity and current token +named entity and current token +current token +the current token +Find axiom type of the +Formulate SPARQL query by +Find axiom type of the related +related token +Propertiesare +using current token, named +Yes +token +found between +entity and properties +named entity +and current +token +No +No +Axiom type of the +Determine answer +current (object) +Find common +type is not hold in +token is object +connected token with +object token and +current token +propertyornot +query cannot be +generated +Yes +Find axiom type of the +Find related token with the +common connected +current token +token +Axiom type of the +Formulate SPARQL query by +current (subject) +Yes +using named entity, related +token is +token and current token +Find connected +individual or not +token with the +current token +Find axiom type of the +common connected token +No +Find axiom type of +Find common connected +the connected +token with the current token +token +Axiom type of the +Determine answer +current (subject) +No +type is not hold in +subject token and +token is object + query cannot be +propertyornot +generated22 + +4.3 Experimental study + +The experimental study was performed on two types of questions (QT1 and QT2: See Section +4.2). A comparison was generated to give the results for QT1 in order to understand more deeply +the contribution of the NLP output while interpreting the user input. Our main contribution is +to show the results of combining the NLP output with semantic technologies so as to build up +a SPARQL query by discovering accurate entities and relationships. Matching the entities and +relationships in the NLP and ontology by double-checking the tokens, and dependencies +between them, is the main focus of the study. Instead of only checking for each token, their +types, and possible relationships between them in the ontology, the hybrid method +disambiguates the possible relationships by applying NLP output. A basic user interface, as +illustrated in Figure 13, was implemented for an experimental study of 100 test questions. + + +Figure 13. User Interface for Experimental Study + + +4.4 Results and Discussion + +Comparison paradigms consist of using a hybrid approach (NLP + ontology-based approach) +and only applying an ontology-based approach. Comparison metrics given in Table 8 are +precision, recall, and F-measure. +Table 8. Results of Comparison +Method +Precision +Recall +F-Measure +Method 1: Hybrid +approach +0.77 +0.68 +0.71 +Method 2: Ontology +based approach +0.64 +0.57 +0.60 + + +图 +Question Answering over GEO-TR +口 +X +Type your question here Turkiye'nin en kalabalik ili hangisidir? +See the answer! +Clear all for the new question! +Morphological Analyzer: +Named Entity Recognition: +Turkiye+Noun+Prop+A3sg+P2sg+Nom +≤DOC> DOC>+BDTag-S>S>+BSTag +0 +Turkiye'nin Turkiye+Noun+Prop+A3s g+Pnon+Gen +en+Adverb +en en+Adverb +0 +en+Noun+A3s g+Pnon+Nom +SPARQL Query: +kalabalik kalabalik+Noun+A3s g+Pnon+Nom +0 +ili il+Noun+A3sg+P3sg+Nom +0 +SELECT ?x ?var +kalabalik+Noun+A3sg+Pnon+Nom +hangisidir? hangisidir?+? +0 +WHERE (?x rdf.type geo_turkce:Sehir. +kalaba+Noun+A3s g+Pnon+Nom*DB+Adj+Fiffor +≤/S> ≤/S>+ ESTagDOC> DOC>+EDTag +?x ins:populasyon ?var . +[SELECT (MAX(?val) as ?var) +il+Noun+A3sg+Pnon+Acc +WHERE { ?x rdf.type geo_ turkce:Sehir. +i+Noun+A3sg+P3sg+Nom +?y rdf.type geo_turkce:Ulke. +?x ins:konumlanir ?y . +hangisidir?+? +?x ins:populasyon ?val. +FILTER(regex(str(?y),"Turkiye",'"i") )) +Morphological Disambiguator: +Dependency Analysis: +ES>S>+BSTag +Turkiye'nin +Turkiye +Noun +Noun +Turkiye'nin Turkiye+Noun+Prop+A3s g+Pnon+Gen +2 +en +en +Adverb +Adverb +en en+Adverb +3 +kalabalik +kalabalik +Noun +Noun +kalabalik kalabalik+Noun+A3sg+Pnon+Nom +4 +ili +Noun +Noun +ili il+Noun+A3sg+P3sg+Nom +5 +hangisidir? +hangisidir? +? +? +hangisidir? hangisidir?+? +/S> /S>+ ESTag +123 + +Results indicate that using morphologically analyzed word forms and their dependencies with +semantic items in GEO-TR contributes to improving the accuracy of the framework. Questions +with possessive constructions are appropriate examples of how NLP output directly contributes +to the meaning by applying dependency analysis to decide on target answer type. Relationships +between words that might be critical to ascertaining the answer type are missed by Method 2. +For instance, for a question such as “Ege Bölgesi'ndeki şehirlerin nüfuslarını gösterir misin? +(Can you show the populations of cities in Aegean Region?)”, which has a possessive +construction, the ontology-based approach failed to disambiguate whether the population of the +Aegean Region (individual in GEO-TR) was intended, or populations of cities that are located +in the Aegean Region separately. The relationship “POSSESSOR” between the tokens +“şehirlerin(cities)” and “nüfuslarını (populations)” disambiguates the user intent as the +population of the cities are asked for. The only advantage of Method 2 relates to capturing +entities, labels that are not recognized by the named entity recognition process. Considering +overall performance, this advantage is not sufficient to compete with the hybrid method. The +ontology-based method simply checks for each, whether it exists in the ontology or not. If a +token is found in the ontology, possible relationships and types of axiom are checked. By using +the same SPARQL patterns with Method 1, fulfilling the items with corresponding tokens is +performed. +Another drawback of Method 2 is that it is challenging to decide on the answer type for +sentences that involve more than one class item. A good example can be given as: “İzmir şehri +hangi bölgededir?”. “Izmir” (individual), “Sehir” (class), and “Bolge” (class) are matched with +semantic items in GEO-TR after processing. Possible extracted relationships between +individuals and classes are “konumlanir (locatedIn)” (Izmir – Bolge) and “komsu +(neighbourOf)” (Izmir – Sehir). An incorrect intention can be produced by using the outcome +of Method 2 to show the neighboring cities of Izmir. Overall, the application of Method 1 results +in more accurate results for all facts compared to Method 2. +A learning model generated by the supervised learning method can be used to predict query +components. Predicted components are target class, entity class, data property, object property, +and aggregate function name to perform quantitative reasoning analysis for the given attributes. +Predicted components are used to formulate the SPARQL query. The accuracy of the learning +model that is built to handle QT2 is 0.72. The experiment is performed by using a train/test split +of 0.8/0.2. + +5. Conclusion +In order to fill a gap in the literature, a Turkish question answering framework over linked data +(GEO-TR) in the geographic domain is proposed in this study. Combining NLP techniques and +an ontology, two types of questions (QT1 and QT2) are handled in this framework. The main +conclusion is that a hybrid approach (Method 1) interprets a sentence in natural language more +accurately than an ontology-based approach (Method 2). Another significant contribution of +this study is the creation of a novel Turkish ontology in the geographical domain, which has +been developed by following the rules of ontology development 101(Noy and McGuinness +2001). + +GEO-TR is extendable and ready to use as a data source for other researchers. Following the +question pre-processing and query formulation, the experimental study demonstrates the main +contribution of hybrid architecture, and results are given by using precision, recall, and F- +measure values. + + +24 + +Currently, this study is not capable of handling more complex queries with more than one +recognized entity, more than one level possessive constructions, or conditional and comparative +expressions. Future work is suggested to apply deep learning techniques to handle complexity +in question forms. Moreover, extending the coverage and creating a multi-ontology platform +could be an additional direction of future study. It is proposed that a pipeline that accepts natural +language input and classifies the sentence according to the domain types and matches with a +corresponding ontology can fit with this architecture. + +Declarations: +Funding This research received no external funding. +Conflicts of interest/Competing interests The authors declare no conflict of interest, financial +or otherwise. +Availability of data and material Not applicable +Code availability Not applicable +Acknowledgments Declared none. +References +Baudiš, P., & Šedivý, J. Modeling of the question answering task in the yodaqa system. In +International Conference of the Cross-Language Evaluation Forum for European Languages, +2015 (pp. 222-228): Springer +Berners-Lee, T., Hendler, J., & Lassila, O. (2001). The Semantic Web - A new form of Web content that +is meaningful to computers will unleash a revolution of new possibilities. 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In CCF International Conference on Natural Language Processing and Chinese +Computing, 2014 (pp. 333-344): Springer + diff --git a/ZNE3T4oBgHgl3EQf2Avb/content/tmp_files/load_file.txt b/ZNE3T4oBgHgl3EQf2Avb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..171631cb35f5a86a6b5a91f32a60d8cc6950cb47 --- /dev/null +++ b/ZNE3T4oBgHgl3EQf2Avb/content/tmp_files/load_file.txt @@ -0,0 +1,1090 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf,len=1089 +page_content='1 Semantic Web Enabled Geographic Question Answering Framework: GeoTR Ceren Ocal Tasar1, Murat Komesli2,*, Murat Osman Unalir3, 1 Yaşar University, Department of Computer Engineering, Universite Cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 35, 35100, Bornova, Izmir, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=', Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' E-mail: ceren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='ocal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ORCID: 0000-0002-0652- 7386 2 Yaşar University, Department of Management Information Systems, Universite Cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 35, 35100, Bornova, Izmir, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' E-mail: murat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='komesli@yasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ORCID: 0000-0002- 8240-5540 3 Ege University, Department of Computer Engineering, Universite Cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 9, 35100, Bornova, Izmir, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' E-mail: murat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='osman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='unalir@ege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ORCID: 0000-0003-4531-0566 *Corresponding Author: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Murat Komesli, Yaşar University, Department of Management Information Systems, Universite Cad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 35, 35100, Bornova, Izmir, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' E- mail: murat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='komesli@yasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Tel (Office): +902325707817;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Mobile: +905055718202 2 Semantic Web Enabled Geographic Question Answering Framework: GeoTR Abstract With the considerable growth of linked data, researchers have focused on how to increase the availability of semantic web technologies to provide practical usages for real-life systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Question answering systems are an example of real-life systems that communicate directly with end-users, understand user intention and generate answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' End users do not care about the structural query language or the vocabulary of the knowledge base where the point of a problem arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In this study, a question-answering framework that converts Turkish natural language input into SPARQL queries in the geographical domain is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Additionally, a novel Turkish ontology, which covers a 10th-grade geography lesson named “Spatial Synthesis: Turkey”, has been developed to be used as a linked data provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Moreover, a gap in the literature on Turkish question answering systems, which utilizes linked data in the geographical domain, is addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A hybrid system architecture that combines natural language processing techniques with linked data technologies to generate answers is also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Further related research areas are suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Key Words: question answering systems, linked data, ontology development, natural language processing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Introduction As the importance of technology and the amount of stored data increases, fast and easy access to information for end-users has become a never-ending challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' One of these challenges is that, as natural language is the most common way for human beings to communicate as end- users, user intention in the form of natural language must be transformed into a structural format and represented with machine-understandable notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Tokenizing natural language input, analysis of each token, converting it into a machine-understandable representation, and generating answers or outcomes based on user intention are the major elements focused on by researchers in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Natural language processing (NLP), one of the research fields within artificial intelligence, can be used to solve some of these aforementioned issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Processes that perform the tasks of tokenizing natural language input and carrying out linguistic analysis on each token and their components are the main phases of NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' However, despite the deep analysis outcomes of NLP, there are some weaknesses in enriching input semantically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Therefore, semantic technologies are required to analyse the outcomes of NLP and to focus on supporting techniques and methodologies to achieve an upper-level understanding of expressions (Berners-Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The technology behind semantic analysis consists of ontologies that are composed of relationships and taxonomies to represent a specific conceptualization in a specific domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontologies play an essential role in sharing and exchanging knowledge between different platforms of different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' An ontology provides flexibility and interoperability opportunities, which has resulted in a significant increase in the number of studies of the ontology concept, OWL or RDF representation, and ontology query languages like RDQL and SPARQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Advances in NLP techniques have also developed on account of improvements in semantic technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Recently, a new discussion has arisen about how to understand natural language input and generate appropriate answers with hybrid solutions, using a combination of NLP techniques and semantic technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Researchers have attempted to improve NLP outcomes 3 by applying semantic enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Another important discussion concerns how NLP contributes to ontology learning, ontology querying, and multilingual ontology mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Hybrid solutions involve a reciprocal relationship between NLP and semantic technologies (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Understanding a sentence involves two main phases: morphological analysis and semantic analysis (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Morphological analysis focuses on determining the structure of words, the relationships between words, POS (part of speech) tags (noun, verb, adjective, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ), and their positions in the sentence (subject, object, predicate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named entity recognition (NER) is one of the NLP techniques for semantic enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Entities in an expression are tagged by using predefined categories, such as person, organization, date, location, number, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' in the NER model (Collobert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The generated outcome is combined with conceptualization and linked data defined in ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The hybrid approach provides an opportunity to use ontologies as data sources for question answering systems that have a natural language input format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A knowledge requirement denoted in natural language is transformed into ontology query language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Without dealing with additional technical details relating to the ontology, such as query language, vocabulary or hierarchical structures, even end-users can utilize the knowledge presented in ontologies (Bernstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' This approach bridges the gap between end-users and semantic technologies and also provides practical implications for ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' This study firstly outlines the Question Answering Framework (QAF) over linked data, presents a detailed comparison between them, and proposes a Turkish geographical question answering framework that combines NLP techniques and semantic web technologies over linked data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The current section introduces the problem, concepts, motivation, and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The following section includes background information and related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Section 3 defines the natural language-aware ontology development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Following this, section 4 describes the proposed framework and developed geographical ontology by dividing it into 3 subsections: question pre-processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' query formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and experimental study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Conclusions and future work are set out in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The contributions of this study to the literature are also discussed in the final section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Background Information and Related Work 12 question answering systems or frameworks over linked data with cutting edge approaches are examined to give an overview of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Habernal and Konopík introduced a Semantic Web Search Using Natural Language (SWSNL) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' SWSNL employs ontologies in order both to store data and domain structure, and to return answers for users’ search queries in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The pipeline described for the system holds the phases for pre-processing, semantic analysis, semantic interpretation, and executing SPARQL to generate answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The authors claim there are significant differences between their approach and other similar systems in that users can formulate natural language queries in more than one sentence, and stochastic semantic analysis model pre-processing is handled without requiring any syntactic parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' SWSNL has been tested for three domains in different languages: accommodation in English, public transportation in Czech, and a subset of the English ATIS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Serving different domains and languages addressed the portability feature of SWSNL in their study (Habernal and Konopik 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Xser is a question answering system that generates answers over DBpedia by transforming natural language questions into linked data query representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Unstructured natural language input is converted into structured SPARQL queries to be executed on the DBpedia endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The main motivation is to understand user intention accurately and to map user intention and 4 ontology conceptualization, which form the model for concepts and their interrelationships in an ontology (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Intui3 is proposed as a multilingual question answering platform over linked data that applies both syntactic and semantic analysis of natural language input while formulating RDF triples to represent user queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dima addresses the problems in traditional keyword-based searches – that results are given only as a ranked list of documents rather than a precise answer - and proposes a solution by offering a new alternative search paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The Intui3 search paradigm is based on a predicate and entity index and descriptions of all entities and predicates are sourced from DBpedia (Dima 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In the study of (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2014), a question answering system over linked data, namely CASIA@V2, is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A Markov Logic Network, a joint learning model, is used to detect and classify phrases and perform mapping of phrases with semantic entities (Richardson and Domingos 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In the first step, the input question is decomposed to detect phrases, followed by mapping the selected candidate phrases and concepts in DBpedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Decomposed phrases that contribute to user intention are grouped to generate triples which are actually the components of SPARQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2015) propose ISOFT as a SPARQL template generator for natural language questions that combines two notable fields: knowledge-based question answering and information retrieval-based question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Semantic similarity is positioned as the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The natural language input query is parsed into constituent phrases to be matched with SPARQL templates by imposing semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Phrases interpreted as a contributor to answer generation are extracted to be mapped with uniform resource identifiers (URIs) that represent concepts in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The study of (Paredes-Valverde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2015) proposes an ontology-based information retrieval system called ONLI (Ontology-based Natural Language Interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The structure and context of a natural language question are represented with an ontology model in DBpedia to address user intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Questions are classified regarding their context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The probability of returning accurate answers is improved by classifying the NL questions to reduce search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The ONLI pipeline has 3 main modules: question processing, knowledge bases, and answer searching and building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The question processing module deals with the pre-processing of natural language input syntactically to further improve the results of semantic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Identifying semantically similar terms or finding synonyms enables semantic enrichment of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' An ontological model, namely the Question Model, is used to organize answer searching and building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Applied search methods vary in different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ONLI provides possible answers with relevance ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' QAnswer is a question answering framework developed by (Ruseti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2015) to convert natural language input questions into a SPARQL query to be executed on a DBpedia endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The first task in their pipeline is to detect the DBpedia entities and the type of dependency relationships between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The outcome of the dependency-parsing phase is a directed graph with vertices and edges, where vertices represent annotated tokens and edges hold collapsed dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' After addressing the dependencies, entity type detection is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' After deciding on the various types and dependencies, candidate matches are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In other words, multiple different directed candidate graphs that represent possible question patterns are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Finally, a scoring algorithm is applied to identify the graph with the highest score to generate SPARQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 5 Baudis and Sedivy propose a question answering system pipeline called YodaQA that generates answers sourced from DBpedia and Freebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Fundamental processes included in the pipeline can be listed as question analysis, answer production, answer analysis, answer merging and scoring, and successive refining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' They formulate each question by naming three dimensions, “Clues”, “Focus” and “LAT (Lexical Answer Type)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Clues represent keywords or expressions that might play essential roles in user intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Focus is the target object to be queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' LAT is derived from focus and stands for answer type description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Semantic endpoints are searched for each clue in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Returned answers are analysed to determine LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Answers are classified with regard to their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A scoring algorithm is applied for further processing with successive refining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The answer with the highest score is selected as the outcome of the pipeline (Baudiš and Šedivý 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Mazzeo and Zaniolo put forward a question answering framework named CANaLI (Context- Aware controlled Natural Language Interface) to return results for controlled natural language (CNL) questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' CNL refers to restricting the grammar to make the language more easily machine-interpretable and creating a formal structure to the input questions while still protecting the natural format of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' CANaLI suggests auto-generated question patterns to end-users typing questions, thereby correcting them semantically, syntactically, and grammatically in accordance with the concepts in the underlying knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' DBpedia, MusicBrainz, DrugBank, Diseasome, and SIDER are the domain knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Encyclopedic, music, and medicine are employed as data sources (Mazzeo and Zaniolo 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' AMAL (Ask Me in Any Language) is a solution that allows users to ask questions in French that are then automatically converted into SPARQL queries to find answers found in DBpedia (Radoev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' AMAL has four main phases in generating an answer: question classification, entity extraction, property identification, and SPARQL query building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For now, it only focuses on simple questions that hold one single entity and a single property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' However, the authors define AMAL as a still-evolving system that has future capacity for complex questions which hold more than one entity-property match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Diefenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2018) proposes a question answering component entitled WDAqua-core0 that finds answers to natural language queries and keyword queries over DBpedia and Wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' DBpedia provides language support only for English, whereas Wikidata extends the range to French, German and Italian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' WDAqua-core0 serves the research community through the Qanary Ecosystem (Diefenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2017b), which is a framework to integrate question- answering components for reusability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The authors apply a combinatorial approach based on the semantic items represented in linked data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Two triple patterns, SELECT and ASK, are handled by WDAqua-core0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Only using the COUNT operator and not being able to handle questions containing comparative and superlative expressions are expressed as limitations of their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A method called SPARQLtoUser is described in another study by (Diefenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' It focuses on generating a user understandable version of SPARQL by converting user intention to a representation of a SPARQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The noteworthy features of SPARQLtoUser are claimed to be its multilingualism and portability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Users can generate answers to questions in different languages from different knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Instead of verbalizing the query, a generic approach to building up a user understandable schematic representation of the query is proposed in their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The above-mentioned studies and this study are compared according to year, domain, language, learning appliance, and employed methods in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Comparison of this study (Geo-TR) and literature studies System Year Domain Language Learning Applied or Not Methods SWSNL 2013 Multidomain (tested on accommodation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and DBpedia) Multilingual (tested in Czech and English) Yes POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lemmatizing Xser 2014 Multidomain (tested on Freebase and DBpedia) Multilingual (tested in English) Yes POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition Intui3 2014 Multidomain (tested on DBpedia) English No POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lemmatizing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Chunking CASIA@V2 2014 Multidomain (tested on DBpedia) English Yes POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lexicon Model ISOFT 2015 Multidomain English No Natural language input query,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Imposing semantic similarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Matching with SPARQL templates ONLI 2015 Multidomain (tested on DBpedia) Multilingual (tested in English) No POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lemmatizing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Synonym Extension QAnswer 2015 Multidomain (tested on DBpedia) English No Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Generate SPARQL YodaQA 2015 Multidomain (tested on Freebase and DBpedia) English Yes POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lexicon Model CANaLI 2016 Multidomain (tested on DBpedia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' music,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and medicine) English No Controlled Natural Language Grammar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontology-Driven Auto-Completion Amal 2017 Multidomain (tested on DBpedia) French Yes Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontology-Driven Property Extraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lexicon Model WDAqua-core0 2017 Multidomain Multilingual (tested in English,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' French,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' No Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontology Development (Natural Language Aware) The term “ontology” comes from the combination of the ancient Greek words “ontos”, which means “being”, and “logos”, which means “word”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In philosophy, ontology is defined as the subject of existence by (Gaševic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2006), and existence categorization for a specific domain by (Sowa 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontologies are applied to represent knowledge by defining a set of concepts and relationships in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' By modelling a specific domain with concepts, including their properties, and relationships, an ontology represents a structural shared vocabulary among multiple systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Underpinning the development of an ontology is the requirement that it allows for the sharing of domain knowledge to provide semantic interoperability and facilitates the reusing of ontological concepts in different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The most critical factor during its development is understanding the set of questions and related answers that an ontology must be able to answer correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' These are competency questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Uschold and Gruninger 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Grüninger and Fox 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' User stories, possible use case scenarios, and normal and alternative processing steps are helpful to determine such competency questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Compared with other steps involved in ontology development, dealing with competency questions is essential, not only to define the context of the ontology but also to underpin the architecture behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Information contexts that cover competency questions provide clues about how to encode semantic items in the ontology and any application platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Additionally, competency questions are dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For long-term sustainability purposes, they must be updated by extending the context of the ontology (Kendall and McGuinness 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Competency questions help an ontology to be natural language-aware, which fits well with question answering systems that take input in the form of natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In this study, a natural language-aware geographical ontology was developed by applying the steps defined in Ontology Development 101 (Noy and McGuinness 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Instead of using an existing ontology encoded in a language other than Turkish and having to deal with the burden of translation, an ontology (named GEO-TR) was developed in respect of the geography of Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' All data and object properties, individuals, and class names are in Turkish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Additionally, the development of a Turkish ontology in the geographical domain contributes to the field and addresses a gap in the current literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The first step in ontology development 101 is determining the domain and scope by defining competency questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The questions: “Which type of domain will the ontology cover?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=', “For what the ontology will be used?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and “What types of questions will be answered by the ontology?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' should be answered during the first step of the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' While designing the GEO-TR ontology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Chapter 6: “Spatial Synthesis of Turkey” from the 10th grade geography (tested on DBpedia and Wikidata) German and Italian) Ontology-Driven Property Extraction SPARQLtoUser 2018 Multidomain (tested in English) Multilingual No The extended version of WDAqua-core0 with user interaction OUR STUDY (Geo-TR) 2019 Multidomain (tested on Geographic domain) Multilingual (tested in Turkish) Yes POS Tagging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ontology-Driven Property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and Instance Validation 8 class from the national curriculum was chosen as the target scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Competency questions were derived from “Characteristics and distribution of landforms”, “Climate of Turkey”, “Features of geospatial model of Turkey” and “Water resources of Turkey (rivers, seas, lakes)” which are the subsections of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" Sample competency questions from these subsections can be listed as: “Lütfen, Türkiye'deki şehirleri listeler misiniz?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Please list the cities in Turkey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, “İzmir’in komşularını gösterir misin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Which provinces border Izmir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, “Akdeniz bölgesinde bulunan dağları gösterir misin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (List the mountains in the Mediterranean region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, “Manisa şehrinin çevresinde hangi şehirler konumlanır?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Which cities are located in the province of Manisa?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, “İzmir’ in en yüksek dağı hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (What is the highest mountain in Izmir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, “Ege Bölgesi’ndeki nehirlerin uzunluklarını gösterir misin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Can you tell the length of the rivers in the Aegean region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ), “Türkiye’ de en fazla yağış alan il hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Which city has the most rainfall in Turkey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Step 2 was the consideration of reusing of existing ontologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' This was rejected for language- related reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Instead, the decision was taken to develop an ontology from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Step 3 concerned the determination of competency questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' This involved determining the main conceptualization, scope, and hierarchies in the ontology to avoid overlap between concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Concepts, corresponding properties, and candidate relations were then defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Important concepts and related terms in the GEO-TR ontology are set out in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Main concepts and related terms Important Concepts Related Terms Ada (Island) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' nufus (population),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Bogaz (Strait) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' uzunluk (length) Bolge (Region) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' konumVar (hasLocations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' nufus (population),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yuzolcumu (surface area) Dag (Mountain) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yukseklik (height) Deniz (Sea) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' derinlik (depth),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' tuzluluk (salinity) Gol (Lake) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' derinlik (depth) Nehir (River) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' uzunluk (length) Ova (Plain) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yuzolcumu (surface area) Sehir (City) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' konumVar (hasLocations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' nufus (population),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yuzolcumu (surface area),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yukseklik (height),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ortalamaYagis (average rainfall),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' komsu (neighbourOf) Ilce (District) (subclass of Sehir) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' nufus (population),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yuzolcumu (surface area) Ulke (Country) konumlanir (locatedIn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' konumVar (hasLocations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' nufus (population),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yuzolcumu (surface area),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' iklim (climate),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' baskent (capital) After determining the main conceptualization and relationships,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Step 4 involved defining classes and applying a corresponding hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Within GEO-TR, important terms are represented as classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Each of these constitutes a subclass of the “Thing” class that represents the root node in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ada (Island), Bogaz (Strait), Bolge (Region), Dag (Mountain), Deniz (Sea), Gol (Lake), Nehir (River), Ova (Plain), Sehir (City), Ilce (District) (and Ulke (Country) were determined as such subclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The only defined hierarchy in the ontology is between the subclasses Sehir (City) and Ilce (District).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The class list and hierarchy are shown in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 9 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Class List in GEO-TR – OntoGraf view Protégé (Gennari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The class hierarchy between Sehir (City) and Ilce (District) – (OntoGraf view Protégé) Step 5 of the ontology development process centred around determining the properties of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Related terms in the main conceptualization were selected as possible properties in the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' There are two types of properties in an ontology, namely data and object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Data properties imply a data holding element, whereas object properties hold object-oriented information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' a mountain has a height property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A river has a length property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The sea class’s property is salinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The city class has the properties locatedIn, neighbourOf, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The object and data properties of each class are illustrated in Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' There is a symmetric relationship between konumlanir (locatedIn) and konumVar (hasLocations), which implies these two properties should also be defined inversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Object properties in GEO-TR ODag OThing Sehir OGol Bogaz OAda + OUike Bolge OOva Nehir DenizO Ada Bogaz Bolge Dag Deniz Gol Nehir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='ova = Sehir Ilce oulke目Sehir Sehir - has subclass - ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ice OllceObject property hierarchy: 日日口卤 X topobjectProperty komsu konumlanir konumVar10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Data properties in GEO-TR Step 6 related to defining the facets of properties that refer to value type, allowed values, cardinality (number of allowed values), and other value features a property may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Property facets are defined as the domain and range of a property in an ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In ontology development terminology, the range is defined for data properties and the domain is defined for object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For instance, the data property nufus (population) should have an integer type that declares the range for this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Another example, the konumlanir (locatedIn) property is valid between specific pairs like Sehir (City) – Ulke (Country) and Sehir (City) – Bolge (Region) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' to represent the domain of this property, which defines “a city is located in a country” or “a city is located in a region”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' An additional example can be given for the object property komsu (neighbourOf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Classes Sehir (City), Ulke (Country), and Bolge (Region) are defined as possible domains to apply the komsu (neighbourOf) property in GEO-TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The final step in the process involved creating instances in the structural environment of the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A sample list of instances for the classes Sehir and Bolge in GEO-TR is shown in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Sample list of instances of Sehir Data property hierarchy: topDataProperty 日 风 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='topDataProperty baskent derinlik iklim koordinat nufus ortYagis tuzluluk uzunluk yukseklik yuzolcumuThing Ankara Sehir O llce O Ulke JzI Antalya →Bolu Sanllurfa nquersi Bursa Van →Manisa11 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Sample list of instances of Bolge Names of all classes, data, and object properties and instances are in Turkish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Thus, GEO-TR is coherent with semantic web-enabled geographic question-answering in Turkish, which is the principal novel contribution of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Methodology A geographical question answering framework over linked data is represented in this study for given natural language sentences in Turkish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The main components and corresponding sub- components in the system architecture are illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' System architecture Three main processes are configured in the system architecture, with the following layer naming: question pre-processing, query formulation, and query execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Question pre- processing is the step in which questions are analysed morphologically and NLP techniques applied by morphologically disambiguating the POS tags of each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In addition, named entities are recognized and dependencies between each word extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Each component in the question pre-processing layer acts as a pipeline, finally generating a pre-processed form of the natural language input, which is prepared to further semantically enrich the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The proposed framework is designed and implemented to answer two types of questions, namely informative and quantitative reasoning involved questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The question pre-processing layer is applied to both two types before deciding on the type of question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' the query formulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Thing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='OBolge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Marmara ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+ Akdeniz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+ DoguAnadolu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='IcAnadolu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='→Karadeniz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Gune ydoguAna dol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Ege1)QuestionProcessing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='2)QueryFormulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='3)QueryExecution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Morphological ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='AnswerType ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Analyzer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='StructuredQuery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='NLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='SPARQL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Morphological ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Disambiguator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='(WEKA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Geo TR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Named Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Relation Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ontology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Recognizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Dependency Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Answer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='ITU NLPPIPELINE12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='layer accepts the processed natural language input that is generated by the question pre- processing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The type of question is determined and further corresponding processing tasks are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A structured query in the form of SPARQL (query language of the ontology) is the outcome produced at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The generated SPARQL query is executed on GEO-TR to return the answer to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='1 Question Pre-processing Understanding user intention requires a combination of syntactic and semantic analysis of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Eliminating tokens that do not have any contribution to achieve meaning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' understanding relationships between tokens to get the focus of the sentence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' tagging named entities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' and extracting possible relations between these entities and the focus are the first steps in converting user intent in an unstructured form to a structured query language in the proposed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Turkish is a complex language that is agglutinative, morphologically different, and has free constituent order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2 types of suffixes contribute significantly to meaning: constructive and inflectional suffixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' By adding constructive suffixes to a word, it is possible to form completely different new words semantically or words with a similar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Inflectional suffixes are used for properly placing a word into a sentence (Erguvanli and Taylan 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Combinations that have a different meaning or proper usage for a given word are generated by placing the suffixes at the end of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The morphological features of Turkish make this language more challenging for pre-processing, necessitating a customized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The Turkish NLP pipeline (Eryiğit 2014), developed and served as SaaS (software as a service) by the NLP research group of Istanbul Technical University (ITU), is used for the question pre- processing layer in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Processing components, namely a “Morphological Analyzer”, “Morphological Disambiguator”, “Named Entity Recognizer” and “Dependency Parser” are utilized in this study, and spotter methods are implemented to convert the input format appropriately for each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For a sample question input, each processing step is described in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='2 Morphological Analysis The morphological analysis step includes two main processing layers, namely determination and disambiguation of POS tags in a question input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The morphological analyzer component of the ITU Turkish NLP pipeline is the first to apply at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The method, which combines the word lemmata lexicon with over 49321 entries and flag diacritics for Turkish to handle exceptions regarding phonetic and morphological rules, is presented for further processing to disambiguate the POS tags of each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In the disambiguation layer, affixes are removed recursively without having an additional lexicon (named affix stripping) to find the accurate POS tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Details of the method are represented in (Sahin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The sample question output of the morphological analysis is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Disambiguated output in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' represents the morphological structure of each word that is ready for further processing to understand their role in the sentence to achieve user intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Named Entity Recognition The pipeline further processes disambiguated output with the named entity recognition method to extract location information by resolving the mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Several NER techniques are applied for different types of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The ITU Turkish NLP tool uses the methodology of the Conditional Random Fields (CRF) technique for statistical modeling (Lafferty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2001) of predefined entity categories, such as person, location, organization, money, number, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Details 13 of their methodology are described in their study (Şeker and Eryiğit 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Sample named entity recognizer output is illustrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' “B-LOCATION” is the first location identifier token in which “B” stands for the beginning of the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 3 types of prefixes exist in NER output format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The first token of a named entity is tagged by using the prefix “B” and continues with other tokens (if possible) that are location identifiers “I-LOCATION”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' “I” stands for in the location expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The last type of output prefix is “O”, which represents out of any named entity tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Morphological analyzer output Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' NER output Ankara iline komsu olan illeri gosterir misin ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ankara+Noun+Prop+A3sg+Pnon+Nom il+Noun+A3sg+P2sg+Dat il+Noun+A3sg+P3sg+Dat komsu+Adj komsu+Noun+NAdj+A3sg+Pnon+Nom ol+Verb+Pos^DB+Adj+PresPart ol+Verb+Pos^DB+Adj+PresPart^DB+Noun+Zero+A3sg+Pnon+Nom il+Noun+A3pl+P3pl+Nom il+Noun+A3pl+Pnon+Acc il+Noun+A3pl+P3sg+Nom il+Noun+A3sg+P3pl+Nom goster+Verb+Pos+Aor+A3sg goster+Verb+Pos^DB+Adj+AorPart mi+Postp+Ques+Pres+A2sg ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Punc + Ankara Ankara+Noun+Prop+A3sg+Pnon+Nom iline il+Noun+A3sg+P3sg+Dat komsu komsu+Adj olan ol+Verb+Pos^DB+Adj+PresPart illeri il+Noun+A3pl+Pnon+Acc gosterirgoster+Verb+Pos+Aor+A3sg misin mi+Postp+Ques+Pres+A2sg ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+PuncAnkaraAnkara+Noun+Prop+A3sg+Pnon+Nom ilineil+Noun+A3sg+P3sg+Dat komsu komsu+Adj olan ol+Verb+Pos^DB+Adj+PresPart illeriil+Noun+A3pl+Pnon+Acc gosterir goster+Verb+Pos+Aor+A3sg misin mi+Postp+Ques+Pres+A2sg ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Punc + AnkaraAnkara+Noun+Prop+A3sg+Pnon+Nom B-LOCATION iline il+Noun+A3sg+P3sg+Dat O komsukomsu+AdjO olan ol+Verb+Pos^DB+Adj+PresPart O illeri il+Noun+A3pl+Pnon+Acc O gosterirgoster+Verb+Pos+Aor+A3sgO misin mi+Postp+Ques+Pres+A2sg O ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Punc O14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='4 Dependency Analysis Dependency analysis comprises tagging relationships between words to understand the roles of each token in the sentence and determining its various components, such as an object, subject, verb, or other modifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Generating a dependency graph composed of dependency nodes (tokens) and relationships is the underlying methodology for most dependency analysis algorithms (Nivre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The Conference on Computational Natural Language Learning (CoNLL-X) (Buchholz and Marsi 2006) input format and tags of the Turkish Dependency TreeBank (subject, object, modifier, classifier, possessor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=') are used in the dependency analysis tool of the ITU Turkish NLP pipeline (Eryiğit 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Eryiğit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The tenth CoNLL (CoNLL-X) promoted a shared training file format for multilingual dependency parsing models that has a standardized structured, column-based form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The dependency analysis result of the sample input sentence is demonstrated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency analysis output of the sample sentence ID FORM LEMMA CPOS TAG POSTAG FEATS HEAD DEPREL PHE AD PDEPREL 1 Ankara Ankara Noun Noun Prop|A3sg|Pnon|No m _ 2 _ POSSESSOR 2 iline il Noun Noun A3sg|P3sg|Dat _ 4 _ MODIFIER 3 komşu komşu Adj Adj _ _ 4 _ MODIFIER 4 olan ol Verb Verb Pos^DB|Adj|PresPar t _ 6 _ MODIFIER 5 illeri il Noun Noun A3pl|Pnon|Acc _ 6 _ OBJECT 6 gösterir göster Verb Verb Pos|Aor|A3sg _ 7 _ ARGUMENT 7 misin mi Postp Postp Ques|Pres|A2sg _ 0 _ PREDICATE 8 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Punc Punc _ _ 7 _ PUNCTUATION Id is the counter to represent a token number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Form is the original token that is in the form of an original word or punctuation symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Lemma is the stem of a word or the same as a form if that token is a punctuation symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Cpostag stands for coarse-grained and Postag is the fine- grained POS tag definition from a specific treebank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Feats is the set of syntactic and morphological structure definitions, separated by “|” symbol or underscore if not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Head and Phead values are eliminated in CoNLL-X format (Nivre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Deprel represents the related token and Pdeprel is the type of relationship, or in other words, type of dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='2 Query Formulation The initial step of query formulation is answering type detection, entailing discovering the mention and user intention that is critically helpful in deciding answer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Answer type classification is performed on 2 main types of questions that hold quantitative reasoning: question type 1 (QT1), or not, question type 2 (QT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A rule-based approach detects quantitative reasoning required expressions such as “kaç tane/kaç” (how many), “ne kadar” (how many) or “en (superlative expression in Turkish - Adverb)” and further checks for the bigram of the words to detect “Adjective + Noun”, “Adverb + Adjective” or “Adverb + Noun” patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The main flow for query formulation is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For a given natural language input, if isQuantitative returns true, the question is determined as QT2 and query components are classified to fulfil the items in SPARQL query patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Target class, entity class, data property, object property, and function name are represented as categorical variables in the training model and these parameters are all matched with semantic 15 Yes items in GEO-TR ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A multilayer perceptron, which is an artificial neural network, is used to generate a learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The attributes and categories of the training model are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Main Flow of Query Formulation Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Structure of training model Attribute Name Categories target_class {Sehir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Bolge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ulke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Dag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Nehir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Gol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Deniz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ilce,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='null} entity_class {Sehir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Bolge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ulke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Dag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Nehir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Gol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Ova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Deniz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Ilce,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='null} data_property {yuzolcumu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' populasyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' yukseklik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' derinlik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' tuzluluk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ortYagis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' sicaklik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' enlemBoylam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' bitkiOrtusu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' baskent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' iklim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' null} object_property {konumlanir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='konumVar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='komsu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='null} function_name {count,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='null} Considering the structure of the training model in Table 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" an instance sentence “Türkiye'nin en derin denizi hangisidir?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Which sea is the deepest in Turkey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )” is modelled as target_class = Deniz (Sea), entity_class = Ulke (Country), data_property = derinlik (depth), object_property = konumlanir (located in) and function_name = max (maximum as aggregate function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A SPARQL query is formulated by using classified components with corresponding query patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 2 types of SPAQL patterns are designed for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The type of aggregate function specifies the type of query pattern by using a sub query-based approach (Type 1) or not (Type 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For the functions min and max, subquery formation is inevitable due to the nature of SPARQL queries, whereas count and sum functions do not require it (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Start Use NLP output to formulate query (Figure 12) Answer requires quantitative reasoning or not (Figure 11) Use learning model to generate query components Formulate SPARQL query by using classified components No 16 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Types of Query Pattern Query Pattern: Type 1 Query Pattern: Type 2 SELECT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='m WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type ontology_name_prefix:target-class .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y property_prefix:data-property ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' { SELECT (function_name(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='var) as ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='m) WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type ontology_name_prefix:entity-class .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type ontology_name_prefix:target-class .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y property_prefix:object-property ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y property_prefix:data-property ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='var FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x),"named entity","i")) } }} SELECT (function_name (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y) as ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='total) WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type ontology_name_prefix: entity-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type ontology_name_prefix: target- class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y property_prefix: data-property ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x),"named entity","i")) } If any quantitative expression is not held in the natural language input, NLP output is employed in a manner based on ontology validation to formulate a SPARQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The algorithm deciding the answer type is indicated in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Answer Type Detection In the case of a sentence that does not hold any quantitative reasoning expression, NLP output is combined with semantic web technologies to convert natural language input into a structured query form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The algorithm designed for the conversion is mainly based on dependency analysis, No Yes No Yes No Yes17 NER output, and ontology-based validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' In the Turkish language, the target intent of the user is generally located in the object or subject of a sentence, or any other connected token with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Following that assumption, which is based on the rules of Turkish grammar, the algorithm was designed to combine NLP output with ontology capabilities to improve accuracy while understanding user intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The algorithm is represented and described with examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Algorithm Algorithm to find the answer type of question in Turkish and generate SPARQL query by using processed output by NLP techniques (Method name: generateSparql) Require: sentence processed by NLP techniques agenda: generate query by using NLP output Ensure: final_query 1: nerEntities = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='getNamedEntities(nerOutput) 2: if dependencyOutput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='contains(“OBJECT”) 3: answerType = objectTerm 4: axiomType = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='checkAxiomType(answerType) 5: if axiomType == “CLASS” then 6: properties = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findProperties(answerType, nerEntities) 7: final_query = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='formulate_query(properties, nerEntities, answerType) 8: end if 9: if axiomType == “DATA PROPERTY” then 10: relatedToken = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findRelatedToken(answerType, dependencyOutput) 11: axiomTypeRelated = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='checkAxiomType(relatedToken) 12: Go back to Step 5 call the method for the input axiomTypeRelated and continue again 13: end if 14: if axiomType == “OBJECT PROPERTY” then 15: relatedClass = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findRelatedToken(answerType,dependencyOutput) 16: final_query = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='formulate_query(answerType, nerEntities, relatedClass) 17: end if 19: if dependencyOutput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='contains(“SUBJECT”) then 20: answerType = subjectTerm 21: axiomType = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='checkAxiomType(answerType) 22: if axiomType == “CLASS” then 23: properties = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findProperties(answerType, nerEntities) 24: if properties == NONE then 25: commonConnected = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findCommonConnected (dependencyOutput, answerType) 26: Go to Step 21 call the method for the input commonConnected and continue again 27: end if 28: end if 29: else 30: final_query = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='formulate_query(properties, nerEntities, answerType) 31: if axiomType == “DATA PROPERTY” then 32: relatedToken = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findRelatedToken(answerType, dependencyOutput) 33: axiomTypeRelated = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='checkAxiomType(relatedToken) 34: Go back to Step 21 call the method for the input axiomTypeRelated and continue again 35: end if 36: if axiomType == “INDIVIDUAL” then 18 37: connectedToken = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findConnectedToken (answerType, dependencyOutput) 38: axiomTypeConnected = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' checkAxiomType(connectedToken) 39: Go back to Step 21 call the method for the input axiomTypeConnected and cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' again 40: end if 41: if axiomType == “OBJECT PROPERTY” then 42: commonConnected = pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='findCommonConnected(dependencyOutput, answerType) 43: Go back to Step 21 call checkAxiomType for the input commonConnected and continue 44: end if 45: end if The first processing step for the sample input question after applying NLP methods is deciding on the type of dependency relationship the sentence holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' As illustrated in the dependency analysis output of the sample sentence (Figure Table 3), token 5 (“il” (city)) is the object of that sentence, and the axiom type of the object is checked to decide whether it is a class, a data or object property, or an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' City is a semantic item and represented as a class in the ontology and determined as a target class for query formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For the condition that the object of the sentence is a class (Algorithm – Step 5), the properties of that class with the named entity (if it exists) in the sentence are found to generate a SPARQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' “Ankara” is the named entity for this sentence (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The entity class for the individual “Ankara” is extracted from the ontology as Sehir (entity class), which is the same class with the object token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Possible relationships with “Ankara” and class Sehir are extracted from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The only relationship is found to be an object property “komsu” (neighbourOf) for the sample case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The generic query pattern for the SPARQL formulation is: SELECT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type ontology_name_prefix: entity-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type ontology_name_prefix: target-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y property_prefix: nameOfproperty ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x),"named entity","i")) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' By using this pattern, the query is generated as follows: SELECT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type geo_turkce:Sehir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type geo_turkce:Sehir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y ins:komsu ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x),"Ankara","i")) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Another sample question includes a subject phrase via a deep-thinking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' For the question “Ege Bölgesi’nin yüzölçümü ne kadardır?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (How much is the total area of the Aegean region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" )”, the dependency analysis output is given in Table 6 and the NER result is: Ege ege+Noun+A3sg+Pnon+Nom B LOCATION Bölgesi'nin bölge+Noun+A3sg+P3sg+Gen I LOCATION yüzölçümü yüzölçüm+Noun+A3sg+P3sg+Nom O 19 ne ne+Pron+Ques+A3sg+Pnon+Nom O kadardır kadar+Postp+PCNom^DB+Noun+Zero+A3sg+Pnon+Nom^DB+Verb+Zero+Pres+A 3sg+Cop O ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Punc O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis Result of Sample Sentence ID FORM LEMMA CPOS TAG POST AG FEATS DEP REL PDEPREL 1 Ege ege Noun Noun A3sg|Pnon|Nom 2 POSSESSOR 2 Bölgesi’nin bölge Noun Noun A3sg|P3sg|Gen 3 POSSESSOR 3 yüzölçümü yüzölçüm Noun Noun A3sg|P3sg|Nom 5 SUBJECT 4 ne ne Pron Pron Ques|A3sg|Pnon|No m 5 ARGUMENT 5 kadardır kadar Postp Postp PCNom^DB|Noun|Ze ro|A3sg|Pnon|Nom^ DB|Verb|Zero|Pres| A3sg|Cop 0 PREDICATE 6 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Punc Punc _ 5 PUNCTUATION Token 3 is the subject of the sentence and axiom type determined as a data property from GEO- TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' This is an indicator that quantitative analysis is required to handle user intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Possible classes that are assigned with the aforementioned data property are detected in the sentence (Algorithm – Step 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' From the NER result, “Ege Bölgesi” (Aegean Region) is the named entity, and the entity class is extracted as “Bölge (Region)” from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Additionally, the dependency parsing result shows that token 2 is directly related to the subject token, and the axiom type of token 2 is a class in GEO-TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' At that point, the algorithm moves back to Step 22 to check for possible properties from the ontology but, for that sample case, the answer type is a property so searching for the commonly connected token with the subject expression is the second thing to do (Step 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The commonly connected token is already found because of the fact that there is no other entity for that case and the algorithm formulates the query as: SELECT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='variable WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type geo_turkce:Bolge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ins:yuzolcumu ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x),"Ege","i")) } Final sample input: “Ege Bölgesi\'ndeki şehirlerin nüfuslarını gösterir misin ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Can you show me the populations of the cities in the Aegean Region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, is more complex and holds a determinative group expression for possessive construction (“zincirleme isim tamlaması”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" The dependency analysis output is shown in Table 7 and the NER result of the sentence is as follows: Ege ege+Noun+A3sg+Pnon+Nom B LOCATION Bölgesi'ndeki bölge+Noun+A3sg+P3sg+Loc^DB+Adj+Rel I LOCATION şehirlerin şehir+Noun+A3pl+Pnon+Gen O nüfuslarını nüfus+Noun+A3pl+P3sg+Acc O 20 gösterir göster+Verb+Pos+Aor+A3sg O misin mi+Postp+Ques+Pres+A2sg O ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+Punc O Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Dependency Analysis Output of Final Sample Sentence ID FORM LEMMA CPOS TAG POST AG FEATS DEP REL PDEPREL 1 Ege ege Noun Noun A3sg|Pnon|Nom 2 POSSESSOR 2 Bölgesi’nd eki bölge Noun Noun A3sg|P3sg|Loc^DB|A dj|Rel 5 MODIFIER 3 şehirlerin şehir Noun Noun A3pl|Pnon|Gen 4 POSSESSOR 4 nüfuslarını nüfus Noun Noun A3pl|P3sg|Acc 5 OBJECT 5 gösterir göster Verb Verb Pos|Aor|A3sg 6 ARGUMENT 6 misin mi Postp Postp Ques|Pres|A2sg 0 PREDICATE 7 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Punc Punc _ 6 PUNCTUATION After determining token 4 (“nüfuslarını” (population)) as the object of the sample sentence, the algorithm firstly checks the axiom type for the stemmed form of the token or synonyms (nüfus (population)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The axiom type is determined as a data property and the algorithm continues with Step 10 by using the findRelatedToken method to find the directly dependent token with the token 3 object (“şehirlerin”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The dependency output is critical for specifically this type of question in order to understand user intent to find the population of cities located in the Aegean region rather than displaying the population of the Aegean region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Token 3 “şehir” (city) is checked for the axiom type from the ontology and the result returns as the class that moves the algorithm to Step 6 by assigning the answerType to the related token (“şehir”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Properties defined between “Ege Bölgesi” and “şehir” are extracted from the ontology and only one object property returns, namely “konumVar (hasLocation)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The named entity, entity class, target class, data, and object properties are assigned to formulate the query as follows: SELECT ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='variable WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf:type geo_turkce:Sehir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf:type geo_turkce:Bolge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y ins:konumVar ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ins:populasyon ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y),"Ege","i")) 21 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Flowchart of Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Outputof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='dependency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='dependency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='Determinethat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='cannot ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' named Yes token found between entity and properties named entity and current token No No Axiom type of the Determine answer current (object) Find common type is not hold in token is object connected token with object token and current token propertyornot query cannot be generated Yes Find axiom type of the Find related token with the common connected current token token Axiom type of the Formulate SPARQL query by current (subject) Yes using named entity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' related token is token and current token Find connected individual or not token with the current token Find axiom type of the common connected token No Find axiom type of Find common connected the connected token with the current token token Axiom type of the Determine answer current (subject) No type is not hold in subject token and token is object query cannot be propertyornot generated22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='3 Experimental study The experimental study was performed on two types of questions (QT1 and QT2: See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A comparison was generated to give the results for QT1 in order to understand more deeply the contribution of the NLP output while interpreting the user input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Our main contribution is to show the results of combining the NLP output with semantic technologies so as to build up a SPARQL query by discovering accurate entities and relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Matching the entities and relationships in the NLP and ontology by double-checking the tokens, and dependencies between them, is the main focus of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Instead of only checking for each token, their types, and possible relationships between them in the ontology, the hybrid method disambiguates the possible relationships by applying NLP output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A basic user interface, as illustrated in Figure 13, was implemented for an experimental study of 100 test questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' User Interface for Experimental Study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='4 Results and Discussion Comparison paradigms consist of using a hybrid approach (NLP + ontology-based approach) and only applying an ontology-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Comparison metrics given in Table 8 are precision, recall, and F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Results of Comparison Method Precision Recall F-Measure Method 1: Hybrid approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='71 Method 2: Ontology based approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content="60 图 Question Answering over GEO-TR 口 X Type your question here Turkiye'nin en kalabalik ili hangisidir?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' See the answer!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Clear all for the new question!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" Morphological Analyzer: Named Entity Recognition: Turkiye+Noun+Prop+A3sg+P2sg+Nom ≤DOC> DOC>+BDTag-S>S>+BSTag 0 Turkiye'nin Turkiye+Noun+Prop+A3s g+Pnon+Gen en+Adverb en en+Adverb 0 en+Noun+A3s g+Pnon+Nom SPARQL Query: kalabalik kalabalik+Noun+A3s g+Pnon+Nom 0 ili il+Noun+A3sg+P3sg+Nom 0 SELECT ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='var kalabalik+Noun+A3sg+Pnon+Nom hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 0 WHERE (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='type geo_turkce:Sehir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' kalaba+Noun+A3s g+Pnon+Nom*DB+Adj+Fiffor ≤/S> ≤/S>+ ESTagDOC> DOC>+EDTag ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ins:populasyon ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='var .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' [SELECT (MAX(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='val) as ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='var) il+Noun+A3sg+Pnon+Acc WHERE { ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x rdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='type geo_ turkce:Sehir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' i+Noun+A3sg+P3sg+Nom ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y rdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='type geo_turkce:Ulke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ins:konumlanir ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='x ins:populasyon ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' FILTER(regex(str(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='y),"Turkiye",\'"i") )) Morphological Disambiguator: Dependency Analysis: ES>S>+BSTag Turkiye\'nin Turkiye Noun Noun Turkiye\'nin Turkiye+Noun+Prop+A3s g+Pnon+Gen 2 en en Adverb Adverb en en+Adverb 3 kalabalik kalabalik Noun Noun kalabalik kalabalik+Noun+A3sg+Pnon+Nom 4 ili Noun Noun ili il+Noun+A3sg+P3sg+Nom 5 hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' hangisidir?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='+?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' /S> /S>+ ESTag 123 Results indicate that using morphologically analyzed word forms and their dependencies with semantic items in GEO-TR contributes to improving the accuracy of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Questions with possessive constructions are appropriate examples of how NLP output directly contributes to the meaning by applying dependency analysis to decide on target answer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Relationships between words that might be critical to ascertaining the answer type are missed by Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=" For instance, for a question such as “Ege Bölgesi'ndeki şehirlerin nüfuslarını gösterir misin?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' (Can you show the populations of cities in Aegean Region?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' )”, which has a possessive construction, the ontology-based approach failed to disambiguate whether the population of the Aegean Region (individual in GEO-TR) was intended, or populations of cities that are located in the Aegean Region separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The relationship “POSSESSOR” between the tokens “şehirlerin(cities)” and “nüfuslarını (populations)” disambiguates the user intent as the population of the cities are asked for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The only advantage of Method 2 relates to capturing entities, labels that are not recognized by the named entity recognition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Considering overall performance, this advantage is not sufficient to compete with the hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The ontology-based method simply checks for each, whether it exists in the ontology or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' If a token is found in the ontology, possible relationships and types of axiom are checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' By using the same SPARQL patterns with Method 1, fulfilling the items with corresponding tokens is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Another drawback of Method 2 is that it is challenging to decide on the answer type for sentences that involve more than one class item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A good example can be given as: “İzmir şehri hangi bölgededir?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' “Izmir” (individual), “Sehir” (class), and “Bolge” (class) are matched with semantic items in GEO-TR after processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Possible extracted relationships between individuals and classes are “konumlanir (locatedIn)” (Izmir – Bolge) and “komsu (neighbourOf)” (Izmir – Sehir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' An incorrect intention can be produced by using the outcome of Method 2 to show the neighboring cities of Izmir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Overall, the application of Method 1 results in more accurate results for all facts compared to Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' A learning model generated by the supervised learning method can be used to predict query components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Predicted components are target class, entity class, data property, object property, and aggregate function name to perform quantitative reasoning analysis for the given attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Predicted components are used to formulate the SPARQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The accuracy of the learning model that is built to handle QT2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The experiment is performed by using a train/test split of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='8/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Conclusion In order to fill a gap in the literature, a Turkish question answering framework over linked data (GEO-TR) in the geographic domain is proposed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Combining NLP techniques and an ontology, two types of questions (QT1 and QT2) are handled in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' The main conclusion is that a hybrid approach (Method 1) interprets a sentence in natural language more accurately than an ontology-based approach (Method 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Another significant contribution of this study is the creation of a novel Turkish ontology in the geographical domain, which has been developed by following the rules of ontology development 101(Noy and McGuinness 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' GEO-TR is extendable and ready to use as a data source for other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Following the question pre-processing and query formulation, the experimental study demonstrates the main contribution of hybrid architecture, and results are given by using precision, recall, and F- measure values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 24 Currently, this study is not capable of handling more complex queries with more than one recognized entity, more than one level possessive constructions, or conditional and comparative expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Future work is suggested to apply deep learning techniques to handle complexity in question forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Moreover, extending the coverage and creating a multi-ontology platform could be an additional direction of future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' It is proposed that a pipeline that accepts natural language input and classifies the sentence according to the domain types and matches with a corresponding ontology can fit with this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Declarations: Funding This research received no external funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Conflicts of interest/Competing interests The authors declare no conflict of interest, financial or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' Availability of data and material Not applicable Code availability Not 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Chinese Computing, 2014 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} +page_content=' 333-344): Springer' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQf2Avb/content/2301.04752v1.pdf'} diff --git a/_NE3T4oBgHgl3EQfTAkN/content/tmp_files/2301.04437v1.pdf.txt b/_NE3T4oBgHgl3EQfTAkN/content/tmp_files/2301.04437v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7986f2dd35e8f13ae233c70b1be6d8f45b5ad205 --- /dev/null +++ b/_NE3T4oBgHgl3EQfTAkN/content/tmp_files/2301.04437v1.pdf.txt @@ -0,0 +1,465 @@ +arXiv:2301.04437v1 [math.CA] 11 Jan 2023 +New fixed point theorem and its +application to ODE +Oleg Zubelevich +Steklov Mathematical Institute of Russian Academy of Sciences +oezubel@gmail.com +Abstract. We prove a fixed point theorem that combines the +contraction mapping principle and some Knaster-Tarski-like theo- +rem. As a consequence we obtain an existence theorem to initial +value problem for ordinary differential equation with discontinuous +vector field. +This theorem generalizes Carath´eodory’s existence +theorem. +1. Introduction +The analysis of ODE with non Lipschitz right hand side has a long +history. Without any claims on completeness of exposition we just note +some principle points of this history. A detailed discussion of further +developments in any of these points requires a separate survey. +The first result belongs to G. Peano (1890). G. Peano considered +an initial value problem +˙x = f(t, x), +x(t0) = x0 +(1.1) +where f is a continuous mapping of some domain +D ⊂ Rm+1 = {(t, x)}, +x = (x1, . . . , xm) +with values in Rm. +G. Peano stated that this problem has a solution that is defined +locally for small |t − t0|. This solution may not be unique. +2000 Mathematics Subject Classification. 06A06, 34A12, 34A36, 34A34, 47H09, +47H10. +Key words and phrases. Discontinuous ODE, initial value problem, fixed points, +partial order. +The research was funded by a grant from the Russian Science Foundation +(Project No. 19-71-30012). +1 + +2 +OLEG ZUBELEVICH +C. Carath´eodory relaxed the conditions of this theorem up to mea- +surability of the function f in t. +Note also that in case of non-Lipschitz equations a solution is in +general not unique and one has to make separate efforts to study the +problem of uniqueness. +When the function f is discontinuous in x then even examples that +are pretty innocuous from the first glance can provide nonexistence. +Indeed, consider a scalar IVP [2], [3] +˙x = h(x), +x(0) = 1, +(1.2) +where +h(x) = +� +1, +if x ≤ 1, +−1, +if x > 1. +This problem does not have a continuous solution x(t) in the sense of +integral equation: +x(t) = 1 + +� t +0 +h(x(ξ))dξ, +t > 0. +To show this assume the converse: this solution exists for some t > 0. +It is clear that it can not be equal to 1 identically. Thus for some t′ > 0 +we have x(t′) > 1. (The case x(t′) < 1 is accomplished similarly.) Then +there exists an interval (t1, t′) such that +t ∈ (t1, t′) =⇒ x(t) > 1 +and x(t1) = 1. +For t ∈ (t1, t′) we can write +x(t) = x(t1) + +� t +t1 +h(x(ξ))dξ = 1 − (t − t1) < 1. +This is a contradiction. +Such examples prompt an idea to change the concept of a solution +itself. Note in addition that if the right side of equation (1.1) is just +a measurable function then even for continuous x(t) a mapping t �→ +f(t, x(t)) is not obliged to be measurable [6]. +The corresponding transformation of the notion of a solution was +proposed by A. Filippov [7]. +Once we have denied the classical concept of a solution then a lot of +reasonable generalizations arise. Filippov’s concept is good for control +and for dry friction mechanics [9], [10]. A very different approach by +DiPerna and Lions is good for PDE and fluid mechanics [4]. +In this article we try to save the classical concept of a solution for +some class of discontinuous ODE. + +NEW FIXED POINT THEOREM +3 +To do that we first prove a fixed point theorem. This theorem is +based on two different fields of ides: contraction mappings and mono- +tone mappings in partially ordered sets. +2. The fixed point theorem +Let (X, ρ) be a complete metric space and (Y, 0 is a constant. +Remark 2. Hypothesis 12 implies that for almost all t the function +U is contimuous in x and left continuous in y. +Take 0 < τ ≤ T so small that +∥w∥L1(Iτ ) < min{R, 1}. +(3.2) +Theorem 2. Assume that hypotheses 8, 6, 11, 12, 9, 10, 7 are +fulfilled. Then problem (3.1) has a solution (x, y) ∈ C(Iτ, Rm+n) in the +following integral sense: +x(t) = +� t +0 +U(s, x(s), y(s))ds, +y(t) = +� t +0 +V (s, x(s), y(s))ds, +t ∈ Iτ. +Theorem 2 is a consequence from theorem 1. Let us discuss it. +Let X stand for a set +{x ∈ C(Iτ, Rm) | x(0) = 0, +∥x∥C(Iτ ) ≤ R}. +The set X is a metric space with the standard metric of C(Iτ, Rm). +Let Y denote the space of lower semicontinuous functions y : Iτ → +Rn such that +∥y(t)∥ ≤ R, +∀t ∈ Iτ. +The space Y is endowed with the following partial order +y′