diff --git a/.gitattributes b/.gitattributes index be561b08eebbc9755ccdd1af929572b4a6401994..c9f9df356eac686fe62510f47c3de25cd7e3ee00 100644 --- a/.gitattributes +++ b/.gitattributes @@ -6352,3 +6352,61 @@ l9E2T4oBgHgl3EQfJAaa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex ZtFJT4oBgHgl3EQf7i35/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text EdE2T4oBgHgl3EQfSgfT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text CdE1T4oBgHgl3EQfpwUV/content/2301.03334v1.pdf filter=lfs diff=lfs merge=lfs -text +hNFKT4oBgHgl3EQfuC7r/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +UNE2T4oBgHgl3EQftQjd/content/2301.04069v1.pdf filter=lfs diff=lfs merge=lfs -text +ZtAzT4oBgHgl3EQfm_2G/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kdE4T4oBgHgl3EQftA0V/content/2301.05220v1.pdf filter=lfs diff=lfs merge=lfs -text +K9E1T4oBgHgl3EQfYwRv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf filter=lfs diff=lfs merge=lfs -text +LtE2T4oBgHgl3EQfBAZB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +TNE2T4oBgHgl3EQfWwfK/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +NNE2T4oBgHgl3EQfqwgI/content/2301.04041v1.pdf filter=lfs diff=lfs merge=lfs -text +ttE3T4oBgHgl3EQfNQnH/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +hNFKT4oBgHgl3EQfuC7r/content/2301.11890v1.pdf filter=lfs diff=lfs merge=lfs -text +qdE5T4oBgHgl3EQflA_S/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +qtFST4oBgHgl3EQfOzjA/content/2301.13753v1.pdf filter=lfs diff=lfs merge=lfs -text +j9E4T4oBgHgl3EQfTQyV/content/2301.05006v1.pdf filter=lfs diff=lfs merge=lfs -text +l9E2T4oBgHgl3EQfJAaa/content/2301.03687v1.pdf filter=lfs diff=lfs merge=lfs -text +XtFOT4oBgHgl3EQf9DQS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf filter=lfs diff=lfs merge=lfs -text +CdE5T4oBgHgl3EQfTw8s/content/2301.05538v1.pdf filter=lfs diff=lfs merge=lfs -text +m9FKT4oBgHgl3EQfEy02/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +z9E2T4oBgHgl3EQfMwaS/content/2301.03728v1.pdf filter=lfs diff=lfs merge=lfs -text +Y9E2T4oBgHgl3EQfEgYU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +z9E2T4oBgHgl3EQfMwaS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ZNE5T4oBgHgl3EQfDA6r/content/2301.05402v1.pdf filter=lfs diff=lfs merge=lfs -text +ZNE5T4oBgHgl3EQfDA6r/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf filter=lfs diff=lfs merge=lfs -text +rdE5T4oBgHgl3EQflg-S/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mdFAT4oBgHgl3EQfcR2T/content/2301.08563v1.pdf filter=lfs diff=lfs merge=lfs -text +PNFKT4oBgHgl3EQfgS6S/content/2301.11833v1.pdf filter=lfs diff=lfs merge=lfs -text +G9E1T4oBgHgl3EQfXQT3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +w9FST4oBgHgl3EQfRzg4/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +WtFLT4oBgHgl3EQfTC_4/content/2301.12044v1.pdf filter=lfs diff=lfs merge=lfs -text +LtE3T4oBgHgl3EQfAwm8/content/2301.04261v1.pdf filter=lfs diff=lfs merge=lfs -text +r9E4T4oBgHgl3EQfwg0d/content/2301.05250v1.pdf filter=lfs diff=lfs merge=lfs -text +s9AzT4oBgHgl3EQfPfs9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kdE4T4oBgHgl3EQftA0V/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +QtE0T4oBgHgl3EQfkQGy/content/2301.02471v1.pdf filter=lfs diff=lfs merge=lfs -text +AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf filter=lfs diff=lfs merge=lfs -text +Y9E2T4oBgHgl3EQfEgYU/content/2301.03635v1.pdf filter=lfs diff=lfs merge=lfs -text +99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf filter=lfs diff=lfs merge=lfs -text +BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf filter=lfs diff=lfs merge=lfs -text +PNFKT4oBgHgl3EQfgS6S/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf filter=lfs diff=lfs merge=lfs -text +mdFIT4oBgHgl3EQftCvn/content/2301.11338v1.pdf filter=lfs diff=lfs merge=lfs -text +hNAyT4oBgHgl3EQfxfmy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +2tE1T4oBgHgl3EQflgTe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +29E1T4oBgHgl3EQflwQn/content/2301.03288v1.pdf filter=lfs diff=lfs merge=lfs -text +WtFLT4oBgHgl3EQfTC_4/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +gtE4T4oBgHgl3EQfRwx9/content/2301.04993v1.pdf filter=lfs diff=lfs merge=lfs -text +V9E1T4oBgHgl3EQfbQST/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +fdE1T4oBgHgl3EQfygXR/content/2301.03435v1.pdf filter=lfs diff=lfs merge=lfs -text +A9AyT4oBgHgl3EQfRvd3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +Y9E4T4oBgHgl3EQfOAzo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +tdE2T4oBgHgl3EQffge_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mdFIT4oBgHgl3EQftCvn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +vNFKT4oBgHgl3EQf3y7o/content/2301.11931v1.pdf filter=lfs diff=lfs merge=lfs -text +2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf filter=lfs diff=lfs merge=lfs -text +1tE0T4oBgHgl3EQfuQHv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +HtA0T4oBgHgl3EQfB_9Y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/2301.03956v1.pdf.txt b/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/2301.03956v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58bb343fafb2837441556bfd95e70aa70696d6d6 --- /dev/null +++ b/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/2301.03956v1.pdf.txt @@ -0,0 +1,895 @@ +Towards High-Definition Maps: a Framework Leveraging Semantic +Segmentation to Improve NDT Map Compression and Descriptivity +Petri Manninen1, Heikki Hyyti1, Ville Kyrki2, Jyri Maanp¨a¨a1, Josef Taher1 and Juha Hyypp¨a1 +Abstract— High-Definition (HD) maps are needed for robust +navigation of autonomous vehicles, limited by the on-board +storage capacity. To solve this, we propose a novel frame- +work, Environment-Aware Normal Distributions Transform +(EA-NDT), that significantly improves compression of standard +NDT map representation. The compressed representation of +EA-NDT is based on semantic-aided clustering of point clouds +resulting in more optimal cells compared to grid cells of +standard NDT. To evaluate EA-NDT, we present an open-source +implementation that extracts planar and cylindrical primitive +features from a point cloud and further divides them into +smaller cells to represent the data as an EA-NDT HD map. We +collected an open suburban environment dataset and evaluated +EA-NDT HD map representation against the standard NDT +representation. Compared to the standard NDT, EA-NDT +achieved consistently at least 1.5× higher map compression +while maintaining the same descriptive capability. Moreover, +we showed that EA-NDT is capable of producing maps with +significantly higher descriptivity score when using the same +number of cells than the standard NDT. +I. INTRODUCTION +The current development of mobile robots and the on- +going competition for the crown of autonomous driving +has increased the demand of accurate positioning services. +Generally, Global Navigation Satellite System (GNSS) can +be used to measure global position of a mobile robot but the +accuracy of satellite navigation alone is typically around a +few meters and because of signal obstruction the satellite +signals may be unavailable [1], [2]. Alternatively, global +position can be solved by fitting the current sensor view +into an existing georeferenced map, that can be computed +e.g. with Simultaneous Localization and Mapping (SLAM) +[3]. Moreover, map-based technique provides a combined +position and rotation estimate in contrast to a global position +measured by GNSS. +Maps used in autonomous driving are typically called +High-Definition (HD) maps [4], [5]. Data compression of +HD maps is of high importance within many applications +that have limited computational resources and storage capac- +ity [6]–[8]. Moreover, real-time localization requires com- +pressed maps to ensure fast processing capability. +*This work was supported by Academy of Finland, decisions 337656, +319011, 318437 and by Henry Ford foundation Finland. +1P. Manninen, H. Hyyti, J. Maanp¨a¨a, J. Taher, J. Hyypp¨a are +with +Department +of +Remote +Sensing +and +Photogrammetry, +Finnish +Geospatial Research Institute (FGI), National Land Survey of Finland +(NLS), +02150 +Espoo, +Finland +petri.manninen@nls.fi, +heikki.hyyti@nls.fi, jyri.maanpaa@nls.fi, +josef.taher@nls.fi, juha.hyyppa@nls.fi +2V. Kyrki is with School of Electrical Engineering, Aalto University, +02150 Espoo, Finland ville.kyrki@aalto.fi +Fig. 1: An illustration of a point cloud (white) and corre- +sponding EA-NDT HD map representation. EA-NDT cells +are visualized with ellipsoids (mass within a standard de- +viation) presenting building (yellow), fence (cyan), ground +(purple), pole (blue), tree trunk (orange) and traffic sign (red) +labels. +Since positioning in real-time with raw point clouds is +infeasible, alternative methods have been developed to over- +come the problem [6], [9]–[12]. One promising approach is +Normal Distributions transform (NDT) [9]. NDT compresses +the three-dimensional point cloud data by dividing the cloud +into equal sized cubical cells that are expressed by their mean +and covariance. To improve the scan registration, Semantic- +assisted Normal Distributions Transform (SE-NDT) [12] +expanded the original NDT with semantic information. +However, both NDT and SE-NDT use a grid structure for +the division of the point cloud, and therefore cannot find +the fundamental geometrical structure of the environment +(e.g. boundaries between object surfaces). Consequently, this +results in an NDT representation where part of the cells have +a high variance in all three dimensions. Magnusson [9] also +presented that the point cloud can alternatively be divided +by K-means clustering [13] and that it improves the scan +registration compared to using a grid structure. +In this work, we address the aforementioned problem of +sub-optimal point cloud division. We propose to solve the +problem by leveraging semantic-aided clustering. We present +a novel framework called Environment-Aware NDT (EA- +NDT) (illustrated in Fig. 1), which provides EA-NDT HD +Map representation. EA-NDT HD Map is a compressed +representation of a point cloud that is based on the leaf cell +representation of the standard NDT, and therefore NDT scan +registration technique is directly applicable with EA-NDT. In +this work, the standard NDT is referred as NDT. In contrast +to the grid structure of NDT, EA-NDT leverages semantic +information to cluster planar and cylindrical primitives of the +©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any +copyrighted component of this work in other works. DOI: 10.1109/IROS47612.2022.9982050 +arXiv:2301.03956v1 [cs.RO] 10 Jan 2023 + +environment to provide a more optimal NDT cell division. +In EA-NDT HD Map, each cell only consists of points that +model the same basic geometrical shape such as a plane +or a pole. Moreover, by adding understanding of semantic +information in the scene, we can compute a map containing +only stable objects that are useful for accurate localization. +The main contributions of this paper are: +1) A novel data-driven framework to compute NDT map +representation without the grid structure. +2) Demonstration of significantly improved data compres- +sion compared to NDT representation. +3) An open-source implementation1 of the proposed EA- +NDT is shared for the community. +4) A registered dataset2 to evaluate the proposed EA-NDT +on data collected with Velodyne VLS-128 LiDAR. +The rest of the paper is organized as follows: The next +section describes the related work in the fields of HD Maps, +scan registration, SLAM, and point cloud semantic segmen- +tation. In Section III, we formalize a pipeline architecture +of the proposed framework to extract planar and cylindrical +primitives of the point cloud. The implementation details of +our proof of concept solution are explained in Section IV +together with an introduction to the data collection setup and +preprocessed dataset, and the evaluation metrics used for the +experiment. In Section V, we compare the proposed EA- +NDT to a map computed with NDT and show that EA-NDT +provides a significant map compression while maintaining +the same descriptive capability. Finally, in Section VI we +consider the advantages and disadvantages of EA-NDT and +provide a discussion over the validity, reliability and gener- +alizability of the experiment. +II. RELATED WORK +HD Maps are one of the key techniques to enable au- +tonomous driving [5]. Seif and Hu [4] have recognized three +challenges to be solved with an HD map: the localization +of the vehicle, reacting to events beyond sight, and driving +according to the needs of the traffic. In this work, we focus +in the localization task. An HD Map can be computed e.g. +with SLAM that is a well established and profoundly studied +problem about how to align subsequent sensor measurements +to incrementally compute a map of the surrounding environ- +ment while simultaneously localizing the sensor [14]. In the +review by Bresson et al [14], they found that an accuracy +of 10 cm has been reported for the built maps but even an +accuracy of 2 cm is possible. +Data compression is a crucial challenge for HD maps in +large environments. For example, the well known Iterative +Closest Point (ICP) [15] algorithm is infeasible in large +point clouds due to the computational cost of finding closest +corresponding points across a measurement and a map. To +improve the computational problems of ICP, Magnusson +proposed Point-to-Distribution (P2D)-NDT in which the +reference cloud is divided by a fixed sized 3D grid into +1https://gitlab.com/fgi_nls/public/hd-map +2https://doi.org/10.5281/zenodo.6796874 +cells modelled by the mean and covariance of the points +[9]. In (P2D)-NDT, each point in the registered scan is +fitted to the cells within a local neighbourhood of the point. +In addition to robust scan registration, NDT representation +provides data compression together with faster registration. +Stoyanov et al. presented Distribution-to-Distribution (D2D)- +NDT that further develops the P2D-NDT to likewise model +the registered scan with normal distributions [10]. +Semantic information can enhance scan registration perfor- +mance of NDT. For example, Semantic-assisted NDT (SE- +NDT) [11], proposed by Zaganidis et al., showed that the use +of even two semantic labels (edges and planes) can improve +the scan registration. To further develop SE-NDT, Zaganidis +et al. presented a complete semantic registration pipeline that +uses a deep neural network for semantic segmentation of the +point cloud [12]. SE-NDT uses the 3D grid cell structure +of NDT but models each semantic label separately to utilize +the division of similar entities in the registration task. For +semantic segmentation, SE-NDT uses PointNet [16] that is +a pioneering solution of a point cloud segmentation network +that consumes raw point cloud data without voxelization +or rendering. Cho et al. proposed that the uncertainty of +semantic information could also be used in the registration +task [17]. +Semantic information can be utilized further than was +proposed in the previous works. In this work, we propose +to replace the aforementioned grid division by leveraging +semantic-aided clustering that finds planar and cylindrical +structures of a point cloud. For semantic segmentation, +we use Random sampling and an effective Local feature +Aggregator-Net (RandLA-Net) [18] that presents a new local +feature aggregation module to support random sampling that +was found a suitable technique for semantic segmentation +of large scale point clouds. They have reported up to 200× +faster processing capacity compared to the existing solutions. +A further review of semantic segmentation of point cloud +data is available e.g. in [19]. +III. ENVIRONMENT-AWARE NDT +Here we propose a framework, called Environment-Aware +NDT (EA-NDT), to divide a semantically segmented point +cloud into NDT cells. The proposed framework is a straight +pipeline process consisting of 4 stages (Fig. 3) that step- +by-step divide the input point cloud into cells which are +ultimately represented as an NDT map. The input of the +pipeline is a Registered Point Cloud, which is processed +in the following order by stages called Semantic Seg- +mentation, Instance Clustering, Primitive Extraction and +Cell Clustering. Finally, the output of the pipeline is +an environment-aware NDT-based HD map representation, +called EA-NDT HD Map, which stores the found cells using +NDT representation. +In the Registered Point Cloud each 3D point has X, +Y, Z Cartesian coordinate (e.g. ETRS-TM35FIN, ECEF) +and an intensity value. Semantic Segmentation appends +semantic information for each point in the cloud to enable +further clustering of the data. In this work, we used road, + +sidewalk, parking, building, fence, pole, traffic sign, and +tree trunk labels to demonstrate the framework but also +other labels could be used. Instance Clustering divides +each semantic segment into instances that are spatially sep- +arated from each other. Primitive Extraction divides each +instance into predefined primitives that can be modeled with +an unimodal distribution. In this work, we have defined +planar and cylindrical primitives but the framework could +be extended to support new types of primitives. However, +large primitives such as trees can not be modelled well with +an uniform distribution. Therefore, Cell Clustering further +divides each primitive into cells of approximately equal size +while minimizing the number of used cells. Ultimately, EA- +NDT HD Map is presented as an octree [20] that in this +work stores the point counter, point sum and upper diagonal +of the covariance matrix for each cell but other attributes +such as semantic segment, instance cluster or primitive type +could be included. +IV. METHODS AND EXPERIMENTS +To demonstrate the proposed framework to build an EA- +NDT HD Map, we used a dataset collected with Velodyne +VLS-128 Alpha Puck [21] LiDAR 7th of September 2020 in +a suburban environment in the area of K¨apyl¨a in Helsinki, the +capital of Finland. The environment in the dataset consists of +a straight two-way asphalt street, called Pohjolankatu, which +starts from a larger controlled intersection at the crossing of +Tuusulanv¨ayl¨a (60.213326° N, 24.942908° E in WGS84) and +passes by three smaller uncontrolled intersections until the +crossing of Metsolantie (60.215537° N, 24.950065° E). It is +a typical suburban street with tram lines, sidewalks, small +buildings, traffic signs, light poles, and cars parked on both +sides of the streets. To collect a reference trajectory and to +synchronize the LiDAR measurements, we have used a No- +vatel PwrPak7-E1 GNSS Inertial Navigation System (INS) +[22]. The sensors were installed on a Ford Mondeo Hybrid +research platform named Autonomous Research Vehicle Ob- +servatory (ARVO) [23]. The sensors were interfaced through +Robotic Operation System (ROS) [24] version Kinetic Kame +and the sensor measurements were saved in rosbag format +for further processing. +A. Preprocessed Dataset +Our open preprocessed dataset2, shown in Fig. 2, consists +of a two-way asphalt paved street with a tram line to both +directions and sidewalks in both sides of the street. The +length of the dataset trajectory is around 640 m and it has +Fig. 2: The complete dataset visualized with semantic labels +in different colors: road (magenta), sidewalk (violet), park- +ing (pink), terrain (green), buildings (yellow), fence (light +brown), tree trunk (brown), traffic sign (red), pole (grey). +TABLE I: RandLA-Net classified dataset label proportions. +Semantic label +No. of points +% of all +% of used +Ground +14,052,836 +34.7 +50.8 +Building +7,650,980 +18.9 +27.7 +Tree trunk +3,560,910 +8.8 +12.9 +Fence +2,120,849 +5.2 +7.7 +Pole +193,516 +0.5 +0.7 +Traffic sign +82,680 +0.2 +0.3 +Labels used here +27,661,771 +68.4 +100.0 +Others +12,799,904 +31.6 +Total +40,461,675 +100.0 +in total more than 40 million points from which 28 million +are used in this work. All the intersections together have +a plenty of traffic signs. The sidewalks are separated from +the road by a row of tall planted trees. The dataset contains +nearly 30 buildings that are mostly wooden and there are +several fences between the houses. Our dataset includes all +the semantic labels classified by RandLA-Net [18] but in this +work we have used only road, sidewalk, parking, building, +fence, tree trunk, traffic sign, and pole labels. In this work, +road, sidewalk, and parking labels were reassigned into a +common ground label. The proportion and the number of +the points of each label are shown in Table I. Half of the +used points consist of ground and roughly a fourth represent +buildings whereas poles and traffic signs together represent +only 1 %. Tree trunks and fences together represent a fifth +of the used points. The preprocessing of the data consists +of three steps; semantic segmentation, scan registration, and +data filtering. +In semantic segmentation of scans, we used a RandLA-Net +model pre-trained with SemanticKITTI dataset which was +collected with Velodyne HDL-64 LiDAR [25]. Instead, we +used VLS-128 which has a longer range and 128 laser beams +instead of 64 [21]. Also, VLS-128 has a wider field of view +(FOV) in vertical direction. Consequently, the measurements +outside of the vertical FOV of HDL-64 were constantly +misclassified so only the measurements within HDL-64 FOV +were used. RandLA-Net outputs a probability estimate vector +of labels for each point. In this work, we call it as label +probabilities. +In scan registration, the motion deformation of each scan +was first fixed according to a GNSS INS trajectory post- +processed with Novatel Inertial Explorer [26], after which +P2D-NDT implementation [27] with 1 m grid cell size was +used for registration. In registration, a local map of 5 last +keyframes was used as a target cloud and a motion threshold +of 10 cm was used to add a new keyframe. Grid cells +containing points from a single ring of the LiDAR were +ignored in the registration. Moreover, points that were con- +sidered possibly unreliable (vehicles, bicycles, pedestrians +and vegetation) or further than 50 m away from the LiDAR, +were ignored. +After the scan registration, the dense Registered Point +Cloud was voxel filtered to average X, Y, Z position and + +Fig. 3: A visualization of the proposed EA-NDT processing pipeline that is based on the framework in Section III. The input +is a semantically segmented point cloud and the intermediate phases before EA-NDT HD Map are instances, primitives and +cells, in which the entities are separated by color. The color mapping of semantic information is explained in Fig. 1. +label probabilities of each 1 cm voxel. To smoothen the +semantic segmentation, the label probabilities of each point +was averaged within a radius of 5 cm. +B. The Implementation +A method1, based on the framework presented in Sec- +tion III, was implemented in C++14 on top of ROS Noetic +Ninjemys. The main functionality of the implementation uses +existing functions and classes of Point Cloud Library (PCL) +[28]. The implemented processing pipeline is demonstrated +in Fig. 3. The 1st stage of the framework, Semantic Segmen- +tation, is explained in Section IV-A. Therefore, our dataset +already includes the semantic information. +Instance clustering was implemented with Euclidean +region growing algorithm [29] to divide each semantic seg- +ment into spatially separate instances shown in Fig. 3. In +general, we require a distance threshold of 30 cm between +the instances and a minimum of 10 points per instance. +For ground label we require a distance threshold of 50 cm +between the instances and a minimum of 3000 points per +instance. In our dataset, there is a significant amount of +outliers and reflected points below the ground plane that are +undesired in a map, Instance clustering is used to filter those +points. +In Primitive Extraction, tree trunk and pole instances +are modeled as individual cylindrical primitives, and traffic +sign instances as individual planar primitives. Both primitive +types are shown in Fig. 3. For other semantic labels, planar +primitives were extracted by Random Sample Consensus +(RANSAC) [30] based normal plane fitting algorithm after +subsampling the instance with an averaging 10 cm voxel grid +and estimating the point normals for each remaining point +from 26 nearest neighbours. For building and fence instances, +a normal distance weight of π/4 and a distance threshold of +15 cm was used for plane fitting. For ground instances, the +procedure differs slightly: 1) an existing implementation [31] +of K-means++ algorithm [32] was used to divide the ground +instances into primitives with an area of approximately 100 +m² (The initialization of number of K-means clusters is +explained later in this section in Cell Clustering), after +which 2) the plane fitting was performed for each primitive +with a 30 cm distance threshold for a coarse noise filtering. +In Cell Clustering, primitives are divided into cells +(shown in Fig. 3) with K-means++ algorithm for which the +number of clusters +NL = ⌈fLnL +gL⌉ +(1) +is initialized for each label L. In (1), ⌈·⌉ is the ceiling +operator, nL is either nα for cylindrical primitives (tree +trunk and pole) or nβ for planar primitives (ground, building, +fence, and traffic sign): +nα = lα +� +sc +and +nβ = Aβ +� +sc +2, +(2) +TABLE II: The final values of the scaling parameters +Semantic label (L) +fL +gL +Ground +1.680 +0.083 +Building +2.708 +0.137 +Tree trunk +4.179 +0.318 +Fence +2.248 +−0.788 +Pole +1.687 +−0.315 +Traffic sign +3.923 +0.317 +Fig. 4: The number of cells Nc after fitting EA-NDT with +NDT shown w.r.t. cell size sc, color indicates the method, +line style the label, and green background the fitted range. + +Fig. 5: The complete map descriptivity score Sd compared +w.r.t. number of cells Nc. The violet line depicts the com- +putation of the NDT compression efficiency η for each Sd. +where lα is the length of a cylindrical primitive and Aβ is +the number of points remaining after projecting the planar +primitive into the eigenspace found by principal component +analysis (PCA) and filtering with a 10 cm voxel grid. +Additionally, after clustering cells in ground, a plane fitting +with a 15 cm threshold is performed for each cell for a finer +noise filtering. In (1), scaling parameters fL and gL, shown +in Table II, were manually fitted for each L over 6 iterations +starting from fL0 = 1 and gL0 = 1 until the number of cells +Nc for EA-NDT (shown in Fig. 4) was sufficiently close to +NDT with cell size sc < 1 m. Despite the cell size, each +primitive is required to have at least one cell. Fig. 4 reveals +how this sets a lower boundary for Nc with larger cells. +Finally, all the computed cells were stored into an octree +structure [20] that represents the EA-NDT HD Map. Each +leaf cell in the octree stores a point counter, point sum +and upper diagonal of the covariance matrix for the cell. +We require a minimum of 6 points for a leaf cell to be +modeled reliably with a normal distribution, hence cells with +less points are ignored. The octree implementation in PCL +requires a minimum leaf cell size parameter, we used sc/4 +to make it sufficiently smaller compared to the required cell +size of EA-NDT. +C. Evaluation +Here, a descriptivity score Sd, in which a higher score +denotes higher similarity, is defined to evaluate how well the +map models the raw point cloud. It is derived using a density +function of a multivariate normal distribution [33], which is +defined for each 3D point xi and jth NDT cell as +fj(xi) = +1 +� +(2π)k|Σj| +e(− 1 +2 (xi−µj)TΣ−1 +j +(xi−µj)). +(3) +In (3), µj is the mean vector of a distribution with a +covariance matrix Σj for jth NDT cell, | · | is the determinant +operator and k = 3 describes dimension of the multivariate +distribution. Restricting to the local neighborhood of each +Np point, the descriptivity score Sd is an average density of +Fig. 6: An alternative comparison of the complete map +descriptivity score Sd w.r.t. cell size sc. The violet line +depicts the computation of the descriptivity ratio Rd for each +sc and green background emphasizes the applicable range. +best fitting NDT cells: +Sd = 1 +Np +Np +� +i=1 +max fj(xi) +∥xi−µj∥2 ≤ 2sc +. +(4) +The maximum distance inside a grid cell is +√ +3sc, and +therefore the radius of 2sc was considered large enough to +contain the highest fit. +We have defined two ratios, descriptivity ratio Rd requir- +ing sEA +c += sNDT +c +(Fig. 6) and data compression ratio Rc: +Rd = SEA +d +� +SNDT +d +and +Rc = Npσp +� +Ncσc, +(5) +where superscripts EA and NDT stand for EA-NDT and +NDT, respectively, σp and σc are the data size of the point +and the cell, respectively. Using (5) while requiring SEA +d += +SNDT +d +, we define an NDT compression efficiency (Fig. 5) +η = RNDT +c +� +REA +c += N NDT +c +� +N EA +c . +(6) +V. RESULTS +In this section, we evaluate the quality between EA- +NDT and NDT map representations and demonstrate the +data compression of EA-NDT. The performance of both +methods was evaluated by stepping the cell size from 0.2 m +to 10 m with 30 values. Note that the computational time +increases exponentially with decreasing cell size. The lower +boundary of 0.2 m was selected since it could still be +computed overnight. Similarly, the upper boundary of 10 m +was considered large enough for this test. In Figs. 6–8, a +practically applicable range of 0.5 – 2.0 m, based on previous +work [9], is highlighted. +The evaluation of complete map representation on the +dataset described in Section IV-A is shown in Fig. 5. It +shows that EA-NDT map representation provides a higher +descriptivity score for any number of cells (note that the min- +imum number of cells is limited for EA-NDT as explained +in Section IV-B). However, typically the results of NDT are +compared as a function of the cell size, and therefore, in + +Fig. 7: Comparison of descriptivity score Sd of each label +w.r.t. cell size sc, line style indicates the method, color the +label, and green background the applicable range. +Fig. 8: Both NDT compression efficiency η (above) and +descriptivity ratio Rd (below) of the proposed method are +visualized for the complete map and all labels w.r.t NDT cell +size sc, green background emphasizes the applicable range. +Fig. 6 we present an alternative comparison for which the +number of cells in EA-NDT were fitted with NDT as ex- +plained in Section IV-B. Likewise, descriptivity of EA-NDT +outperforms NDT with any cell size. By comparing Fig. 5 +and Fig. 6, one can note that both plots are equally capable of +showing the differences between the compared methods. In +general, it can be noticed that the descriptivity score of NDT +approaches EA-NDT with smaller cells. However, this is an +expected phenomenon of grid cell division; the probability +of multiple objects to be associated within one cell decreases +with smaller cells. +In Table I in Section IV-B, it is shown that around 78.5 % +of the data consists of points labelled as ground or building, +which reflects a similar proportion to the number of cells +shown in Fig. 4. The descriptivity score of the complete map +is dominated by these abundant labels leaving the effect of +other labels imperceptible. Therefore, in Fig. 7, we present +an equivalent descriptivity score comparison separated for +each label, which in case of NDT is equivalent to SE-NDT +representation [11]. Similarly to the comparison of complete +map representation, EA-NDT descriptivity score of each +separate label is higher except for tree trunks with 3 – 6 +m cells, for which the descriptivity score equals with NDT. +The low descriptivity of EA-NDT is most likely caused by +the use of K-means clustering because if a cluster is large or +a diameter of a trunk is small, a single cluster can contain +points within the entire circumference of the trunk resulting +in a non-Gaussian distribution. Moreover, the use of HDL- +64 vertical FOV limits the height of tree trunks and poles +in to a range of 2.5 – 3 m (as explained in Section IV-A) +and when the required cell size exceeds half of that height, a +large portion of the primitives is assigned into a one cluster +instead of two causing the observed discontinuity. However, +because of scaling the number of cells, the effect does not +appear exactly with the expected cell size. With tree trunk +and pole labels it is also observable that the descriptivity +does not decrease with the largest cells, because the size of +the primitive limits the cluster size from increasing. +The descriptivity ratio between EA-NDT and NDT is +shown in the lower part of Fig. 8. In general, the descriptivity +ratio increases for all the labels towards the greater cell +sizes, tree trunk and pole labels make an exception that was +already covered above. Within the applicable cell range, the +improvement in descriptivity ratio is more constant for all +labels. For the complete map with 2 m cells, the descriptivity +is 2× higher compared to NDT and for 10 m cells the +descriptivity is 20× higher. Especially, building and fence +labels show relatively higher descriptivity scores, which +suggests that the plane extraction is advantageous. +Map compression is a direct consequence of EA-NDT’s +higher descriptivity scores; EA-NDT achieves the same de- +scriptivity with a larger cell size which means a smaller num- +ber of cells. The NDT compression efficiency η, visualized +in the upper part of Fig. 8, was used to compare compression +of EA-NDT with NDT (note that η, shown in Fig. 5, can be +computed only when a corresponding score exists for both +methods). For the complete map representation, EA-NDT +provides 1.5 – 1.75× better compression within the whole +examined range. The compression of the complete EA-NDT +is mainly defined by the ground label, which is about 1.5× +better within the whole range. EA-NDT’s compression of +traffic sign and pole labels is more than 2.1× higher than +NDT for the smallest cells but drops steeply towards greater +cell sizes, though, remaining higher compression even for the +largest cells. For building and fence labels, the compression +is more than 2.2× higher around 0.5 m cell size, for smaller + +and larger cells the NDT compression efficiency decreases. +This suggests that EA-NDT’s technique of modeling the +planes and excluding the other points is beneficial until cell +size of about 0.5 m but with smaller cells, NDT reaches +the difference by modeling the excluded points. Finally, we +can state that the complete map representation of EA-NDT +achieves the highest compression improvement around 0.7 m +cell size which is also within the applicable cell size range. +VI. DISCUSSION +The proposed EA-NDT achieves 1) at least 1.5× higher +compression, and 2) always a higher descriptivity score +with the same number of cells compared to NDT as shown +in Fig. 8. For separately tested semantic labels, EA-NDT +achieves 1) always a higher compression, and 2) a higher +descriptivity score in the applicable cell size range of 0.5 – +2.0 m. However, we suggest to use cell sizes of 0.5 – 1.0 +m for EA-NDT in a suburban environment since our results +(Fig. 8) indicate that this range provides better compression. +Due to the semantic-aided instance clustering and primi- +tive extraction, the proposed EA-NDT is able to find the most +significant planar and cylindrical primitives in the environ- +ment. NDT representation is especially informative within +planar and thin cylindrical structures that can be modeled +with a unimodal distribution, and therefore, EA-NDT is +able to model the environment more optimally compared +to NDT. Moreover, the use of semantic information enables +selection of the stable objects that should be modeled in the +map. Finally, K-means clustering of the primitives ensures +data efficient placement of cells where they are needed. +The advantage of EA-NDT is a result of improved point +cloud division. Therefore, the advantage is prominent in +small objects such as poles or complicated structures such as +buildings or fences. In the ground plane, the advantage is less +evident because it is in any case a one large plane and the +benefit can be almost completely explained by the removal +of outliers and by the efficiency gained from clustering the +ground plane. +Finding planar primitives in building and fence instances +removes some points which are not modeled by EA-NDT +HD Map. As shown in our results in Fig. 8, that is beneficial +for compression with cell sizes above 0.5 m but for smaller +cell sizes it could be beneficial to model those points with +additional NDT cells. However, as shown in this work, +the described effect is not significant within the range of +the suggested cell sizes. Moreover, the suggested correction +should be justified only if it improves also the performance +of scan registration. +The classification accuracy of the pre-trained RandLA-Net +(see Section IV-A) was a limiting factor for the quality of se- +mantic segmentation. The misclassification increases the total +number of cells when overlapping cells of different semantic +labels model the same object (see Fig. 1), which reduces the +compression of EA-NDT representation. In future work, the +classification accuracy of the semantic segmentation could be +improved by using a more advanced model [34], [35] and +by retraining the model for the used LiDAR. +The proposed EA-NDT was tested in a suburban envi- +ronment that consist of 1) a flat ground, buildings, fences, +and traffic signs, which are modeled as planar surfaces, and +2) poles and tree trunks, which are modeled as cylindrical +objects. The tested environment contains enough samples of +all the tested semantic labels to demonstrate that EA-NDT is +able to compress the data more than NDT. However, our tests +did not concern vegetation, tree canopies, water, significant +height variations, nor high rise buildings. In the future, a +larger variety of environments should be studied. +EA-NDT provides map compression within the tested +semantic labels in environments where 1) the used semantic +labels exist in the environment, 2) the reliability of semantic +segmentation is high enough, and 3) instances are separated +by sufficient distance. In order to use the proposed EA-NDT, +the following assumptions need to hold: 1) ground, buildings, +fences, and traffic signs must be composed of planar surfaces, +and 2) tree trunks and poles need to be cylindrical. In future +work, for other semantic labels, the type of primitive would +need to be defined according to the properties of that label. +Semantic information is a powerful tool and a key enabler +of HD Maps. In this work, we have shown that semantic +information enables separate processes for each semantic +label which results into more optimal clustering of point +cloud data. Furthermore, in previous works semantic infor- +mation has been used to improve positioning [11], [12], +[17]. Moreover, semantic information enables the removal of +unwanted dynamic objects from the map. In future work, the +use of semantic information opens a possibility to study the +positioning accuracy and reliability of different object types +over time. That could be especially useful when navigating +in constantly changing environments such as arctic areas. +This work was outlined on evaluating compression and +descriptivity properties of EA-NDT HD Map, and therefore, +the positioning performance of the proposed framework +remains an open question for the future work. Although, +the positioning was not evaluated, the well established scan +registration and cell representation of NDT is integrated into +positioning of EA-NDT. Moreover, the data compression +of an HD map is a desired property of any mobile robot +application. Another open question is how the proposed +EA-NDT HD map can be efficiently updated with new +information. Also, currently the computation of EA-NDT +is very slow and the computational optimization is left for +future work. +VII. CONCLUSIONS +In this work, we proposed EA-NDT, that is a novel +framework to compute a compressed map representation +based on NDT formulation. The fundamental concept of +EA-NDT is semantic-aided clustering to find planar and +cylindrical primitive features of a point cloud to model them +as planar or elongated normal distributions in a 3D space, +respectively. +We showed that compared to NDT, the data-driven ap- +proach of EA-NDT achieves consistently at least 1.5× higher +map descriptivity score, and therefore enables a significant + +map compression without deteriorating the descriptive ca- +pability of the map. The best compression in comparison +to NDT is obtained within cell sizes of 0.5 – 2 m, which +is an applicable range for real-time positioning. Moreover, +the results show that compared to NDT, the representation +achieves a higher data compression within all the tested +semantic labels, that is a desired property for mobile robots +such as autonomous vehicles. +When data compression is a required property of an HD +map, we recommend the use of EA-NDT instead of NDT. +Based on the results of this work, it seems likely that the +positioning accuracy using EA-NDT maps exceeds that of +standard NDT maps of same size. However, this warrants +future studies because there are several interacting factors +such as potentially varying contribution of different semantic +labels to the positioning accuracy. +ACKNOWLEDGMENT +In addition, the authors would like to thank Paula Litkey +and Eero Ahokas from FGI for data management and col- +lection and Antero Kukko and Harri Kaartinen from FGI +for assistance and advices. We would also like to thank Leo +Pakola for participation in the research vehicle development. +REFERENCES +[1] A. Zaidi and M. Suddle, “Global navigation satellite systems: a +survey,” in 2006 International Conference on Advances in Space +Technologies, pp. 84–87, 2006. +[2] J. Wang, “Pseudolite applications in positioning and navigation: +Progress and problems,” Journal of Global Positioning Systems, vol. 1, +no. 03, pp. 48–56, 2002. +[3] C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, +I. Reid, and J. J. Leonard, “Past, present, and future of simultaneous +localization and mapping: Toward the robust-perception age,” IEEE +Transactions on Robotics, vol. 32, no. 6, pp. 1309–1332, 2016. +[4] H. G. Seif and X. Hu, “Autonomous driving in the iCity—HD maps +as a key challenge of the automotive industry,” Engineering, vol. 2, +no. 2, pp. 159–162, 2016. +[5] R. Liu, J. Wang, and B. Zhang, “High definition map for automated +driving: Overview and analysis,” The Journal of Navigation, vol. 73, +no. 2, pp. 324–341, 2020. +[6] R. Dub´e, A. Cramariuc, D. Dugas, H. Sommer, M. Dymczyk, J. Nieto, +R. Siegwart, and C. Cadena, “SegMap: Segment-based mapping and +localization using data-driven descriptors,” The International Journal +of Robotics Research, vol. 39, no. 2-3, pp. 339–355, 2020. +[7] H. Yin, Y. Wang, L. Tang, X. Ding, S. Huang, and R. Xiong, +“3D LiDAR map compression for efficient localization on resource +constrained vehicles,” IEEE Transactions on Intelligent Transportation +Systems, vol. 22, no. 2, pp. 837–852, 2020. +[8] M.-F. Chang, W. Dong, J. Mangelson, M. Kaess, and S. Lucey, “Map +compressibility assessment for LiDAR registration,” in 2021 IEEE/RSJ +International Conference on Intelligent Robots and Systems (IROS), +pp. 5560–5567, 2021. +[9] M. Magnusson, The three-dimensional normal-distributions transform: +an efficient representation for registration, surface analysis, and loop +detection. PhD thesis, ¨Orebro University, 2009. +[10] T. Stoyanov, M. Magnusson, H. Andreasson, and A. J. Lilienthal, “Fast +and accurate scan registration through minimization of the distance +between compact 3D NDT representations,” The International Journal +of Robotics Research, vol. 31, no. 12, pp. 1377–1393, 2012. +[11] A. Zaganidis, M. Magnusson, T. Duckett, and G. Cielniak, “Semantic- +assisted 3D normal distributions transform for scan registration in +environments with limited structure,” in 2017 IEEE/RSJ International +Conference on Intelligent Robots and Systems (IROS), pp. 4064–4069, +2017. +[12] A. Zaganidis, L. Sun, T. Duckett, and G. Cielniak, “Integrating +deep semantic segmentation into 3-D point cloud registration,” IEEE +Robotics and Automation Letters, vol. 3, no. 4, pp. 2942–2949, 2018. +[13] C. Sammut and G. I. Webb, Encyclopedia of machine learning and +data mining. Springer Publishing Company, Incorporated, 2017. +[14] G. Bresson, Z. Alsayed, L. Yu, and S. Glaser, “Simultaneous localiza- +tion and mapping: A survey of current trends in autonomous driving,” +IEEE Transactions on Intelligent Vehicles, vol. 2, no. 3, pp. 194–220, +2017. +[15] P. J. Besl and N. D. McKay, “Method for registration of 3-D shapes,” in +Sensor Fusion IV: Control Paradigms and Data Structures, vol. 1611, +pp. 586–606, 1992. +[16] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “PointNet: Deep learning +on point sets for 3D classification and segmentation,” in Proceedings +of the IEEE/CVF Conference on Computer Vision and Pattern Recog- +nition (CVPR), pp. 652–660, 2017. +[17] S. Cho, C. Kim, J. Park, M. Sunwoo, and K. Jo, “Semantic point +cloud mapping of LiDAR based on probabilistic uncertainty modeling +for autonomous driving,” Sensors, vol. 20, no. 20, p. 5900, 2020. +[18] Q. Hu, B. Yang, L. Xie, S. Rosa, Y. Guo, Z. Wang, N. Trigoni, and +A. Markham, “RandLA-Net: Efficient semantic segmentation of large- +scale point clouds,” in Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition (CVPR), pp. 11108–11117, +2020. +[19] J. Zhang, X. Zhao, Z. Chen, and Z. Lu, “A review of deep learning- +based semantic segmentation for point cloud,” IEEE Access, vol. 7, +pp. 179118–179133, 2019. +[20] J. Kammerl, N. Blodow, R. B. Rusu, S. Gedikli, M. Beetz, and +E. Steinbach, “Real-time compression of point cloud streams,” in 2012 +IEEE International Conference on Robotics and Automation, pp. 778– +785, 2012. +[21] Velodyne, VLS-128 Alpha Puck, 2019. 63-9480 Rev-3 datasheet. +[22] Novatel, PwrPak7-E1, October 2020. D18496 Version 7 datasheet. +[23] J. Maanp¨a¨a, J. Taher, P. Manninen, L. Pakola, I. Melekhov, and +J. Hyypp¨a, “Multimodal end-to-end learning for autonomous steering +in adverse road and weather conditions,” in 2020 25th International +Conference on Pattern Recognition (ICPR), pp. 699–706, 2021. +[24] M. Quigley, B. Gerkey, K. Conley, J. Faust, T. Foote, J. Leibs, +E. Berger, R. Wheeler, and A. Ng, “ROS: an open-source robot +operating system,” in ICRA Workshop on Open Source Software in +Robotics, 2009. +[25] J. Behley, M. Garbade, A. Milioto, J. Quenzel, S. Behnke, C. Stach- +niss, and J. Gall, “SemanticKITTI: A dataset for semantic scene +understanding of LiDAR sequences,” in Proceedings of the IEEE/CVF +International Conference on Computer Vision (ICCV), pp. 9297–9307, +2019. +[26] Novatel, Inertial Explorer, February 2020. +D18034 Version 9 +brochure. +[27] K. +Koide, +“OpenMP-boosted +normal +distributions +transform.” +https://github.com/koide3/ndt_omp, 2017. +Referenced +26 January 2022. +[28] R. B. Rusu and S. Cousins, “3D is here: Point cloud library (PCL),” +in 2011 IEEE International Conference on Robotics and Automation, +2011. +[29] A. J. Trevor, S. Gedikli, R. B. Rusu, and H. I. Christensen, “Efficient +organized point cloud segmentation with connected components,” in +3rd Workshop on Semantic Perception, Mapping and Exploration +(SPME), 2013. +[30] M. A. Fischler and R. C. Bolles, “Random sample consensus: a +paradigm for model fitting with applications to image analysis and +automated cartography,” Communications of the ACM, vol. 24, no. 6, +pp. 381–395, 1981. +[31] N. Sarten, “A generic C++11 k-means clustering implementation.” +https://github.com/genbattle/dkm, 2015. Referenced 26 +January 2022. +[32] D. Arthur and S. Vassilvitskii, “k-means++: The advantages of careful +seeding,” in SODA ’07: Proceedings of the Eighteenth Annual ACM- +SIAM Symposium on Discrete Algorithms, pp. 1027–1035, 2007. +[33] S. J. Prince, Computer vision: models, learning, and inference. Cam- +bridge University Press, 2012. +[34] H. Tang, Z. Liu, S. Zhao, Y. Lin, J. Lin, H. Wang, and S. Han, “Search- +ing efficient 3D architectures with sparse point-voxel convolution,” in +European Conference on Computer Vision, pp. 685–702, 2020. +[35] X. Zhu, H. Zhou, T. Wang, F. Hong, Y. Ma, W. Li, H. Li, and +D. Lin, “Cylindrical and asymmetrical 3D convolution networks for +LiDAR segmentation,” in Proceedings of the IEEE/CVF Conference +on Computer Vision and Pattern Recognition (CVPR), pp. 9939–9948, +2021. + diff --git a/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/load_file.txt b/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f8874fe8804576d2d1e2b48c7e8086f3141e71d --- /dev/null +++ b/0dE2T4oBgHgl3EQfiQdl/content/tmp_files/load_file.txt @@ -0,0 +1,632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf,len=631 +page_content='Towards High-Definition Maps: a Framework Leveraging Semantic Segmentation to Improve NDT Map Compression and Descriptivity Petri Manninen1, Heikki Hyyti1, Ville Kyrki2, Jyri Maanp¨a¨a1, Josef Taher1 and Juha Hyypp¨a1 Abstract— High-Definition (HD) maps are needed for robust navigation of autonomous vehicles, limited by the on-board storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To solve this, we propose a novel frame- work, Environment-Aware Normal Distributions Transform (EA-NDT), that significantly improves compression of standard NDT map representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The compressed representation of EA-NDT is based on semantic-aided clustering of point clouds resulting in more optimal cells compared to grid cells of standard NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To evaluate EA-NDT, we present an open-source implementation that extracts planar and cylindrical primitive features from a point cloud and further divides them into smaller cells to represent the data as an EA-NDT HD map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We collected an open suburban environment dataset and evaluated EA-NDT HD map representation against the standard NDT representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Compared to the standard NDT, EA-NDT achieved consistently at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5× higher map compression while maintaining the same descriptive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, we showed that EA-NDT is capable of producing maps with significantly higher descriptivity score when using the same number of cells than the standard NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' INTRODUCTION The current development of mobile robots and the on- going competition for the crown of autonomous driving has increased the demand of accurate positioning services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Generally, Global Navigation Satellite System (GNSS) can be used to measure global position of a mobile robot but the accuracy of satellite navigation alone is typically around a few meters and because of signal obstruction the satellite signals may be unavailable [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Alternatively, global position can be solved by fitting the current sensor view into an existing georeferenced map, that can be computed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' with Simultaneous Localization and Mapping (SLAM) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, map-based technique provides a combined position and rotation estimate in contrast to a global position measured by GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Maps used in autonomous driving are typically called High-Definition (HD) maps [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Data compression of HD maps is of high importance within many applications that have limited computational resources and storage capac- ity [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, real-time localization requires com- pressed maps to ensure fast processing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' This work was supported by Academy of Finland, decisions 337656, 319011, 318437 and by Henry Ford foundation Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Manninen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hyyti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Maanp¨a¨a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Taher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hyypp¨a are with Department of Remote Sensing and Photogrammetry, Finnish Geospatial Research Institute (FGI), National Land Survey of Finland (NLS), 02150 Espoo, Finland petri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='manninen@nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi, heikki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='hyyti@nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi, jyri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='maanpaa@nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi, josef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='taher@nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi, juha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='hyyppa@nls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi 2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Kyrki is with School of Electrical Engineering, Aalto University, 02150 Espoo, Finland ville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='kyrki@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='fi Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1: An illustration of a point cloud (white) and corre- sponding EA-NDT HD map representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' EA-NDT cells are visualized with ellipsoids (mass within a standard de- viation) presenting building (yellow), fence (cyan), ground (purple), pole (blue), tree trunk (orange) and traffic sign (red) labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Since positioning in real-time with raw point clouds is infeasible, alternative methods have been developed to over- come the problem [6], [9]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' One promising approach is Normal Distributions transform (NDT) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' NDT compresses the three-dimensional point cloud data by dividing the cloud into equal sized cubical cells that are expressed by their mean and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To improve the scan registration, Semantic- assisted Normal Distributions Transform (SE-NDT) [12] expanded the original NDT with semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, both NDT and SE-NDT use a grid structure for the division of the point cloud, and therefore cannot find the fundamental geometrical structure of the environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' boundaries between object surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Consequently, this results in an NDT representation where part of the cells have a high variance in all three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Magnusson [9] also presented that the point cloud can alternatively be divided by K-means clustering [13] and that it improves the scan registration compared to using a grid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we address the aforementioned problem of sub-optimal point cloud division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We propose to solve the problem by leveraging semantic-aided clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We present a novel framework called Environment-Aware NDT (EA- NDT) (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1), which provides EA-NDT HD Map representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' EA-NDT HD Map is a compressed representation of a point cloud that is based on the leaf cell representation of the standard NDT, and therefore NDT scan registration technique is directly applicable with EA-NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, the standard NDT is referred as NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In contrast to the grid structure of NDT, EA-NDT leverages semantic information to cluster planar and cylindrical primitives of the ©2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='1109/IROS47612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='9982050 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='03956v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='RO] 10 Jan 2023 environment to provide a more optimal NDT cell division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In EA-NDT HD Map, each cell only consists of points that model the same basic geometrical shape such as a plane or a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, by adding understanding of semantic information in the scene, we can compute a map containing only stable objects that are useful for accurate localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The main contributions of this paper are: 1) A novel data-driven framework to compute NDT map representation without the grid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2) Demonstration of significantly improved data compres- sion compared to NDT representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3) An open-source implementation1 of the proposed EA- NDT is shared for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4) A registered dataset2 to evaluate the proposed EA-NDT on data collected with Velodyne VLS-128 LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The rest of the paper is organized as follows: The next section describes the related work in the fields of HD Maps, scan registration, SLAM, and point cloud semantic segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Section III, we formalize a pipeline architecture of the proposed framework to extract planar and cylindrical primitives of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The implementation details of our proof of concept solution are explained in Section IV together with an introduction to the data collection setup and preprocessed dataset, and the evaluation metrics used for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Section V, we compare the proposed EA- NDT to a map computed with NDT and show that EA-NDT provides a significant map compression while maintaining the same descriptive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finally, in Section VI we consider the advantages and disadvantages of EA-NDT and provide a discussion over the validity, reliability and gener- alizability of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' RELATED WORK HD Maps are one of the key techniques to enable au- tonomous driving [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Seif and Hu [4] have recognized three challenges to be solved with an HD map: the localization of the vehicle, reacting to events beyond sight, and driving according to the needs of the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we focus in the localization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' An HD Map can be computed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' with SLAM that is a well established and profoundly studied problem about how to align subsequent sensor measurements to incrementally compute a map of the surrounding environ- ment while simultaneously localizing the sensor [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In the review by Bresson et al [14], they found that an accuracy of 10 cm has been reported for the built maps but even an accuracy of 2 cm is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Data compression is a crucial challenge for HD maps in large environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For example, the well known Iterative Closest Point (ICP) [15] algorithm is infeasible in large point clouds due to the computational cost of finding closest corresponding points across a measurement and a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To improve the computational problems of ICP, Magnusson proposed Point-to-Distribution (P2D)-NDT in which the reference cloud is divided by a fixed sized 3D grid into 1https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='com/fgi_nls/public/hd-map 2https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='6796874 cells modelled by the mean and covariance of the points [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In (P2D)-NDT, each point in the registered scan is fitted to the cells within a local neighbourhood of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In addition to robust scan registration, NDT representation provides data compression together with faster registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Stoyanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' presented Distribution-to-Distribution (D2D)- NDT that further develops the P2D-NDT to likewise model the registered scan with normal distributions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Semantic information can enhance scan registration perfor- mance of NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For example, Semantic-assisted NDT (SE- NDT) [11], proposed by Zaganidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=', showed that the use of even two semantic labels (edges and planes) can improve the scan registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To further develop SE-NDT, Zaganidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' presented a complete semantic registration pipeline that uses a deep neural network for semantic segmentation of the point cloud [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' SE-NDT uses the 3D grid cell structure of NDT but models each semantic label separately to utilize the division of similar entities in the registration task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For semantic segmentation, SE-NDT uses PointNet [16] that is a pioneering solution of a point cloud segmentation network that consumes raw point cloud data without voxelization or rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' proposed that the uncertainty of semantic information could also be used in the registration task [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Semantic information can be utilized further than was proposed in the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we propose to replace the aforementioned grid division by leveraging semantic-aided clustering that finds planar and cylindrical structures of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For semantic segmentation, we use Random sampling and an effective Local feature Aggregator-Net (RandLA-Net) [18] that presents a new local feature aggregation module to support random sampling that was found a suitable technique for semantic segmentation of large scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' They have reported up to 200× faster processing capacity compared to the existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' A further review of semantic segmentation of point cloud data is available e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' ENVIRONMENT-AWARE NDT Here we propose a framework, called Environment-Aware NDT (EA-NDT), to divide a semantically segmented point cloud into NDT cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The proposed framework is a straight pipeline process consisting of 4 stages (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3) that step- by-step divide the input point cloud into cells which are ultimately represented as an NDT map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The input of the pipeline is a Registered Point Cloud, which is processed in the following order by stages called Semantic Seg- mentation, Instance Clustering, Primitive Extraction and Cell Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finally, the output of the pipeline is an environment-aware NDT-based HD map representation, called EA-NDT HD Map, which stores the found cells using NDT representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In the Registered Point Cloud each 3D point has X, Y, Z Cartesian coordinate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' ETRS-TM35FIN, ECEF) and an intensity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Semantic Segmentation appends semantic information for each point in the cloud to enable further clustering of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we used road, sidewalk, parking, building, fence, pole, traffic sign, and tree trunk labels to demonstrate the framework but also other labels could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Instance Clustering divides each semantic segment into instances that are spatially sep- arated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Primitive Extraction divides each instance into predefined primitives that can be modeled with an unimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we have defined planar and cylindrical primitives but the framework could be extended to support new types of primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, large primitives such as trees can not be modelled well with an uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Therefore, Cell Clustering further divides each primitive into cells of approximately equal size while minimizing the number of used cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Ultimately, EA- NDT HD Map is presented as an octree [20] that in this work stores the point counter, point sum and upper diagonal of the covariance matrix for each cell but other attributes such as semantic segment, instance cluster or primitive type could be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' METHODS AND EXPERIMENTS To demonstrate the proposed framework to build an EA- NDT HD Map, we used a dataset collected with Velodyne VLS-128 Alpha Puck [21] LiDAR 7th of September 2020 in a suburban environment in the area of K¨apyl¨a in Helsinki, the capital of Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The environment in the dataset consists of a straight two-way asphalt street, called Pohjolankatu, which starts from a larger controlled intersection at the crossing of Tuusulanv¨ayl¨a (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='213326° N, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='942908° E in WGS84) and passes by three smaller uncontrolled intersections until the crossing of Metsolantie (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='215537° N, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='950065° E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' It is a typical suburban street with tram lines, sidewalks, small buildings, traffic signs, light poles, and cars parked on both sides of the streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To collect a reference trajectory and to synchronize the LiDAR measurements, we have used a No- vatel PwrPak7-E1 GNSS Inertial Navigation System (INS) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The sensors were installed on a Ford Mondeo Hybrid research platform named Autonomous Research Vehicle Ob- servatory (ARVO) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The sensors were interfaced through Robotic Operation System (ROS) [24] version Kinetic Kame and the sensor measurements were saved in rosbag format for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Preprocessed Dataset Our open preprocessed dataset2, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, consists of a two-way asphalt paved street with a tram line to both directions and sidewalks in both sides of the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The length of the dataset trajectory is around 640 m and it has Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2: The complete dataset visualized with semantic labels in different colors: road (magenta), sidewalk (violet), park- ing (pink), terrain (green), buildings (yellow), fence (light brown), tree trunk (brown), traffic sign (red), pole (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' TABLE I: RandLA-Net classified dataset label proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Semantic label No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' of points % of all % of used Ground 14,052,836 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='8 Building 7,650,980 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='9 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='7 Tree trunk 3,560,910 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='9 Fence 2,120,849 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='7 Pole 193,516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='7 Traffic sign 82,680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='3 Labels used here 27,661,771 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='4 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='0 Others 12,799,904 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='6 Total 40,461,675 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='0 in total more than 40 million points from which 28 million are used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' All the intersections together have a plenty of traffic signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The sidewalks are separated from the road by a row of tall planted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The dataset contains nearly 30 buildings that are mostly wooden and there are several fences between the houses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Our dataset includes all the semantic labels classified by RandLA-Net [18] but in this work we have used only road, sidewalk, parking, building, fence, tree trunk, traffic sign, and pole labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, road, sidewalk, and parking labels were reassigned into a common ground label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The proportion and the number of the points of each label are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Half of the used points consist of ground and roughly a fourth represent buildings whereas poles and traffic signs together represent only 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Tree trunks and fences together represent a fifth of the used points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The preprocessing of the data consists of three steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' semantic segmentation, scan registration, and data filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In semantic segmentation of scans, we used a RandLA-Net model pre-trained with SemanticKITTI dataset which was collected with Velodyne HDL-64 LiDAR [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Instead, we used VLS-128 which has a longer range and 128 laser beams instead of 64 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Also, VLS-128 has a wider field of view (FOV) in vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Consequently, the measurements outside of the vertical FOV of HDL-64 were constantly misclassified so only the measurements within HDL-64 FOV were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' RandLA-Net outputs a probability estimate vector of labels for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we call it as label probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In scan registration, the motion deformation of each scan was first fixed according to a GNSS INS trajectory post- processed with Novatel Inertial Explorer [26], after which P2D-NDT implementation [27] with 1 m grid cell size was used for registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In registration, a local map of 5 last keyframes was used as a target cloud and a motion threshold of 10 cm was used to add a new keyframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Grid cells containing points from a single ring of the LiDAR were ignored in the registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, points that were con- sidered possibly unreliable (vehicles, bicycles, pedestrians and vegetation) or further than 50 m away from the LiDAR, were ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' After the scan registration, the dense Registered Point Cloud was voxel filtered to average X, Y, Z position and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3: A visualization of the proposed EA-NDT processing pipeline that is based on the framework in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The input is a semantically segmented point cloud and the intermediate phases before EA-NDT HD Map are instances, primitives and cells, in which the entities are separated by color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The color mapping of semantic information is explained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' label probabilities of each 1 cm voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' To smoothen the semantic segmentation, the label probabilities of each point was averaged within a radius of 5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The Implementation A method1, based on the framework presented in Sec- tion III, was implemented in C++14 on top of ROS Noetic Ninjemys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The main functionality of the implementation uses existing functions and classes of Point Cloud Library (PCL) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The implemented processing pipeline is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The 1st stage of the framework, Semantic Segmen- tation, is explained in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Therefore, our dataset already includes the semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Instance clustering was implemented with Euclidean region growing algorithm [29] to divide each semantic seg- ment into spatially separate instances shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In general, we require a distance threshold of 30 cm between the instances and a minimum of 10 points per instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For ground label we require a distance threshold of 50 cm between the instances and a minimum of 3000 points per instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In our dataset, there is a significant amount of outliers and reflected points below the ground plane that are undesired in a map, Instance clustering is used to filter those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Primitive Extraction, tree trunk and pole instances are modeled as individual cylindrical primitives, and traffic sign instances as individual planar primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Both primitive types are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For other semantic labels, planar primitives were extracted by Random Sample Consensus (RANSAC) [30] based normal plane fitting algorithm after subsampling the instance with an averaging 10 cm voxel grid and estimating the point normals for each remaining point from 26 nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For building and fence instances, a normal distance weight of π/4 and a distance threshold of 15 cm was used for plane fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For ground instances, the procedure differs slightly: 1) an existing implementation [31] of K-means++ algorithm [32] was used to divide the ground instances into primitives with an area of approximately 100 m² (The initialization of number of K-means clusters is explained later in this section in Cell Clustering), after which 2) the plane fitting was performed for each primitive with a 30 cm distance threshold for a coarse noise filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Cell Clustering, primitives are divided into cells (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3) with K-means++ algorithm for which the number of clusters NL = ⌈fLnL gL⌉ (1) is initialized for each label L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In (1), ⌈·⌉ is the ceiling operator, nL is either nα for cylindrical primitives (tree trunk and pole) or nβ for planar primitives (ground, building, fence, and traffic sign): nα = lα � sc and nβ = Aβ � sc 2, (2) TABLE II: The final values of the scaling parameters Semantic label (L) fL gL Ground 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='083 Building 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='137 Tree trunk 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='318 Fence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='248 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='788 Pole 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='687 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='315 Traffic sign 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='317 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4: The number of cells Nc after fitting EA-NDT with NDT shown w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' cell size sc, color indicates the method, line style the label, and green background the fitted range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5: The complete map descriptivity score Sd compared w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' number of cells Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The violet line depicts the com- putation of the NDT compression efficiency η for each Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' where lα is the length of a cylindrical primitive and Aβ is the number of points remaining after projecting the planar primitive into the eigenspace found by principal component analysis (PCA) and filtering with a 10 cm voxel grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Additionally, after clustering cells in ground, a plane fitting with a 15 cm threshold is performed for each cell for a finer noise filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In (1), scaling parameters fL and gL, shown in Table II, were manually fitted for each L over 6 iterations starting from fL0 = 1 and gL0 = 1 until the number of cells Nc for EA-NDT (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4) was sufficiently close to NDT with cell size sc < 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Despite the cell size, each primitive is required to have at least one cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4 reveals how this sets a lower boundary for Nc with larger cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finally, all the computed cells were stored into an octree structure [20] that represents the EA-NDT HD Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Each leaf cell in the octree stores a point counter, point sum and upper diagonal of the covariance matrix for the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We require a minimum of 6 points for a leaf cell to be modeled reliably with a normal distribution, hence cells with less points are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The octree implementation in PCL requires a minimum leaf cell size parameter, we used sc/4 to make it sufficiently smaller compared to the required cell size of EA-NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Evaluation Here, a descriptivity score Sd, in which a higher score denotes higher similarity, is defined to evaluate how well the map models the raw point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' It is derived using a density function of a multivariate normal distribution [33], which is defined for each 3D point xi and jth NDT cell as fj(xi) = 1 � (2π)k|Σj| e(− 1 2 (xi−µj)TΣ−1 j (xi−µj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' (3) In (3), µj is the mean vector of a distribution with a covariance matrix Σj for jth NDT cell, | · | is the determinant operator and k = 3 describes dimension of the multivariate distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Restricting to the local neighborhood of each Np point, the descriptivity score Sd is an average density of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6: An alternative comparison of the complete map descriptivity score Sd w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' cell size sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The violet line depicts the computation of the descriptivity ratio Rd for each sc and green background emphasizes the applicable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' best fitting NDT cells: Sd = 1 Np Np � i=1 max fj(xi) ∥xi−µj∥2 ≤ 2sc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' (4) The maximum distance inside a grid cell is √ 3sc, and therefore the radius of 2sc was considered large enough to contain the highest fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We have defined two ratios, descriptivity ratio Rd requir- ing sEA c = sNDT c (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6) and data compression ratio Rc: Rd = SEA d � SNDT d and Rc = Npσp � Ncσc, (5) where superscripts EA and NDT stand for EA-NDT and NDT, respectively, σp and σc are the data size of the point and the cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Using (5) while requiring SEA d = SNDT d , we define an NDT compression efficiency (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5) η = RNDT c � REA c = N NDT c � N EA c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' (6) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' RESULTS In this section, we evaluate the quality between EA- NDT and NDT map representations and demonstrate the data compression of EA-NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The performance of both methods was evaluated by stepping the cell size from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2 m to 10 m with 30 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Note that the computational time increases exponentially with decreasing cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The lower boundary of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2 m was selected since it could still be computed overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Similarly, the upper boundary of 10 m was considered large enough for this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6–8, a practically applicable range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='0 m, based on previous work [9], is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The evaluation of complete map representation on the dataset described in Section IV-A is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' It shows that EA-NDT map representation provides a higher descriptivity score for any number of cells (note that the min- imum number of cells is limited for EA-NDT as explained in Section IV-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, typically the results of NDT are compared as a function of the cell size, and therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 7: Comparison of descriptivity score Sd of each label w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' cell size sc, line style indicates the method, color the label, and green background the applicable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8: Both NDT compression efficiency η (above) and descriptivity ratio Rd (below) of the proposed method are visualized for the complete map and all labels w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='t NDT cell size sc, green background emphasizes the applicable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6 we present an alternative comparison for which the number of cells in EA-NDT were fitted with NDT as ex- plained in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Likewise, descriptivity of EA-NDT outperforms NDT with any cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6, one can note that both plots are equally capable of showing the differences between the compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In general, it can be noticed that the descriptivity score of NDT approaches EA-NDT with smaller cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, this is an expected phenomenon of grid cell division;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' the probability of multiple objects to be associated within one cell decreases with smaller cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In Table I in Section IV-B, it is shown that around 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 % of the data consists of points labelled as ground or building, which reflects a similar proportion to the number of cells shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The descriptivity score of the complete map is dominated by these abundant labels leaving the effect of other labels imperceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 7, we present an equivalent descriptivity score comparison separated for each label, which in case of NDT is equivalent to SE-NDT representation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Similarly to the comparison of complete map representation, EA-NDT descriptivity score of each separate label is higher except for tree trunks with 3 – 6 m cells, for which the descriptivity score equals with NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The low descriptivity of EA-NDT is most likely caused by the use of K-means clustering because if a cluster is large or a diameter of a trunk is small, a single cluster can contain points within the entire circumference of the trunk resulting in a non-Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, the use of HDL- 64 vertical FOV limits the height of tree trunks and poles in to a range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 3 m (as explained in Section IV-A) and when the required cell size exceeds half of that height, a large portion of the primitives is assigned into a one cluster instead of two causing the observed discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, because of scaling the number of cells, the effect does not appear exactly with the expected cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' With tree trunk and pole labels it is also observable that the descriptivity does not decrease with the largest cells, because the size of the primitive limits the cluster size from increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The descriptivity ratio between EA-NDT and NDT is shown in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In general, the descriptivity ratio increases for all the labels towards the greater cell sizes, tree trunk and pole labels make an exception that was already covered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Within the applicable cell range, the improvement in descriptivity ratio is more constant for all labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For the complete map with 2 m cells, the descriptivity is 2× higher compared to NDT and for 10 m cells the descriptivity is 20× higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Especially, building and fence labels show relatively higher descriptivity scores, which suggests that the plane extraction is advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Map compression is a direct consequence of EA-NDT’s higher descriptivity scores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' EA-NDT achieves the same de- scriptivity with a larger cell size which means a smaller num- ber of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The NDT compression efficiency η, visualized in the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8, was used to compare compression of EA-NDT with NDT (note that η, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5, can be computed only when a corresponding score exists for both methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For the complete map representation, EA-NDT provides 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='75× better compression within the whole examined range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The compression of the complete EA-NDT is mainly defined by the ground label, which is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5× better within the whole range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' EA-NDT’s compression of traffic sign and pole labels is more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='1× higher than NDT for the smallest cells but drops steeply towards greater cell sizes, though, remaining higher compression even for the largest cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For building and fence labels, the compression is more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='2× higher around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 m cell size, for smaller and larger cells the NDT compression efficiency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' This suggests that EA-NDT’s technique of modeling the planes and excluding the other points is beneficial until cell size of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 m but with smaller cells, NDT reaches the difference by modeling the excluded points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finally, we can state that the complete map representation of EA-NDT achieves the highest compression improvement around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='7 m cell size which is also within the applicable cell size range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' DISCUSSION The proposed EA-NDT achieves 1) at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5× higher compression, and 2) always a higher descriptivity score with the same number of cells compared to NDT as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' For separately tested semantic labels, EA-NDT achieves 1) always a higher compression, and 2) a higher descriptivity score in the applicable cell size range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, we suggest to use cell sizes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='0 m for EA-NDT in a suburban environment since our results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8) indicate that this range provides better compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Due to the semantic-aided instance clustering and primi- tive extraction, the proposed EA-NDT is able to find the most significant planar and cylindrical primitives in the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' NDT representation is especially informative within planar and thin cylindrical structures that can be modeled with a unimodal distribution, and therefore, EA-NDT is able to model the environment more optimally compared to NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, the use of semantic information enables selection of the stable objects that should be modeled in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finally, K-means clustering of the primitives ensures data efficient placement of cells where they are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The advantage of EA-NDT is a result of improved point cloud division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Therefore, the advantage is prominent in small objects such as poles or complicated structures such as buildings or fences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In the ground plane, the advantage is less evident because it is in any case a one large plane and the benefit can be almost completely explained by the removal of outliers and by the efficiency gained from clustering the ground plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Finding planar primitives in building and fence instances removes some points which are not modeled by EA-NDT HD Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' As shown in our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 8, that is beneficial for compression with cell sizes above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 m but for smaller cell sizes it could be beneficial to model those points with additional NDT cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, as shown in this work, the described effect is not significant within the range of the suggested cell sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, the suggested correction should be justified only if it improves also the performance of scan registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The classification accuracy of the pre-trained RandLA-Net (see Section IV-A) was a limiting factor for the quality of se- mantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The misclassification increases the total number of cells when overlapping cells of different semantic labels model the same object (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1), which reduces the compression of EA-NDT representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In future work, the classification accuracy of the semantic segmentation could be improved by using a more advanced model [34], [35] and by retraining the model for the used LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The proposed EA-NDT was tested in a suburban envi- ronment that consist of 1) a flat ground, buildings, fences, and traffic signs, which are modeled as planar surfaces, and 2) poles and tree trunks, which are modeled as cylindrical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The tested environment contains enough samples of all the tested semantic labels to demonstrate that EA-NDT is able to compress the data more than NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, our tests did not concern vegetation, tree canopies, water, significant height variations, nor high rise buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In the future, a larger variety of environments should be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' EA-NDT provides map compression within the tested semantic labels in environments where 1) the used semantic labels exist in the environment, 2) the reliability of semantic segmentation is high enough, and 3) instances are separated by sufficient distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In order to use the proposed EA-NDT, the following assumptions need to hold: 1) ground, buildings, fences, and traffic signs must be composed of planar surfaces, and 2) tree trunks and poles need to be cylindrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In future work, for other semantic labels, the type of primitive would need to be defined according to the properties of that label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Semantic information is a powerful tool and a key enabler of HD Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In this work, we have shown that semantic information enables separate processes for each semantic label which results into more optimal clustering of point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Furthermore, in previous works semantic infor- mation has been used to improve positioning [11], [12], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, semantic information enables the removal of unwanted dynamic objects from the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' In future work, the use of semantic information opens a possibility to study the positioning accuracy and reliability of different object types over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' That could be especially useful when navigating in constantly changing environments such as arctic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' This work was outlined on evaluating compression and descriptivity properties of EA-NDT HD Map, and therefore, the positioning performance of the proposed framework remains an open question for the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Although, the positioning was not evaluated, the well established scan registration and cell representation of NDT is integrated into positioning of EA-NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, the data compression of an HD map is a desired property of any mobile robot application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Another open question is how the proposed EA-NDT HD map can be efficiently updated with new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Also, currently the computation of EA-NDT is very slow and the computational optimization is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' CONCLUSIONS In this work, we proposed EA-NDT, that is a novel framework to compute a compressed map representation based on NDT formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The fundamental concept of EA-NDT is semantic-aided clustering to find planar and cylindrical primitive features of a point cloud to model them as planar or elongated normal distributions in a 3D space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We showed that compared to NDT, the data-driven ap- proach of EA-NDT achieves consistently at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5× higher map descriptivity score, and therefore enables a significant map compression without deteriorating the descriptive ca- pability of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' The best compression in comparison to NDT is obtained within cell sizes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='5 – 2 m, which is an applicable range for real-time positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Moreover, the results show that compared to NDT, the representation achieves a higher data compression within all the tested semantic labels, that is a desired property for mobile robots such as autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' When data compression is a required property of an HD map, we recommend the use of EA-NDT instead of NDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Based on the results of this work, it seems likely that the positioning accuracy using EA-NDT maps exceeds that of standard NDT maps of same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' However, this warrants future studies because there are several interacting factors such as potentially varying contribution of different semantic labels to the positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' ACKNOWLEDGMENT In addition, the authors would like to thank Paula Litkey and Eero Ahokas from FGI for data management and col- lection and Antero Kukko and Harri Kaartinen from FGI for assistance and advices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' We would also like to thank Leo Pakola for participation in the research vehicle development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zaidi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Suddle, “Global navigation satellite systems: a survey,” in 2006 International Conference on Advances in Space Technologies, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 84–87, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, “Pseudolite applications in positioning and navigation: Progress and problems,” Journal of Global Positioning Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 03, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 48–56, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cadena, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Carlone, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Carrillo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Latif, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Scaramuzza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Neira, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Reid, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Leonard, “Past, present, and future of simultaneous localization and mapping: Toward the robust-perception age,” IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1309–1332, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Seif and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hu, “Autonomous driving in the iCity—HD maps as a key challenge of the automotive industry,” Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 159–162, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhang, “High definition map for automated driving: Overview and analysis,” The Journal of Navigation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 73, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 324–341, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Dub´e, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cramariuc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Dugas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Sommer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Dymczyk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Nieto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Siegwart, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cadena, “SegMap: Segment-based mapping and localization using data-driven descriptors,” The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2-3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 339–355, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Ding, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Huang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Xiong, “3D LiDAR map compression for efficient localization on resource constrained vehicles,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 837–852, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Dong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Mangelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Kaess, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lucey, “Map compressibility assessment for LiDAR registration,” in 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5560–5567, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Magnusson, The three-dimensional normal-distributions transform: an efficient representation for registration, surface analysis, and loop detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' PhD thesis, ¨Orebro University, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Stoyanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Magnusson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Andreasson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lilienthal, “Fast and accurate scan registration through minimization of the distance between compact 3D NDT representations,” The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1377–1393, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zaganidis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Magnusson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Duckett, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cielniak, “Semantic- assisted 3D normal distributions transform for scan registration in environments with limited structure,” in 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4064–4069, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zaganidis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Duckett, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cielniak, “Integrating deep semantic segmentation into 3-D point cloud registration,” IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2942–2949, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Sammut and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Webb, Encyclopedia of machine learning and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Springer Publishing Company, Incorporated, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Bresson, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Alsayed, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Yu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Glaser, “Simultaneous localiza- tion and mapping: A survey of current trends in autonomous driving,” IEEE Transactions on Intelligent Vehicles, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 194–220, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Besl and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' McKay, “Method for registration of 3-D shapes,” in Sensor Fusion IV: Control Paradigms and Data Structures, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1611, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 586–606, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Qi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Su, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Mo, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Guibas, “PointNet: Deep learning on point sets for 3D classification and segmentation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recog- nition (CVPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 652–660, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Park, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Sunwoo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Jo, “Semantic point cloud mapping of LiDAR based on probabilistic uncertainty modeling for autonomous driving,” Sensors, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 5900, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [18] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Rosa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Trigoni, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Markham, “RandLA-Net: Efficient semantic segmentation of large- scale point clouds,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 11108–11117, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lu, “A review of deep learning- based semantic segmentation for point cloud,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 179118–179133, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Kammerl, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Blodow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Rusu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Gedikli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Beetz, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Steinbach, “Real-time compression of point cloud streams,” in 2012 IEEE International Conference on Robotics and Automation, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 778– 785, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [21] Velodyne, VLS-128 Alpha Puck, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 63-9480 Rev-3 datasheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [22] Novatel, PwrPak7-E1, October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' D18496 Version 7 datasheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Maanp¨a¨a, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Taher, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Manninen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Pakola, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Melekhov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hyypp¨a, “Multimodal end-to-end learning for autonomous steering in adverse road and weather conditions,” in 2020 25th International Conference on Pattern Recognition (ICPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 699–706, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Quigley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Gerkey, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Conley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Faust, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Foote, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Leibs, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Berger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wheeler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Ng, “ROS: an open-source robot operating system,” in ICRA Workshop on Open Source Software in Robotics, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Behley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Garbade, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Milioto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Quenzel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Behnke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Stach- niss, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Gall, “SemanticKITTI: A dataset for semantic scene understanding of LiDAR sequences,” in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 9297–9307, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [26] Novatel, Inertial Explorer, February 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' D18034 Version 9 brochure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Koide, “OpenMP-boosted normal distributions transform.” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='com/koide3/ndt_omp, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Referenced 26 January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [28] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Rusu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cousins, “3D is here: Point cloud library (PCL),” in 2011 IEEE International Conference on Robotics and Automation, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Trevor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Gedikli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Rusu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Christensen, “Efficient organized point cloud segmentation with connected components,” in 3rd Workshop on Semantic Perception, Mapping and Exploration (SPME), 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Fischler and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 381–395, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [31] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Sarten, “A generic C++11 k-means clustering implementation.” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content='com/genbattle/dkm, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Referenced 26 January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Arthur and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Vassilvitskii, “k-means++: The advantages of careful seeding,” in SODA ’07: Proceedings of the Eighteenth Annual ACM- SIAM Symposium on Discrete Algorithms, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 1027–1035, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Prince, Computer vision: models, learning, and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Cam- bridge University Press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Tang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Han, “Search- ing efficient 3D architectures with sparse point-voxel convolution,” in European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 685–702, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' [35] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Ma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Li, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' Lin, “Cylindrical and asymmetrical 3D convolution networks for LiDAR segmentation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} +page_content=' 9939–9948, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQfiQdl/content/2301.03956v1.pdf'} diff --git a/1tE0T4oBgHgl3EQfuQHv/vector_store/index.faiss b/1tE0T4oBgHgl3EQfuQHv/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9bae51946c352b62cfa1005c6523ddcf96be5c31 --- /dev/null +++ b/1tE0T4oBgHgl3EQfuQHv/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:55829734e8880a799b280ace52c650cdedc27f7b438f3845f5aaa0c9debae349 +size 5046317 diff --git a/1tE0T4oBgHgl3EQfuQHv/vector_store/index.pkl b/1tE0T4oBgHgl3EQfuQHv/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..796d7431d8e3ec632d514ce1d7533e99a86e1d15 --- /dev/null +++ b/1tE0T4oBgHgl3EQfuQHv/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6fc48272f6b37ad96892cfa6a1864234586ad656e0c04e27e718cc22a6971620 +size 184595 diff --git a/29E1T4oBgHgl3EQflwQn/content/2301.03288v1.pdf b/29E1T4oBgHgl3EQflwQn/content/2301.03288v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..779f169097ba3045c3b13114e0a3f0b34c0ddfaf --- /dev/null +++ b/29E1T4oBgHgl3EQflwQn/content/2301.03288v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6b14abc3abf8f8cc6c7904c344fc4dfab4c52aa2d579abb8dc17a5537a7b24e +size 1933910 diff --git a/29E1T4oBgHgl3EQflwQn/vector_store/index.pkl b/29E1T4oBgHgl3EQflwQn/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f3b1e730ecdd242d7ca6666ca57ad703325d9d97 --- /dev/null +++ b/29E1T4oBgHgl3EQflwQn/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1a81d5dd18fec218494c6771ea9f48608b837c25f53460693541307cf13bf2fb +size 104835 diff --git a/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/2301.01446v1.pdf.txt b/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/2301.01446v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e02e2bd360822734b4be56c7a59a874edb479fc0 --- /dev/null +++ b/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/2301.01446v1.pdf.txt @@ -0,0 +1,691 @@ +Radio Frequency Fingerprints Extraction for +LTE-V2X: A Channel Estimation Based Methodology +Tianshu Chen∗, Hong Shen∗, Aiqun Hu∗†, Weihang He‡, Jie Xu‡, Hongxing Hu§ +∗National Mobile Communications Research Laboratory, Southeast University, Nanjing, China +†The Purple Mountain Laboratories for Network and Communication Security, Nanjing, China +‡School of Cyber Science and Engineering, Southeast University, Nanjing, China +§China Automotive Innovation Corporation, Nanjing, China +Email: {iamtianshu, shhseu, aqhu, 220205165, 220205095}@seu.edu.cn, huhongxing@t3caic.com +Abstract—The vehicular-to-everything (V2X) technology has +recently drawn a number of attentions from both academic and +industrial areas. However, the openness of the wireless communi- +cation system makes it more vulnerable to identity impersonation +and information tampering. How to employ the powerful radio +frequency fingerprint (RFF) identification technology in V2X +systems turns out to be a vital and also challenging task. In +this paper, we propose a novel RFF extraction method for Long +Term Evolution-V2X (LTE-V2X) systems. In order to conquer the +difficulty of extracting transmitter RFF in the presence of wireless +channel and receiver noise, we first estimate the wireless channel +which excludes the RFF. Then, we remove the impact of the +wireless channel based on the channel estimate and obtain initial +RFF features. Finally, we conduct RFF denoising to enhance the +quality of the initial RFF. Simulation and experiment results both +demonstrate that our proposed RFF extraction scheme achieves +a high identification accuracy. Furthermore, the performance is +also robust to the vehicle speed. +Index Terms—Vehicular-to-everything (V2X), radio frequency +fingerprint (RFF), device identification, channel estimation, RFF +denoising +I. INTRODUCTION +Vehicular-to-everything (V2X) has become a promising +technique for intelligent transportation and autonomous driv- +ing. In particular, the cellular-V2X (C-V2X) has been widely +acknowledged as a key V2X communication standard due to +its superior performance [1], [2]. +Since V2X relies on wireless transmission, the information +is easy to be eavesdropped, forged or tampered with, which +imposes great challenges on the safety of vehicles, pedestrians +and road infrastructures in the V2X communication network +[3]. To deal with the security threats faced by wireless +communications, there are usually two widely used authen- +tication strategies: key-based cryptographic authentication and +physical layer security-based non-cryptographic authentication +[4]. The cryptographic authentication technology needs to +© 2022 IEEE. Personal use of this material is permitted. Permission from +IEEE must be obtained for all other uses, in any current or future media, +including reprinting/republishing this material for advertising or promotional +purposes, creating new collective works, for resale or redistribution to servers +or lists, or reuse of any copyrighted component of this work in other works. +distribute and manage abundant communication keys, which +occupies computing resources and leads to additional overhead +and delays. Moreover, with the rapid development of comput- +ing capability of the computers, especially the emergence of +quantum computers, traditional cryptography technologies are +more vulnerable to brute-force attacks [5]. On the contrary, the +physical layer security based authentication has lower com- +plexity and network overhead with lower latency compared +to traditional cryptography-based authentication methods, and +can achieve non-perceptual authentication without third-party +facilities. One typical example is the radio frequency fin- +gerprint (RFF) based authentication, which fully exploits the +hardware differences between any two devices. Since the +hardware characteristic of each device is unique and difficult +to clone, the RFF based authentication can better resist the +identity attacks and spoofing [6]. +In literature, a variety of RFF extraction and identification +methods have been advocated. Early works mainly focus on +the characteristics of transient signals, such as instantaneous +amplitude, frequency, and phase responses [7]. Concerning +the steady-state signal, such as preamble signals, researchers +consider extracting the RFF features including I/Q offset +[8], power spectral density [9], differential constellation trace +figure [10]. Furthermore, some universal RFF extraction meth- +ods which are independent of data, channel or modulation +modes have also been studied. Concretely, Shen et al. [11] +constructed channel independent spectrogram and utilized data +augmentation for RFF extraction and identification of Lora +devices, which achieves good performance under different +channel conditions. Alternatively, Yang et al. [12] used random +data segments to extract the tap coefficients of the least mean +square (LMS) adaptive filter as data independent RFF. Sun et +al. [13] verified the locality and inhomogeneity of the RFF +distribution with the analysis in the cepstral domain, which +yields modulation mode independent RFF. +The aforementioned works mainly consider the RFF extrac- +tion for low mobility and narrowband systems. However, for +the V2X system, the channel typically varies fast due to the +high mobility vehicles. In addition, the V2X signal usually +arXiv:2301.01446v1 [eess.SP] 4 Jan 2023 + +Signal preprocessing +RFF feature extraction +and denoising +Device identification +Digital baseband +signal +Mixer +up-conversion +Transmitter RFF model +RF front-end power amplifier +DAC +Baseband low-pass filter +TX antenna +OBU +OBU +RSU +V2P +V2I +Receiver +Channel estimation +and equalization +RFF identification +and access system +I/Q DC offset +Non-linearity +Gain imbalance +and phase deviation +Frequency response +deviation +Fig. 1. LTE-V2X RFF extraction and identification system framework and RFF model at the transmitter. +has a large bandwidth which is more vulnerable to multipath +environment. Therefore, the current RFF extraction methods +for narrowband systems such as ZigBee and Lora cannot be +directly applied for the V2X system because they do not +take into account the impact of multipath and time-varying +channels. +In this work, we propose a channel estimation based RFF +extraction method for Long Term Evolution-V2X (LTE-V2X) +systems, which, to the best of our knowledge, has not been +investigated in existing works. Specifically, we first estimate +the experienced wireless channel using an improved least +square (LS) channel estimation method. Then, we perform +channel equalization based on the channel estimate to obtain +channel dependent RFF. The RFF quality is further enhanced +via conducting time-domain denoising. It is worthwhile noting +that the developed method eliminates the effect of the channel +and the noise on the RFF with low implementation complex- +ity, and can be extended to various broadband multi-carrier +wireless communication systems. +This paper is organized as follows. Section II introduces the +system model and signal preprocessing. Section III presents +the details of the proposed RFF extraction methodology based +on wireless channel estimation. Section IV evaluates the +performance of the proposed RFF extraction method through +simulations and experiments. Section V concludes this work. +II. SYSTEM MODEL AND SIGNAL PREPROCESSING +A. System Model +Fig. 1 demonstrates the framework of the considered LTE- +V2X RFF extraction and identification system together with +the RFF model at the transmitter. More concretely, one V2X +terminal, e.g., on board unit (OBU) or road side unit (RSU), +first transmits data to other devices, where the transmitted +signal includes the RFF of the transmitter. Then, the receiver +preprocesses the received signal which consists of converting +the RF signal to the baseband signal and performing time- +frequency synchronization. Subsequently, the RFF features are +extracted based on the synchronized signal, where the effects +of the wireless channel and the noise on the RFF need to be +mitigated. Finally, the device identification is performed using +the extracted RFF features. +It is necessary to note that the considered RFF refers to all +the characteristics of the circuits at the transmitter, which, as +shown in Fig. 1, include the I/Q DC offsets of the digital-to- +analog converter (DAC), the frequency response deviation of +the filter, the gain imbalance and the carrier phase quadrature +deviation of the mixer, and the non-linearity of the power +amplifier [14]. +B. LTE-V2X PSBCH +We adopt the physical sidelink broadcast channel (PSBCH) +in LTE-V2X systems for RFF extraction. According to [15], +PSBCH is transmitted every 160 ms occupying the central 6 + +PSBCH +PSBCH +PSSS +PSSS +DMRS +PSBCH +DMRS +DMRS +PSBCH +PSBCH +PSBCH +SSSS +SSSS +GUARD +1 2 3 4 5 6 7 8 9 10 11 12 13 14 time +1 ms +frequency +6 RBs +Fig. 2. LTE-V2X PSBCH format. +resource blocks (RBs), i.e., 72 subcarriers and 14 single-carrier +frequency division multiple access (SC-FDMA) symbols. +The detailed format of PSBCH is shown in Fig. 2, where +primary sidelink synchronization signal (PSSS), secondary +sidelink synchronization signal (SSSS), and demodulation +reference signal (DMRS) all depend on the currently used +sidelink synchronization signal (SLSS) ID. Since the SLSS +ID can be estimated [15], we can readily obtain ideal PSSS, +SSSS, and DMRS at the receiver which are used for extracting +transmitter RFF. +C. Signal Preprocessing +In order to ensure the stability of the extracted RFF, the +signal preprocessing procedure includes time synchronization +and carrier frequency offset (CFO) estimation and compensa- +tion after the received signal is down-converted from the RF +band to the baseband. +The time synchronization is realized by utilizing two identi- +cal training symbols, e.g., two repeated PSSS or SSSS symbols +in LTE-V2X PSBCH, and the cross-correlation between the +received signal r(n) and the training signal x(n) as +P(d)= +N−1 +� +n=0 +|r(n+d)x∗(n)|2+ +N−1 +� +n=0 +|r(n+d+N +NCP )x∗(n)|2 , +(1) +where N = 2048 for LTE-V2X systems and NCP denotes +the length of the cyclic prefix (CP). When P(d) exceeds a +given threshold PTH and reaches the maximum, we obtain the +estimated starting position of the training symbol [16], which +is expressed by +ˆd = arg +max +d∈{d|P (d)>PTH} P(d). +(2) +Afterwards, the CFO is estimated by performing auto- +correlation between adjacent two identical PSSS symbols and +two identical SSSS symbols [17], which is expressed as +ˆε = +1 +2π(N +NCP )angle +�N−1 +� +n=0 +[r(n+ ˆd)r∗(n+ ˆd+N +NCP )] ++ +N−1 +� +n=0 +[r(n+∆n+ ˆd)r∗(n+∆n+ ˆd+N +NCP )] +� +, +(3) +where angle{·} returns the phase angle of the input complex +number and ∆n represents the number of the sampling points +(a) initial time domain channel esti- +mate h5(n) +(b) windowed time domain channel +estimate ˆh5(n) +Fig. 3. The initial and windowed time domain channel estimates of the DMRS +symbol. +between the first PSSS and the first SSSS. Accordingly, we +obtain the CFO compensated signal by +y(n) = ˜r(n)e−j2πnˆε, +(4) +where ˜r(n) denotes the time synchronized signal. +III. PROPOSED RFF EXTRACTION METHOD +In this section, we propose a novel PSBCH based RFF ex- +traction method for LTE-V2X systems, which mainly includes +channel estimation, channel equalization, and RFF denoising. +A. Channel Estimation +We adopt the improved LS algorithm [18] for channel +estimation. The main idea of the algorithm is to obtain the +initial frequency domain channel estimate through the LS +algorithm, which is then transformed into the time domain +via inverse discrete Fourier transform (IDFT). Afterwards, we +perform time-domain windowing to exclude the noise and +the RFF. The resultant signal is finally transformed into the +frequency domain via discrete Fourier transform (DFT). The +detailed steps of channel estimation for the PSBCH subframe +are described as follows. +Denote the i-th time-domain SC-FDMA symbol of the +received PSBCH after preprocessing and CP removal by yi(n), +which carries RFF information and channel information. Then, +we transform the time-domain received signals corresponding +to the PSSS, the SSSS, and the DMRS symbols into the +frequency domain by performing DFT, which is expressed as +Yi(k) = DFTN{yi(n)}, 0 ≤ k ≤ N − 1, +(5) +where DFTN{·} denotes the N-point DFT and i = 2, 3, 5, 7, +10, 12, 13. Denote the frequency domain received signal cor- +responding to the effective bandwidth occupied by the PSSS, +the SSSS, and the DMRS as ÙYi(k). Then, the initial frequency +domain channel estimate of the i-th symbol ˆHi(k) containing +the RFF and the noise is calculated by +ˆHi(k) = +ÙYi(k) +Ù +Xi(k) +, k ∈ Ni, +(6) + +where Ù +Xi(k) denotes the PSSS, the SSSS, or the DMRS, and +Ni is defined by +Ni = +®[5, 66], i = 2, 3, 12, 13 +[0, 71], i = 5, 7, 10 +. +(7) +Subsequently, based on ˆHi(k), we obtain the initial time +domain channel estimate by +ˆhi(n) = IDFTNi{ ˆHi(k)}, n ∈ Ni, +(8) +where IDFTNi{·} denotes the Ni-point IDFT and Ni is +defined by +Ni = +®62, i = 2, 3, 12, 13 +72, i = 5, 7, 10 +. +(9) +Since the channel impulse response is concentrated in a +few time domain samples while the noise and the RFF are +distributed over the entire time domain, we can apply an +appropriate window on ˆhi(n) to obtain an improved time +domain channel estimate by +˘hi(n) = ˆhi(n)wi(n), n ∈ Ni, +(10) +where wi(n) denotes the window function. Fig. 3 illustrates +the windowing operation, where a rectangular window is used. +Since most noises and RFFs are removed by the windowing +operation, the resultant channel estimate becomes more accu- +rate. +After obtaining ˘hi(n), we further acquire the corresponding +frequency domain channel estimate as +˘Hi(k) = DFTNi{˘hi(n)}, k ∈ Ni, +(11) +Considering the fact that the channels experienced by adjacent +symbols are approximately identical, especially when the +vehicle speed is not very high, we can further average adjacent +˘Hi(k)’s to suppress the noise, thus improving the channel +estimation accuracy. For instance, if the channel variation in +one subframe is negligible, the ultimate frequency domain +channel estimate can be calculated by +˜H(k)= +� +� +� +� +� +� +� +˘HPSSS(k) + ˘HDMRS(k) + ˘HSSSS(k) +7 +, 5 ≤ k ≤ 66 +˘HDMRS(k) +3 +, 0 ≤ k ≤ 71 +, +(12) +where +˘HPSSS(k) = ˘H2(k) + ˘H3(k), +(13) +˘HDMRS(k) = ˘H5(k) + ˘H7(k) + ˘H10(k), +(14) +˘HSSSS(k) = ˘H12(k) + ˘H13(k). +(15) +B. Channel Equalization +After acquiring the channel estimate ˜H(k), we can perform +channel equalization to remove the channel information and +achieve the initial RFF features Ri(k) by +Ri(k) = +ÙYi(k) +˜H(k) +, k ∈ Ni. +(16) +Note that the above channel equalization will not lead to a loss +of RFF information since most RFFs have been removed by +the windowing operation during the channel estimation stage. +C. RFF Denoising +According to (16), the initial RFF feature is still affected +by the noise in ÙYi(k). To alleviate the impact of noise +on the extracted RFF, we further average the initial RFFs +corresponding to the same data sequence. Specifically, the +denoised RFFs for the PSSS, the DMRS, and the SSSS are +given by +RPSSS(k) = R2(k) + R3(k) +2 +, 5 ≤ k ≤ 66, +(17) +RDMRS(k)= +� +� +� +� +� +R5(k) + R7(k) + R10(k) +3 +, N SL +ID mod 2=0 +R5(k) + R10(k) +2 +, +N SL +ID mod 2=1 +, +0 ≤ k ≤ 71, +(18) +RSSSS(k) = R12(k) + R13(k) +2 +, 5 ≤ k ≤ 66. +(19) +Note that the DMRS sequence on the 7th symbol differs from +those on the 5th and 10th symbols when the SLSS ID N SL +ID is +odd. Hence, for this case, we only calculate the mean of R5(k) +and R10(k) which have the same data sequence. Finally, we +obtain ultimate RFF features R(k) as +R(k) = +®RDMRS(k), +0 ≤ k ≤ 4, 67 ≤ k ≤ 71 +[RPSSS(k), RDMRS(k), RSSSS(k)] , 5 ≤ k ≤ 66 . +(20) +IV. SIMULATION AND EXPERIMENT RESULTS +In the experiment, we employ 10 simulated LTE-V2X +terminals with different RFF parameters and 6 actual LTE- +V2X modules to generate PSBCH subframes, respectively, +and evaluate the classification performance of different devices +based on our proposed RFF extraction scheme. +A. Simulation Verification +For the simulation, we set different RFF parameters for 10 +terminals, including the I/Q DC offsets, the baseband low-pass +filter coefficients, the gain imbalance, the phase quadrature +deviation, and the RF front-end power amplifier coefficients, +which are specifically shown in Table I, to ensure the modu- +lation domain error vector magnitude (EVM) is within 17.5% +[19]. +Next, the PSBCH signals carrying the RFFs generated by +the 10 terminals pass through the simulated extended typical +urban (ETU) multipath channel [20], where the vehicle speed +ranges from 0 to 120 km/h. Moreover, the SNR ranges from +0 to 30 dB. +Then, we conduct classification experiments on 10 terminals +using random forest algorithm. The 700 received PSBCH +subframes of each terminal constitute the training set, where +the SNR is 30 dB and the vehicle speed is 30 km/h. The +test set consists of 300 other subframes from each terminal. + +TABLE I +RFF PARAMETERS OF 10 SIMULATED LTE-V2X TERMINALS +Terminal index +DC offset +Filter coefficients +Gain imbalance +Phase deviation +Power amplifier coefficient +1 +DI=0, DQ=0 +hI=[1 0], hQ=[1 0] +0.1 +0.1 +[1 0 0] +2 +DI=0.01, DQ=0 +hI=[1 0], hQ=[1 0] +0.01 +0.01 +[1 0 0] +3 +DI=0, DQ=-0.01 +hI=[1 0], hQ=[1 0] +0 +0 +[1 0 0] +4 +DI=-0.005, DQ=0.005 +hI=[1 0], hQ=[1 0] +0.01 +0.01 +[1 0 0] +5 +DI=0.005, DQ=-0.005 +hI=[1 0], hQ=[1 0] +0 +0 +[1 0 0] +6 +DI=0, DQ=0 +hI=[1 0], hQ=[1 0] +0.05 +0 +[0.9+0.15j 0.1 0.1-0.15j] +7 +DI=0, DQ=0 +hI=[1 0], hQ=[1 0] +0 +0.05 +[1.15 -0.2 0] +8 +DI=0, DQ=0 +hI=[0.825 0], hQ=[1.175 0] +0 +0 +[1 0 0] +9 +DI=0, DQ=0 +hI=[1 0.175], hQ=[1 -0.175] +0 +0 +[1 0 0] +10 +DI=0.005, DQ=0 +hI=[0.95 0], hQ=[1 0.05] +0.05 +0.05 +[0.95-0.05j 0 0] +Accuracy (%) +Fig. 4. Identification accuracy of 10 simulated LTE-V2X terminals based +on the proposed RFF extraction method under different SNRs and different +vehicle speeds. +The identification accuracy of the 10 terminals under different +SNRs and different vehicle speeds is depicted in Fig. 4. It +can be found that the vehicle speed has little effect on the +RFF identification accuracy rate. When the SNR exceeds 10 +dB, the accuracy always remains above 97% regardless of +the speed, while the accuracy decreases significantly when +the SNR drops below 10 dB mainly because we only use +one PSBCH subframe for RFF extraction. It reveals that the +proposed RFF extraction method has excellent classification +performance under medium and high SNRs. +Fig. 5 compares the RFF identification performances of the +methods with and without channel equalization, where the +SNR is 30 dB. When the speed increases from 0 to 120 km/h, +there is no obvious loss in the accuracy rate for the channel +equalization based method, which always remains over 99%, +while the identification accuracy without channel equalization +falls rapidly especially at high speeds, which indicates that our +proposed method based on channel estimation can effectively +mitigate the impact of wireless channels on the RFF extraction. +Accuracy (%) +Fig. 5. Comparison of the identification accuracy of 10 simulated LTE-V2X +terminals with and without channel equalization (SNR = 30 dB). +(a) +(b) +USRP B205 +LTE-V2X module +GPS + Receiver +Transmitter +GPS +Fig. 6. Experiment setup: (a) receiving device (USRP B205); (b) transmitting +device (LTE-V2X module). +B. Experiment Verification +For the experiment, we use 6 LTE-V2X modules to transmit +PSBCH subframes and utilize USRP B205 to receive the +signals. The experiment setup is shown in Fig. 6. First, we +collect 400 PSBCH subframes for each module as training set + +TABLE II +RFF IDENTIFICATION ACCURACY OF 6 LTE-V2X +MODULES UNDER DIFFERENT SPEEDS +Device +Accuracy +Speed +0 km/h +10 km/h +20 km/h +30 km/h +Module 1 +92% +93% +90% +91% +Module 2 +69% +71% +69% +68% +Module 3 +92% +90% +93% +93% +Module 4 +100% +100% +100% +97% +Module 5 +100% +100% +100% +100% +Module 6 +100% +100% +100% +100% +Average +92.2% +92.3% +92% +91.5% +under static state and low-speed moving state. Subsequently, +100 other subframes are captured from each module as test set, +where the speed ranges from 10 to 30 km/h. The classification +accuracy of the 6 LTE-V2X modules are shown in Table II. It +can be seen that the average accuracy exceeds 90%. Moreover, +the accuracy rate does not drop significantly after the speed +increases. Note that modules 1 to 4 belong to the same type +with very similar RFF features. Hence, the corresponding +classification accuracy is relatively low. +V. CONCLUSION +In this paper, we proposed a novel RFF extraction method +for LTE-V2X systems. Focusing on the PSSS, the SSSS, +and the DMRS of PSBCH, we successfully obtained highly +distinguishable RFF features by performing channel estima- +tion, channel equalization, and RFF denoising. As verified +via both simulations and experiments, our method displays +robust performance under challenging time-varying and mul- +tipath channels. The proposed method can also be applied +to any broadband multi-carrier communication systems that +have fixed sequences. In the future work, more terminals can +be tested in practical high mobility channel environments to +further verify the effectiveness of this method. +REFERENCES +[1] S. Gyawali, S. Xu, Y. Qian, and R. Q. Hu, “Challenges and solutions +for cellular based V2X communications,” IEEE Commun. Surveys Tuts., +vol. 23, no. 1, pp. 222–255, 1st Quart., 2021. +[2] W. Anwar, N. Franchi, and G. Fettweis, “Physical layer evaluation +of V2X communications technologies: 5G NR-V2X, LTE-V2X, IEEE +802.11bd, and IEEE 802.11p,” in Proc. IEEE 90th Veh. Technol. Conf. +(VTC-Fall), Honolulu, HI, USA, Sept. 2019, pp. 1–7. +[3] C. Wang, Z. Li, X.-G. Xia, J. Shi, J. Si, and Y. Zou, “Physical +layer security enhancement using artificial noise in cellular vehicle- +to-everything (C-V2X) networks,” IEEE Trans. Veh. Technol., vol. 69, +no. 12, pp. 15 253–15 268, Dec. 2020. +[4] X. Luo, Y. Liu, H.-H. Chen, and Q. Guo, “Physical layer security in +intelligently connected vehicle networks,” IEEE Netw., vol. 34, no. 5, +pp. 232–239, Sept./Oct. 2020. +[5] M. Mosca, “Cybersecurity in an era with quantum computers: Will we +be ready?” IEEE Security Privacy, vol. 16, no. 5, pp. 38–41, Sept./Oct. +2018. +[6] K. Zeng, K. Govindan, and P. Mohapatra, “Non-cryptographic authen- +tication and identification in wireless networks [security and privacy in +emerging wireless networks],” IEEE Wireless Commun., vol. 17, no. 5, +pp. 56–62, Oct. 2010. +[7] A. M. Ali, E. Uzundurukan, and A. Kara, “Improvements on transient +signal detection for RF fingerprinting,” in Proc. 25th Signal Process. +Commun. Appl. Conf. (SIU), Antalya, Turkey, May. 2017, pp. 1–4. +[8] Y. Shi and M. A. Jensen, “Improved radiometric identification of +wireless devices using MIMO transmission,” IEEE Trans. Inf. Forensics +Security, vol. 6, no. 4, pp. 1346–1354, Dec. 2011. +[9] W. C. Suski II, M. A. Temple, M. J. Mendenhall, and R. F. Mills, +“Using spectral fingerprints to improve wireless network security,” in +Proc. IEEE Global Telecommun. Conf. (GLOBECOM), New Orleans, +LO, Nov. 2008, pp. 1–5. +[10] L. Peng, A. Hu, J. Zhang, Y. Jiang, J. Yu, and Y. Yan, “Design of a +hybrid RF fingerprint extraction and device classification scheme,” IEEE +Internet Things J., vol. 6, no. 1, pp. 349–360, Feb. 2019. +[11] G. Shen, J. Zhang, A. Marshall, and J. R. Cavallaro, “Towards scalable +and channel-robust radio frequency fingerprint identification for LoRa,” +IEEE Trans. Inf. Forensics Security, vol. 17, pp. 774–787, Feb. 2022. +[12] Y. Yang, A. Hu, Y. Xing, J. Yu, and Z. Zhang, “A data-independent radio +frequency fingerprint extraction scheme,” IEEE Wireless Commun. Lett., +vol. 10, no. 11, pp. 2524–2527, Nov. 2021. +[13] L. Sun, X. Wang, Z. Huang, and B. Li, “Radio frequency fingerprint +extraction based on feature inhomogeneity,” IEEE Internet Things J., +early access, Feb. 25, 2022, doi: 10.1109/JIOT.2022.3154595. +[14] W. Wang, Z. Sun, S. Piao, B. Zhu, and K. Ren, “Wireless physical- +layer identification: Modeling and validation,” IEEE Trans. Inf. Forensics +Security, vol. 11, no. 9, pp. 2091–2106, Sept. 2016. +[15] ETSI 3rd Generation Partnership Project, “LTE; Evolved Universal +Terrestrial Radio Access (E-UTRA); Physical channels and modulation +(Release 14),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS +36.211 version 14.2.0, 2016. +[16] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchro- +nization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613– +1621, Dec. 1997. +[17] J. J. van de Beek and M. Sandell, “ML estimation of time and frequency +offset in OFDM systems,” IEEE Trans. Signal Process., vol. 45, no. 7, +pp. 1800–1805, Jul. 1997. +[18] J. J. van de Beek, O. Edfors, M. Sandell, S. Wilson, and P. Borjesson, +“On channel estimation in OFDM systems,” in Proc. IEEE 45th Veh. +Technol. Conf., vol. 2, Chicago, IL, USA, Jul. 1995, pp. 815–819. +[19] ETSI 3rd Generation Partnership Project, “Technical Specification +Group Radio Access Network; Evolved Universal Terrestrial Radio Ac- +cess (E-UTRA); User Equipment (UE) radio transmission and reception +(Release 9),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS 36.101 +version 9.4.0, 2010. +[20] ETSI 3rd Generation Partnership Project, “Technical Specification +Group Radio Access Network; Evolved Universal Terrestrial Radio +Access (E-UTRA); Base Station (BS) radio transmission and reception +(Release 14),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS +36.104 version 14.3.0, 2017. + diff --git a/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/load_file.txt b/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..530b96cac6109d3f35005830f1280ae14b9f8da2 --- /dev/null +++ b/2NAzT4oBgHgl3EQfe_yo/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf,len=433 +page_content='Radio Frequency Fingerprints Extraction for LTE-V2X: A Channel Estimation Based Methodology Tianshu Chen∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hong Shen∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Aiqun Hu∗†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Weihang He‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Jie Xu‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hongxing Hu§ ∗National Mobile Communications Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Southeast University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' China †The Purple Mountain Laboratories for Network and Communication Security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' China ‡School of Cyber Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Southeast University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' China §China Automotive Innovation Corporation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' China Email: {iamtianshu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' shhseu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' aqhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 220205165,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 220205095}@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='cn, huhongxing@t3caic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='com Abstract—The vehicular-to-everything (V2X) technology has recently drawn a number of attentions from both academic and industrial areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' However, the openness of the wireless communi- cation system makes it more vulnerable to identity impersonation and information tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' How to employ the powerful radio frequency fingerprint (RFF) identification technology in V2X systems turns out to be a vital and also challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In this paper, we propose a novel RFF extraction method for Long Term Evolution-V2X (LTE-V2X) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In order to conquer the difficulty of extracting transmitter RFF in the presence of wireless channel and receiver noise, we first estimate the wireless channel which excludes the RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, we remove the impact of the wireless channel based on the channel estimate and obtain initial RFF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Finally, we conduct RFF denoising to enhance the quality of the initial RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Simulation and experiment results both demonstrate that our proposed RFF extraction scheme achieves a high identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Furthermore, the performance is also robust to the vehicle speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Index Terms—Vehicular-to-everything (V2X), radio frequency fingerprint (RFF), device identification, channel estimation, RFF denoising I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' INTRODUCTION Vehicular-to-everything (V2X) has become a promising technique for intelligent transportation and autonomous driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In particular, the cellular-V2X (C-V2X) has been widely acknowledged as a key V2X communication standard due to its superior performance [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Since V2X relies on wireless transmission, the information is easy to be eavesdropped, forged or tampered with, which imposes great challenges on the safety of vehicles, pedestrians and road infrastructures in the V2X communication network [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' To deal with the security threats faced by wireless communications, there are usually two widely used authen- tication strategies: key-based cryptographic authentication and physical layer security-based non-cryptographic authentication [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The cryptographic authentication technology needs to © 2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' distribute and manage abundant communication keys, which occupies computing resources and leads to additional overhead and delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Moreover, with the rapid development of comput- ing capability of the computers, especially the emergence of quantum computers, traditional cryptography technologies are more vulnerable to brute-force attacks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' On the contrary, the physical layer security based authentication has lower com- plexity and network overhead with lower latency compared to traditional cryptography-based authentication methods, and can achieve non-perceptual authentication without third-party facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' One typical example is the radio frequency fin- gerprint (RFF) based authentication, which fully exploits the hardware differences between any two devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Since the hardware characteristic of each device is unique and difficult to clone, the RFF based authentication can better resist the identity attacks and spoofing [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In literature, a variety of RFF extraction and identification methods have been advocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Early works mainly focus on the characteristics of transient signals, such as instantaneous amplitude, frequency, and phase responses [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Concerning the steady-state signal, such as preamble signals, researchers consider extracting the RFF features including I/Q offset [8], power spectral density [9], differential constellation trace figure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Furthermore, some universal RFF extraction meth- ods which are independent of data, channel or modulation modes have also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Concretely, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [11] constructed channel independent spectrogram and utilized data augmentation for RFF extraction and identification of Lora devices, which achieves good performance under different channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Alternatively, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [12] used random data segments to extract the tap coefficients of the least mean square (LMS) adaptive filter as data independent RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [13] verified the locality and inhomogeneity of the RFF distribution with the analysis in the cepstral domain, which yields modulation mode independent RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The aforementioned works mainly consider the RFF extrac- tion for low mobility and narrowband systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' However, for the V2X system, the channel typically varies fast due to the high mobility vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In addition, the V2X signal usually arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01446v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='SP] 4 Jan 2023 Signal preprocessing RFF feature extraction and denoising Device identification Digital baseband signal Mixer up-conversion Transmitter RFF model RF front-end power amplifier DAC Baseband low-pass filter TX antenna OBU OBU RSU V2P V2I Receiver Channel estimation and equalization RFF identification and access system I/Q DC offset Non-linearity Gain imbalance and phase deviation Frequency response deviation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' LTE-V2X RFF extraction and identification system framework and RFF model at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' has a large bandwidth which is more vulnerable to multipath environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Therefore, the current RFF extraction methods for narrowband systems such as ZigBee and Lora cannot be directly applied for the V2X system because they do not take into account the impact of multipath and time-varying channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In this work, we propose a channel estimation based RFF extraction method for Long Term Evolution-V2X (LTE-V2X) systems, which, to the best of our knowledge, has not been investigated in existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Specifically, we first estimate the experienced wireless channel using an improved least square (LS) channel estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, we perform channel equalization based on the channel estimate to obtain channel dependent RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The RFF quality is further enhanced via conducting time-domain denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' It is worthwhile noting that the developed method eliminates the effect of the channel and the noise on the RFF with low implementation complex- ity, and can be extended to various broadband multi-carrier wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Section II introduces the system model and signal preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Section III presents the details of the proposed RFF extraction methodology based on wireless channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Section IV evaluates the performance of the proposed RFF extraction method through simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Section V concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' SYSTEM MODEL AND SIGNAL PREPROCESSING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' System Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1 demonstrates the framework of the considered LTE- V2X RFF extraction and identification system together with the RFF model at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' More concretely, one V2X terminal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', on board unit (OBU) or road side unit (RSU), first transmits data to other devices, where the transmitted signal includes the RFF of the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, the receiver preprocesses the received signal which consists of converting the RF signal to the baseband signal and performing time- frequency synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Subsequently, the RFF features are extracted based on the synchronized signal, where the effects of the wireless channel and the noise on the RFF need to be mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Finally, the device identification is performed using the extracted RFF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' It is necessary to note that the considered RFF refers to all the characteristics of the circuits at the transmitter, which, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1, include the I/Q DC offsets of the digital-to- analog converter (DAC), the frequency response deviation of the filter, the gain imbalance and the carrier phase quadrature deviation of the mixer, and the non-linearity of the power amplifier [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' LTE-V2X PSBCH We adopt the physical sidelink broadcast channel (PSBCH) in LTE-V2X systems for RFF extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' According to [15], PSBCH is transmitted every 160 ms occupying the central 6 PSBCH PSBCH PSSS PSSS DMRS PSBCH DMRS DMRS PSBCH PSBCH PSBCH SSSS SSSS GUARD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 time 1 ms frequency 6 RBs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' LTE-V2X PSBCH format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' resource blocks (RBs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', 72 subcarriers and 14 single-carrier frequency division multiple access (SC-FDMA) symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The detailed format of PSBCH is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2, where primary sidelink synchronization signal (PSSS), secondary sidelink synchronization signal (SSSS), and demodulation reference signal (DMRS) all depend on the currently used sidelink synchronization signal (SLSS) ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Since the SLSS ID can be estimated [15], we can readily obtain ideal PSSS, SSSS, and DMRS at the receiver which are used for extracting transmitter RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Signal Preprocessing In order to ensure the stability of the extracted RFF, the signal preprocessing procedure includes time synchronization and carrier frequency offset (CFO) estimation and compensa- tion after the received signal is down-converted from the RF band to the baseband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The time synchronization is realized by utilizing two identi- cal training symbols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', two repeated PSSS or SSSS symbols in LTE-V2X PSBCH, and the cross-correlation between the received signal r(n) and the training signal x(n) as P(d)= N−1 � n=0 |r(n+d)x∗(n)|2+ N−1 � n=0 |r(n+d+N +NCP )x∗(n)|2 , (1) where N = 2048 for LTE-V2X systems and NCP denotes the length of the cyclic prefix (CP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' When P(d) exceeds a given threshold PTH and reaches the maximum, we obtain the estimated starting position of the training symbol [16], which is expressed by ˆd = arg max d∈{d|P (d)>PTH} P(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (2) Afterwards,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' the CFO is estimated by performing auto- correlation between adjacent two identical PSSS symbols and two identical SSSS symbols [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' which is expressed as ˆε = 1 2π(N +NCP )angle �N−1 � n=0 [r(n+ ˆd)r∗(n+ ˆd+N +NCP )] + N−1 � n=0 [r(n+∆n+ ˆd)r∗(n+∆n+ ˆd+N +NCP )] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (3) where angle{·} returns the phase angle of the input complex number and ∆n represents the number of the sampling points (a) initial time domain channel esti- mate h5(n) (b) windowed time domain channel estimate ˆh5(n) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The initial and windowed time domain channel estimates of the DMRS symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' between the first PSSS and the first SSSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Accordingly, we obtain the CFO compensated signal by y(n) = ˜r(n)e−j2πnˆε, (4) where ˜r(n) denotes the time synchronized signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' PROPOSED RFF EXTRACTION METHOD In this section, we propose a novel PSBCH based RFF ex- traction method for LTE-V2X systems, which mainly includes channel estimation, channel equalization, and RFF denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Channel Estimation We adopt the improved LS algorithm [18] for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The main idea of the algorithm is to obtain the initial frequency domain channel estimate through the LS algorithm, which is then transformed into the time domain via inverse discrete Fourier transform (IDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Afterwards, we perform time-domain windowing to exclude the noise and the RFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The resultant signal is finally transformed into the frequency domain via discrete Fourier transform (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The detailed steps of channel estimation for the PSBCH subframe are described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Denote the i-th time-domain SC-FDMA symbol of the received PSBCH after preprocessing and CP removal by yi(n), which carries RFF information and channel information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, we transform the time-domain received signals corresponding to the PSSS, the SSSS, and the DMRS symbols into the frequency domain by performing DFT, which is expressed as Yi(k) = DFTN{yi(n)}, 0 ≤ k ≤ N − 1, (5) where DFTN{·} denotes the N-point DFT and i = 2, 3, 5, 7, 10, 12, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Denote the frequency domain received signal cor- responding to the effective bandwidth occupied by the PSSS, the SSSS, and the DMRS as ÙYi(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, the initial frequency domain channel estimate of the i-th symbol ˆHi(k) containing the RFF and the noise is calculated by ˆHi(k) = ÙYi(k) Ù Xi(k) , k ∈ Ni, (6) where Ù Xi(k) denotes the PSSS, the SSSS, or the DMRS, and Ni is defined by Ni = ®[5, 66], i = 2, 3, 12, 13 [0, 71], i = 5, 7, 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (7) Subsequently, based on ˆHi(k), we obtain the initial time domain channel estimate by ˆhi(n) = IDFTNi{ ˆHi(k)}, n ∈ Ni, (8) where IDFTNi{·} denotes the Ni-point IDFT and Ni is defined by Ni = ®62, i = 2, 3, 12, 13 72, i = 5, 7, 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (9) Since the channel impulse response is concentrated in a few time domain samples while the noise and the RFF are distributed over the entire time domain, we can apply an appropriate window on ˆhi(n) to obtain an improved time domain channel estimate by ˘hi(n) = ˆhi(n)wi(n), n ∈ Ni, (10) where wi(n) denotes the window function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 3 illustrates the windowing operation, where a rectangular window is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Since most noises and RFFs are removed by the windowing operation, the resultant channel estimate becomes more accu- rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' After obtaining ˘hi(n), we further acquire the corresponding frequency domain channel estimate as ˘Hi(k) = DFTNi{˘hi(n)}, k ∈ Ni, (11) Considering the fact that the channels experienced by adjacent symbols are approximately identical, especially when the vehicle speed is not very high, we can further average adjacent ˘Hi(k)’s to suppress the noise, thus improving the channel estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' For instance, if the channel variation in one subframe is negligible, the ultimate frequency domain channel estimate can be calculated by ˜H(k)= � � � � � � � ˘HPSSS(k) + ˘HDMRS(k) + ˘HSSSS(k) 7 , 5 ≤ k ≤ 66 ˘HDMRS(k) 3 , 0 ≤ k ≤ 71 , (12) where ˘HPSSS(k) = ˘H2(k) + ˘H3(k), (13) ˘HDMRS(k) = ˘H5(k) + ˘H7(k) + ˘H10(k), (14) ˘HSSSS(k) = ˘H12(k) + ˘H13(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (15) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Channel Equalization After acquiring the channel estimate ˜H(k), we can perform channel equalization to remove the channel information and achieve the initial RFF features Ri(k) by Ri(k) = ÙYi(k) ˜H(k) , k ∈ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (16) Note that the above channel equalization will not lead to a loss of RFF information since most RFFs have been removed by the windowing operation during the channel estimation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' RFF Denoising According to (16), the initial RFF feature is still affected by the noise in ÙYi(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' To alleviate the impact of noise on the extracted RFF, we further average the initial RFFs corresponding to the same data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Specifically, the denoised RFFs for the PSSS, the DMRS, and the SSSS are given by RPSSS(k) = R2(k) + R3(k) 2 , 5 ≤ k ≤ 66, (17) RDMRS(k)= � � � � � R5(k) + R7(k) + R10(k) 3 , N SL ID mod 2=0 R5(k) + R10(k) 2 , N SL ID mod 2=1 , 0 ≤ k ≤ 71, (18) RSSSS(k) = R12(k) + R13(k) 2 , 5 ≤ k ≤ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (19) Note that the DMRS sequence on the 7th symbol differs from those on the 5th and 10th symbols when the SLSS ID N SL ID is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hence, for this case, we only calculate the mean of R5(k) and R10(k) which have the same data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Finally, we obtain ultimate RFF features R(k) as R(k) = ®RDMRS(k), 0 ≤ k ≤ 4, 67 ≤ k ≤ 71 [RPSSS(k), RDMRS(k), RSSSS(k)] , 5 ≤ k ≤ 66 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (20) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' SIMULATION AND EXPERIMENT RESULTS In the experiment, we employ 10 simulated LTE-V2X terminals with different RFF parameters and 6 actual LTE- V2X modules to generate PSBCH subframes, respectively, and evaluate the classification performance of different devices based on our proposed RFF extraction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Simulation Verification For the simulation, we set different RFF parameters for 10 terminals, including the I/Q DC offsets, the baseband low-pass filter coefficients, the gain imbalance, the phase quadrature deviation, and the RF front-end power amplifier coefficients, which are specifically shown in Table I, to ensure the modu- lation domain error vector magnitude (EVM) is within 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='5% [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Next, the PSBCH signals carrying the RFFs generated by the 10 terminals pass through the simulated extended typical urban (ETU) multipath channel [20], where the vehicle speed ranges from 0 to 120 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Moreover, the SNR ranges from 0 to 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Then, we conduct classification experiments on 10 terminals using random forest algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The 700 received PSBCH subframes of each terminal constitute the training set, where the SNR is 30 dB and the vehicle speed is 30 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The test set consists of 300 other subframes from each terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' TABLE I RFF PARAMETERS OF 10 SIMULATED LTE-V2X TERMINALS Terminal index DC offset Filter coefficients Gain imbalance Phase deviation Power amplifier coefficient 1 DI=0, DQ=0 hI=[1 0], hQ=[1 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='1 [1 0 0] 2 DI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01, DQ=0 hI=[1 0], hQ=[1 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01 [1 0 0] 3 DI=0, DQ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01 hI=[1 0], hQ=[1 0] 0 0 [1 0 0] 4 DI=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='005, DQ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='005 hI=[1 0], hQ=[1 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='01 [1 0 0] 5 DI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='005, DQ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='005 hI=[1 0], hQ=[1 0] 0 0 [1 0 0] 6 DI=0, DQ=0 hI=[1 0], hQ=[1 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05 0 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='15j 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='15j] 7 DI=0, DQ=0 hI=[1 0], hQ=[1 0] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='15 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='2 0] 8 DI=0, DQ=0 hI=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='825 0], hQ=[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='175 0] 0 0 [1 0 0] 9 DI=0, DQ=0 hI=[1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='175], hQ=[1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='175] 0 0 [1 0 0] 10 DI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='005, DQ=0 hI=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='95 0], hQ=[1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='95-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='05j 0 0] Accuracy (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Identification accuracy of 10 simulated LTE-V2X terminals based on the proposed RFF extraction method under different SNRs and different vehicle speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The identification accuracy of the 10 terminals under different SNRs and different vehicle speeds is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' It can be found that the vehicle speed has little effect on the RFF identification accuracy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' When the SNR exceeds 10 dB, the accuracy always remains above 97% regardless of the speed, while the accuracy decreases significantly when the SNR drops below 10 dB mainly because we only use one PSBCH subframe for RFF extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' It reveals that the proposed RFF extraction method has excellent classification performance under medium and high SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 5 compares the RFF identification performances of the methods with and without channel equalization, where the SNR is 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' When the speed increases from 0 to 120 km/h, there is no obvious loss in the accuracy rate for the channel equalization based method, which always remains over 99%, while the identification accuracy without channel equalization falls rapidly especially at high speeds, which indicates that our proposed method based on channel estimation can effectively mitigate the impact of wireless channels on the RFF extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Accuracy (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Comparison of the identification accuracy of 10 simulated LTE-V2X terminals with and without channel equalization (SNR = 30 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (a) (b) USRP B205 LTE-V2X module GPS Receiver Transmitter GPS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Experiment setup: (a) receiving device (USRP B205);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (b) transmitting device (LTE-V2X module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Experiment Verification For the experiment, we use 6 LTE-V2X modules to transmit PSBCH subframes and utilize USRP B205 to receive the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The experiment setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' First, we collect 400 PSBCH subframes for each module as training set TABLE II RFF IDENTIFICATION ACCURACY OF 6 LTE-V2X MODULES UNDER DIFFERENT SPEEDS Device Accuracy Speed 0 km/h 10 km/h 20 km/h 30 km/h Module 1 92% 93% 90% 91% Module 2 69% 71% 69% 68% Module 3 92% 90% 93% 93% Module 4 100% 100% 100% 97% Module 5 100% 100% 100% 100% Module 6 100% 100% 100% 100% Average 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='2% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='3% 92% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='5% under static state and low-speed moving state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Subsequently, 100 other subframes are captured from each module as test set, where the speed ranges from 10 to 30 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The classification accuracy of the 6 LTE-V2X modules are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' It can be seen that the average accuracy exceeds 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Moreover, the accuracy rate does not drop significantly after the speed increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Note that modules 1 to 4 belong to the same type with very similar RFF features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hence, the corresponding classification accuracy is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a novel RFF extraction method for LTE-V2X systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Focusing on the PSSS, the SSSS, and the DMRS of PSBCH, we successfully obtained highly distinguishable RFF features by performing channel estima- tion, channel equalization, and RFF denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' As verified via both simulations and experiments, our method displays robust performance under challenging time-varying and mul- tipath channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' The proposed method can also be applied to any broadband multi-carrier communication systems that have fixed sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' In the future work, more terminals can be tested in practical high mobility channel environments to further verify the effectiveness of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Gyawali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Qian, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hu, “Challenges and solutions for cellular based V2X communications,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Surveys Tuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 222–255, 1st Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Anwar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Franchi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Fettweis, “Physical layer evaluation of V2X communications technologies: 5G NR-V2X, LTE-V2X, IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='11bd, and IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='11p,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' IEEE 90th Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (VTC-Fall), Honolulu, HI, USA, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Xia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Si, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zou, “Physical layer security enhancement using artificial noise in cellular vehicle- to-everything (C-V2X) networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 15 253–15 268, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Chen, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Guo, “Physical layer security in intelligently connected vehicle networks,” IEEE Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 232–239, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='/Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Mosca, “Cybersecurity in an era with quantum computers: Will we be ready?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' IEEE Security Privacy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 38–41, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='/Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zeng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Govindan, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Mohapatra, “Non-cryptographic authen- tication and identification in wireless networks [security and privacy in emerging wireless networks],” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 56–62, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Ali, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Uzundurukan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Kara, “Improvements on transient signal detection for RF fingerprinting,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 25th Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (SIU), Antalya, Turkey, May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Shi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Jensen, “Improved radiometric identification of wireless devices using MIMO transmission,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Forensics Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1346–1354, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Suski II, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Temple, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Mendenhall, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Mills, “Using spectral fingerprints to improve wireless network security,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' IEEE Global Telecommun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' (GLOBECOM), New Orleans, LO, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Peng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Yu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Yan, “Design of a hybrid RF fingerprint extraction and device classification scheme,” IEEE Internet Things J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 349–360, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Marshall, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Cavallaro, “Towards scalable and channel-robust radio frequency fingerprint identification for LoRa,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Forensics Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 774–787, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Yang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Xing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Yu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zhang, “A data-independent radio frequency fingerprint extraction scheme,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2524–2527, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Huang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Li, “Radio frequency fingerprint extraction based on feature inhomogeneity,” IEEE Internet Things J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', early access, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 25, 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='1109/JIOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='3154595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Piao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Zhu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Ren, “Wireless physical- layer identification: Modeling and validation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Forensics Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2091–2106, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [15] ETSI 3rd Generation Partnership Project, “LTE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Evolved Universal Terrestrial Radio Access (E-UTRA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Physical channels and modulation (Release 14),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='211 version 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='0, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Schmidl and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Cox, “Robust frequency and timing synchro- nization for OFDM,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1613– 1621, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' van de Beek and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Sandell, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 45, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1800–1805, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' van de Beek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Edfors, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Sandell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Wilson, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Borjesson, “On channel estimation in OFDM systems,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' IEEE 45th Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 2, Chicago, IL, USA, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' 815–819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [19] ETSI 3rd Generation Partnership Project, “Technical Specification Group Radio Access Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Evolved Universal Terrestrial Radio Ac- cess (E-UTRA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' User Equipment (UE) radio transmission and reception (Release 9),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='101 version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='0, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' [20] ETSI 3rd Generation Partnership Project, “Technical Specification Group Radio Access Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Evolved Universal Terrestrial Radio Access (E-UTRA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content=' Base Station (BS) radio transmission and reception (Release 14),” Sophia Antipolis Cedex, Biarritz, France, 3GPP TS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='104 version 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} +page_content='0, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfe_yo/content/2301.01446v1.pdf'} diff --git a/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/2301.00981v1.pdf.txt b/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/2301.00981v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da9f5ddbc4d387e68791b019a221d59b2b3cf5d5 --- /dev/null +++ b/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/2301.00981v1.pdf.txt @@ -0,0 +1,655 @@ +Transfer Generative Adversarial Networks +(T-GAN)-based Terahertz Channel Modeling +Zhengdong Hu, Yuanbo Li, and Chong Han +Terahertz Wireless Communications (TWC) Laboratory, Shanghai Jiao Tong University, China. +Email: {huzhengdong, yuanbo.li, chong.han}@sjtu.edu.cn +Abstract—Terahertz (THz) communications are envisioned as +a promising technology for 6G and beyond wireless systems, +providing ultra-broad bandwidth and thus Terabit-per-second +(Tbps) data rates. However, as foundation of designing THz +communications, channel modeling and characterization are +fundamental to scrutinize the potential of the new spectrum. +Relied on physical measurements, traditional statistical channel +modeling methods suffer from the problem of low accuracy +with the assumed certain distributions and empirical parameters. +Moreover, it is time-consuming and expensive to acquire extensive +channel measurement in the THz band. In this paper, a transfer +generative adversarial network (T-GAN) based modeling method +is proposed in the THz band, which exploits the advantage of +GAN in modeling the complex distribution, and the benefit of +transfer learning in transferring the knowledge from a source +task to improve generalization about the target task with limited +training data. Specifically, to start with, the proposed GAN is pre- +trained using the simulated dataset, generated by the standard +channel model from 3rd generation partnerships project (3GPP). +Furthermore, by transferring the knowledge and fine-tuning the +pre-trained GAN, the T-GAN is developed by using the THz mea- +sured dataset with a small amount. Experimental results reveal +that the distribution of PDPs generated by the proposed T-GAN +method shows good agreement with measurement. Moreover, T- +GAN achieves good performance in channel modeling, with 9 dB +improved root-mean-square error (RMSE) and higher Structure +Similarity Index Measure (SSIM), compared with traditional +3GPP method. +I. INTRODUCTION +With the exponential growth of the number of intercon- +nected devices, the sixth generation (6G) is expected to achieve +intelligent connections of everything, anywhere, anytime [1], +which demands Tbit/s wireless data rates. To fulfill the de- +mand, Terahertz (THz) communications gain increasing atten- +tion as a vital technology of 6G systems, thanks to the ultra- +broad bandwidth ranging from tens of GHz to hundreds of +GHz [2]. The THz band is promising to address the spectrum +scarcity and capacity limitations of current wireless systems, +and realize long-awaited applications, extending from wireless +cognition, localization/positioning, to integrated sensing and +communication [3]. +To design reliable THz wireless systems, one fundamental +challenge lies in developing an accurate channel model to por- +tray the propagation phenomena. Due to the high frequencies, +new characteristics occur in the THz band, such as frequency- +selective absorption loss and rough-surface scattering. At- +tribute to these new characteristics, THz channel modeling is +required to capture these characteristics. However, traditional +statistical channel modeling methods suffer from the prob- +lem of low accuracy with the assumed certain distributions +and empirical parameters. For example, a geometric based +stochastic channel model (GSCM) assumes that the positions +of scatters follow certain statistical distributions, such as the +uniform distribution within a circle around the transmitters +and receivers [4]. However, the positions of scatters are hard +to characterize by certain statistical distributions, making the +GSCM not accurate for utilization in the THz band. Moreover, +it is time-consuming and costly to acquire extensive channel +measurement for THz channel modeling. To this end, an +accurate channel modeling method with limited measurement +data for the THz band is needed. +Recently, deep learning (DL) is popular and widely applied +in wireless communications, such as channel estimation [5], +[6] and channel state information (CSI) feedback [7]. Among +different kinds of DL methods, the generative adversarial +network (GAN) has the advantage of modeling complex dis- +tribution accurately without any statistical assumptions, based +on which GAN can be utilized to develop channel models. The +authors in [8] train GAN to approximate the probability distri- +bution functions (PDFs) of stochastic channel response. In [9], +GAN is applied to generate synthetic channel samples close +to the distribution of real channel samples. The researchers +in [10] model the channel with GAN through channel input- +output measurements. In [11], a model-driven GAN-based +channel modeling method is developed in intelligent reflecting +surface (IRS) aided communication system. These methods +achieve good performance in modeling the channel and prove +high consistency between the target channel distribution and +the generated channel distribution. However, the GAN based +channel modeling method has not been exploited in the THz +band. Moreover, it is a challenge to train GAN for channel +modeling with the scarce THz channel measurement dataset. +In this paper, a transfer GAN (T-GAN)-based THz channel +modeling method is proposed, which can learn the distribution +of power delay profile (PDP) of the THz channel. Moreover, +to tackle the challenge of limited channel measurement in +the THz band, the transfer learning technique is introduced +in T-GAN, which reduces the size requirement of channel +dataset for training and enhances the performance of channel +modeling, through transferring the knowledge stored in a pre- +trained model to a new model [12], [13]. Furthermore, the +performance of T-GAN in modeling the channel distribution +is validated by real measurements [14]. +The contributions of this paper are listed as follows. +• We propose a T-GAN based THz channel modeling +arXiv:2301.00981v1 [eess.SP] 3 Jan 2023 + +method, in which a GAN is designed to capture the +distribution of PDPs of the THz channel, by training on +the dataset of PDP samples. +• To tackle the challenge of limited measurement data +for THz channel modeling, transfer learning is further +exploited by T-GAN, which reduces the size requirement +of training dataset, and enhances the performance of +GAN, through transferring the knowledge stored in a pre- +trained model to a new model. +The rest of the sections are organized as follows. Sec. II +details the proposed T-GAN based channel modeling method. +Sec. III demonstrates the performance of the proposed T-GAN +method. The paper is concluded in Sec. IV. +Notation: a is a scalar. a denotes a vector. A represents a +matrix. E{·} describes the expectation. ∇ denotes the gradient +operation. ∥·∥ represent the L2 norm. IN defines an N dimen- +sional identity matrix. N denotes the normal distribution. +II. TRANSFER GAN (T-GAN) BASED CHANNEL +MODELING +In this section, the channel modeling problem is first for- +mulated into a channel distribution learning problem. Then, +the proposed GAN in T-GAN method is elaborated. Finally, +T-GAN is presented. +A. Problem Formulation +The channel impulse response (CIR) can be represented as +h(τ) = +L−1 +� +l=0 +αlejφlδ(τ − τl), +(1) +where τl denotes the delay of the lth multi-path components +(MPCs), L denotes the number of MPC, αl refers to the path +gain and φl represents the phase of the corresponding MPC. +To characterize the channel, PDP is an important feature, +which indicates the dispersion of power over the time delay, +specifically, the received power with respect to the delay in +a multi-path channel. It can be extracted from the channel +impulse response by +P(τ) = |h(τ)|2, +(2) +Then, the channel modeling problem is exploited by learning +the distribution of PDPs denoted by pr, which is difficult to +be analytically represented. Instead, the distribution pr can be +captured by generating fake PDP samples with distribution pg, +such that the generated distribution pg of PDPs can match the +actual distribution pr. +B. Proposed GAN +The GAN can be utilized to learn the distribution of +PDPs denoted by pr, with the framework depicted in Fig 1. +The GAN consists of two sub-networks, namely, generator +and discriminator. The generator is aimed at generating fake +samples G(z) to fool the discriminator, in which z is the noise +sample, by mapping the input noise distribution pz(z) to the +generated distribution pg = p(G(z)). The discriminator tries to +Fig. 1. Framework of GAN. +distinguish between real samples x from pr and fake samples +G(z) from pg, and the output of the discriminator D(x) and +D(G(z)) can be treated as the probability of being a real +sample. The two networks are trained in an adversarial manner, +which can be considered as a two-player zero-sum minimax +game. Specifically, the training objective can be represented +by +min +G max +D Ex∼pr[log D(x)] + Ez∼pz[log(1 − D(G(z)))], (3) +where the generator minimizes the probability (1 − D(G(z)) +that the generated sample is detected as fake by the dis- +criminator, while the discriminator maximizes this probability. +Therefore, the generator and discriminator compete against +each other with the opposite objectives in the training process. +Through the adversarial training, the Nash equilibrium can +be achieved, such that the generator and discriminator cannot +improve their objectives by changing only their own network. +Moreover, the global optimum of the training objective can be +achieved in the equilibrium when pg = pr. However, training +with the objective function in (3) is unstable, since the training +objective is potentially not continuous with respect to the +generator’s parameters [15]. Therefore, the improved version +of GAN, namely, Wasserstein GAN with gradient penalty +(WGAN-GP) [15] is adopted. The modified objective function +is expressed as +min +G max +D Ex∼pr[D(x)]+Ez∼pz[(1 − D(G(z)))] ++λE˜x[(∥∇˜xD(˜x)∥ − 1)2)], +(4) +where the last term is the gradient penalty term to enforce +Lipschitz constraint that the gradient of the GAN network +is upper-bounded by a maximum value, the symbol ˜x is the +uniformly sampled point between the points of x and G(z). +Moreover, the parameter λ is the penalty coefficient. +After introducing the framework of GAN, the detailed +architecture of proposed GAN network is presented. The +structures of generator G and discriminator D are depicted in +Fig. 2, where the number in the bracket denotes the dimension. +The input to the generator is a noise vector z with dimension +nz = 100, which is sampled from the probability density +function N(0, σ2Inz). The generator consists of five dense +layers, and the numbers of neurons in the dense layers are +128, 128, 128, 128, 401, respectively. It is worth noting that +the size of the output layer is equal to the size of PDP. The + +Fig. 2. Structure of generator and discriminator. +activation function of the first four dense layers is LeakyReLU +function, which can speed up the convergence and avoid the +gradient vanishing problem. The formula of the LeakyReLU +function is expressed as +f(x) = +� +x, +if x ≥ 0 +αx, +if x < 0 , +(5) +where α is the slope coefficient when the value of neuron x is +negative. In addition to the LeakyReLU function, a Sigmoid +function is utilized in the last layer, which maps the output to +the range of [0, 1]. The Sigmoid function is defined as +f(x) = +1 +1 + e−x . +(6) +After going through the dense layers and activation functions +in the generator, the input noise vectors are transformed into +the generated samples. Then, the generated samples together +with real samples are passed to the discriminator. +The discriminator is designed to distinguish between gen- +erated samples and real samples. The numbers of neurons +for the five dense layers in the discriminator are 512, 256, +128, 64, 1, respectively. The activation function chosen for +the first 4 layers is the LeakyReLU function introduced +before. The activation function for the output layer is linear +activation function, which is decided by the objective function +of WGAN-GP introduced in (4). +C. Proposed T-GAN +The framework for the proposed T-GAN is depicted in +Fig. 3, in which the transfer learning is conducted between the +measurement and 3GPP TR 38.901 model [16]. The measured +PDPs denote the PDPs extracted from measurement with +a small size, while the simulated PDPs refer to the PDPs +simulated by the 3GPP model, which is implemented with +the extracted statistics from measurement. Then, the proposed +GAN and T-GAN with the same network structure, are trained +on the simulated PDPs and measured PDPs, respectively, to +capture the distribution of PDPs. Since the size of measured +PDPs is quite small for the training of T-GAN, which can +cause the difficulty of converging or the over-fitting problem, +the transfer learning is exploited to tackle these problems. +Fig. 3. Framework for T-GAN. +To describe the transfer learning formally, a domain denoted +by D consists of a feature space X and a marginal probability +distribution P(X) defined on X = {x1, x2, · · · , xN} ∈ X, +where N is the number of feature vectors in X. As depicted +in Fig. 3, the target domain Dt and source domain Ds are +defined on measurement and 3GPP model, respectively. The +feature spaces for the two domains are both constructed by +PDPs, with different marginal probability distributions defined +on measured PDPs Xt and simulated PDPs Xs. +Moreover, given a domain D(X, P(X)), a task denoted by +T is defined by a label space L and a predictive function f(·), +and the predictive function is learned from the pairs (xn, ln) +with xn ∈ X and ln ∈ L. In the target domain Dt and source +domain Ds, the tasks are the same to capture the distribution +of PDPs, and the label space is L = {0, 1} representing +whether the PDP sample is generated by the proposed GAN +or from the training dataset. The T-GAN and GAN serve as +the predictive functions ft and fs. Then, transfer learning is +aimed at learning the function ft in target domain Dt with +the knowledge of Ts in source domain Ds, i.e., transferring +the knowledge stored in GAN trained on simulated PDPs to +T-GAN trained on the measured PDPs. +The method of fine-tuning [13] is adopted for the transfer +learning. The T-GAN is initialized with the weights of the +GAN trained on the simulated PDPs, and is then fine-tuned +on the measured PDPs with small size. It is worth noting that +both the generator and discriminator in the GAN are trans- +ferred, which can yield the better performance in generating +high quality samples and fast convergence, compared with +transferring only the generator or the discriminator [13]. +With transfer learning, the performance of T-GAN can be +largely enhanced. Specifically, the channel statistics extracted +for 3GPP method are captured by the proposed GAN trained +on simulated PDPs, which are further transferred to T-GAN. +Moreover, T-GAN can learn the features of PDPs that are not +captured by 3GPP method, directly from measurement, which +further improves the performance of T-GAN in modeling the +distribution of PDPs. + +Target Domain +Source DomainFig. 4. Measurement layout in the indoor corridor scenario [14]. +III. EXPERIMENT AND PERFORMANCE EVALUATION +In this section, the experiment settings are elaborated. +Moreover, the performance of the T-GAN are evaluated by +comparing the generated distribution of PDPs with measure- +ment. +A. Dataset and Setup +The dataset is collected from the measurement campaign +in [14]. which is conducted in an indoor corridor scenario +at 306-321 GHz with 400 ns maximum delay, as depicted in +Fig. 4. With the measurement data, the PDPs can be extracted +to characterize the channel in the 21 receiver points. Since +the sample frequency interval is relatively small, as 2.5 MHz, +the measured PDPs are very long, including 6001 sample +points, which results in extraordinary computation and time +consumption to train the GANs. To address this problem, +we only use the measured channel transfer functions in the +frequency band from 314 to 315 GHz, based on which the +PDPs can be shorten to 401 sample points. +The PDPs of the 21 measured channels make up the mea- +sured dataset. In addition to the measured dataset, the dataset +of simulated PDPs can be generated by 3GPP model with +the extracted statistics from the measurement, which consists +of 10000 channels. Compared to the measured dataset, the +simulated dataset has larger data size with the channel statistics +embedded. Moreover, the PDPs in two datasets are normalized +into the range of [0, 1] by the min-max normalization method. +The training procedure of the GAN network is explained +in detail as follows. Firstly, the input noise vector z of size +100 is generated by the multivariate normal distribution, which +can provide the capabilities to transform into the desired +distribution. The gradient penalty parameter λ in (4) is set +as 10, which works well in the training process. Moreover, +the stochastic gradient descent (SGD) optimizer is applied for +the generator network, and the adaptive moment estimation +(Adam) optimizer is chosen for the discriminator network. In +addition, the learning rates of the two optimizers are both set +as 0.0002 to stabilize the training. +All the experimental results are implemented on a PC with +AMD Ryzen Threadripper 3990X @ 2.19 GHz and four +Nvidia GeForce RTX 3090 Ti GPUs. In addition, the training +of GAN network is carried out in the Pytorch framework. +0 +2000 +4000 +6000 +8000 +10000 +Epoch +-3 +-2 +-1 +0 +1 +2 +3 +4 +Loss +G_loss (simulated dataset) +D_loss(simulated dataset) +G_loss (measured dataset) +D_loss (measured dataset) +TG_loss (measured dataset) +TD_loss (measured dataset) +Fig. 5. Loss of the generator and discriminator in the GAN network. +B. Convergence +The proposed GAN is first trained on the simulated dataset, +and is then fine-tuned on the measured dataset with transfer +learning to develop the T-GAN. The numbers of epochs for +training the proposed GAN and T-GAN are both set as 10000. +A epoch is defined as a complete training cycle through the +training dataset, in which the generator and discriminator are +iteratively trained for once. To demonstrate the benefits of +transfer learning, the GAN is also trained on the measured +dataset without transfer learning for comparison. The loss of +generator denoted by G loss and loss of discriminator denoted +by D loss are shown in the Fig. 5, in which the TG loss and +TD loss correspond to the losses for T-GAN. For the simu- +lated dataset, it is clear that the generator and discriminator +reach the equilibrium in the end. For the measured dataset, the +loss of T-GAN is close to the loss for the simulated dataset +except for some small fluctuations. The fluctuations are due +to the small size of the measured dataset. By comparison, the +training is not stable for the GAN network without transfer +leaning. There is large fluctuation in the discriminator loss, +and the absolute values of G loss and D loss are quite +large compared to the losses for the simulated dataset. The +comparison demonstrates the benefits of the transfer learning +in the training of GAN network, which enables T-GAN to +converge with a small training dataset. Moreover, it takes +only 4000 epochs for T-GAN to converge, compared to 6000 +epochs for GAN trained on the simulated dataset. The training +time of T-GAN on the measured dataset is also small, which +is only 114 seconds compared to 7 hours for GAN trained +on the simulated dataset. From these results, it is clear that +the transfer learning technique can improve the convergence +rate of T-GAN, and reduce the training overhead with the +knowledge from the pre-trained model. + +nDoor - + Wooden wall + Glass wall +Concrete wall Metal pillars +a +g +ka +D +业下 +不业 +Rx 1 ～ Rx 15 +Rx 16 ～ Rx 21 +0.86 m 1.26 m +N +TX +0.93 m +5.88 m +F +5 m +3 +19 m +S +31 m +58 m +D +buD +ID +C +9000 +100 +200 +300 +400 + [ns] +-85 +-80 +-75 +-70 +-65 +-60 +-55 +-50 +Power [dB] +Measurement +3GPP +GAN +T-GAN +(a) Samples of PDP. +0 +100 +200 +300 +400 + [ns] +-82 +-80 +-78 +-76 +-74 +-72 +-70 +-68 +-66 +Power [dB] +Measurement +3GPP +GAN +T-GAN +(b) Average PDP. +Fig. 6. Plot of PDPs generated by measurement, 3GPP, the proposed GAN and T-GAN. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +SSIM +0 +0.2 +0.4 +0.6 +0.8 +1 +Culmultative probability function +3GPP +GAN +T-GAN +Fig. 7. SSIM of PDP for 3GPP, the proposed GAN and T-GAN. +C. Power Delay Profile +In the experiment, the samples of PDP from measurement, +3GPP method, the proposed GAN and T-GAN are compared +as in Fig. 6(a). It is clear that the PDPs are similar to each +other, which proves that the proposed GAN and T-GAN can +learn the features of PDPs. Moreover, it is observed that PDP +of measurement is more complex than PDP of 3GPP method. +There are more peaks and fluctuations in the temporal domain. +This shows that 3GPP cannot well capture the channel effects +embedded in PDP. Comparing PDPs generated by the proposed +GAN and T-GAN, the PDP generated by T-GAN is close to +measurement, while the PDP generated by the proposed GAN +is similar to the 3GPP approach. This is reasonable, since the +T-GAN can capture the features of PDP from measurement +0 +20 +40 +60 +80 +100 +120 +140 +Delay spread [ns] +0 +0.2 +0.4 +0.6 +0.8 +1 +Culmulative probability function +Measurement +3GPP +GAN +T-GAN +Fig. 8. Delay spread for 3GPP, the proposed GAN and T-GAN. +through transfer learning, while the propose GAN can only +learn the features of the simulated PDPs by 3GPP method. +In addition, the average PDPs for these method are plotted +in Fig. 6(b). It is clear that T-GAN shows good agreement with +measurement, while 3GPP and GAN have large deviations +from measurement. The deviations can be measured by root- +mean-square error (RMSE), calculated as +RMSE = +� +1 +Nτ +� +(Pm(i∆τ) − Pg(i∆τ))2, +(7) +where Nτ denotes the number of sampling points in PDP, i +indexs temporal sample points of PDPs, Nτ represents the +number of sampling points and ∆τ is the sampling interval. +Moreover, Pm(i∆τ) and Pg(i∆τ) are the average power in the + +ith sample point of measured PDPs and generated PDPs, re- +spectively. The results of RMSE for 3GPP, the proposed GAN +and T-GAN are 4.29 dB, 4.12 dB and -4.82 dB, respectively. +The T-GAN improves the performance of RMSE by about 9 +dB, compared with other methods, which demonstrates that the +T-GAN outperforms the other methods in terms of modeling +the average power of PDP. This is attributed to the powerful +capability of GAN in modeling the complex distribution, and +the benefits of transfer learning in better utilizing the small +measurement dataset. +Moreover, to measure the similarity quantitatively, Structure +Similarity Index Measure (SSIM) is introduced, which is +widely applied to evaluate the quality and similarity of images. +The range of SSIM is from 0 to 1, and the value of SSIM is +larger when the similarity between images is higher. The PDPs +generated by 3GPP method, the proposed GAN and T-GAN +are compared with measurement. The cumulative probability +functions (CDFs) of SSIM for these method are shown in +Fig. 7. It can be observed that the proposed T-GAN can +achieve higher SSIM values compared with other methods. +More than 40% of SSIM values are higher than 0.6 for T- +GAN, compared to only 20% for 3GPP and the proposed +GAN. This further demonstrates the better performance of T- +GAN in modeling the PDPs. +D. Delay Spread +Delay spread characterizes the power dispersion of multi- +path components in the temporal domain, which can be +calculated as the second central moment of PDPs, by +¯τ = +�Nτ +i=0 i∆τP(i∆τ)∆τ +�Nτ +i=0 P(i∆τ)∆τ +, +τrms = +� +� +� +� +�Nτ +i=0(i∆τ − ¯τ)2P(i∆τ)∆τ +�Nτ +i=0 P(i∆τ)∆τ +, +(8) +where ¯τ denotes the mean delay weighted by the power, +τrms refers to the root-mean-square (RMS) delay spread, and +P(i∆τ) are the power in the ith sample point of PDPs. +Then, the CDF plot of delay spread for measurement, 3GPP, +the proposed GAN and T-GAN is depicted in Fig. 8. It can be +observed that the CDFs of delay spread for 3GPP, the proposed +GAN and T-GAN match the measurement well. +IV. CONCLUSION +In this paper, we proposed a T-GAN based THz channel +modeling method, which can capture the distribution of PDPs +for the THz channel with the designed GAN. Moreover, the +transfer learning is exploited in T-GAN to reduce the size +requirement of training dataset and enhance the performance +of GAN, through transferring the knowledge stored in the +pre-trained GAN on the simulated dataset to the target T- +GAN trained on limited measurement. Finally, we validate +the performance of T-GAN with measurement. T-GAN can +generate the PDPs that have good agreement with measure- +ment. Compared with conventional methods, T-GAN has better +performance in modeling the distribution of PDPs, with 9 dB +improved RMSE and higher SSIM. More than 40% of SSIM +values are higher than 0.6 for T-GAN, compared to only 20% +for 3GPP and the proposed GAN. +REFERENCES +[1] I. F. Akyildiz, C. Han, Z. Hu, S. Nie, and J. M. Jornet, “Terahertz band +communication: An old problem revisited and research directions for +the next decade (invited paper),” IEEE Trans. Commun., vol. 70, no. 6, +pp. 4250–4285, 2022. +[2] Z. Chen et al., “Terahertz wireless communications for 2030 and beyond: +A cutting-edge frontier,” IEEE Commun. Mag., vol. 59, no. 11, pp. 66– +72, 2021. +[3] C. Han, Y. Wu, Z. Chen, Y. Chen, and G. Wang, “THz ISAC: A Physical- +Layer Perspective of Terahertz Integrated Sensing and Communication,” +arXiv preprint:2209.03145, 2022. +[4] P. Petrus, J. Reed, and T. Rappaport, “Geometrical-based statistical +macrocell channel model for mobile environments,” IEEE Trans. Com- +mun., vol. 50, no. 3, pp. 495–502, 2002. +[5] Y. Chen, L. Yan, and C. Han, “Hybrid spherical- and planar-wave +modeling and DCNN-powered estimation of Terahertz Ultra-Massive +MIMO channels,” IEEE Trans. Commun., vol. 69, no. 10, pp. 7063– +7076, 2021. +[6] Y. Chen, L. Yan, C. Han, and M. Tao, “Millidegree-level direction- +of-arrival estimation and tracking for terahertz ultra-massive mimo +systems,” IEEE Trans. Wirel. Commun., vol. 21, no. 2, pp. 869–883, +2022. +[7] T. Wang, C.-K. Wen, S. Jin, and G. Y. Li, “Deep learning-based CSI +feedback approach for time-varying massive MIMO channels,” IEEE +Wirel. Commun. Lett., vol. 8, no. 2, pp. 416–419, 2019. +[8] T. J. O’Shea, T. Roy, and N. West, “Approximating the void: Learning +stochastic channel models from observation with variational generative +adversarial networks,” in Proc. Int. Conf. Comput., Netw. Commun., +2019, pp. 681–686. +[9] H. Xiao, W. Tian, W. Liu, and J. Shen, “ChannelGan: Deep learning- +based channel modeling and generating,” IEEE Wirel. Commun. Lett., +vol. 11, no. 3, pp. 650–654, 2022. +[10] T. Orekondy, A. Behboodi, and J. B. Soriaga, “MIMO-GAN: Generative +MIMO channel modeling,” in Proc. IEEE Int. Conf. Commun., 2022, pp. +5322–5328. +[11] Y. Wei, M.-M. Zhao, and M.-J. Zhao, “Channel distribution learning: +Model-driven gan-based channel modeling for irs-aided wireless com- +munication,” IEEE Trans. Commun., vol. 70, no. 7, pp. 4482–4497, +2022. +[12] N. V. Huynh and G. Y. Li, “Transfer learning for signal detection in +wireless networks,” IEEE Wirel. Commun. Lett., pp. 1–1, 2022. +[13] Y. Wang, C. Wu, L. Herranz, J. van de Weijer, A. Gonzalez-Garcia, and +B. Raducanu, “Transferring gans: generating images from limited data,” +in ECCV, 2018. +[14] Y. Li, Y. Wang, Y. Chen, Z. Yu, and C. Han, “Channel measurement +and analysis in an indoor corridor scenario at 300 ghz,” in Proc. IEEE +Int. Conf. Commun., 2022, pp. 2888–2893. +[15] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. Courville, +“Improved training of Wasserstein GANs,” in Proc. Int. Conf. Neural +Inf. Process. Syst., 2017, p. 5769–5779. +[16] Study on Channel Model for Frequencies From 0.5 to 100 GHz (Release +15), document TR 38.901, 3GPP, 2018. + diff --git a/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/load_file.txt b/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5db9fbfa1a48331c416b930a50a9bb396f49f483 --- /dev/null +++ b/2tAzT4oBgHgl3EQfDvpk/content/tmp_files/load_file.txt @@ -0,0 +1,405 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf,len=404 +page_content='Transfer Generative Adversarial Networks (T-GAN)-based Terahertz Channel Modeling Zhengdong Hu, Yuanbo Li, and Chong Han Terahertz Wireless Communications (TWC) Laboratory, Shanghai Jiao Tong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Email: {huzhengdong, yuanbo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='li, chong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='han}@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='cn Abstract—Terahertz (THz) communications are envisioned as a promising technology for 6G and beyond wireless systems, providing ultra-broad bandwidth and thus Terabit-per-second (Tbps) data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' However, as foundation of designing THz communications, channel modeling and characterization are fundamental to scrutinize the potential of the new spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Relied on physical measurements, traditional statistical channel modeling methods suffer from the problem of low accuracy with the assumed certain distributions and empirical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, it is time-consuming and expensive to acquire extensive channel measurement in the THz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In this paper, a transfer generative adversarial network (T-GAN) based modeling method is proposed in the THz band, which exploits the advantage of GAN in modeling the complex distribution, and the benefit of transfer learning in transferring the knowledge from a source task to improve generalization about the target task with limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Specifically, to start with, the proposed GAN is pre- trained using the simulated dataset, generated by the standard channel model from 3rd generation partnerships project (3GPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Furthermore, by transferring the knowledge and fine-tuning the pre-trained GAN, the T-GAN is developed by using the THz mea- sured dataset with a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Experimental results reveal that the distribution of PDPs generated by the proposed T-GAN method shows good agreement with measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, T- GAN achieves good performance in channel modeling, with 9 dB improved root-mean-square error (RMSE) and higher Structure Similarity Index Measure (SSIM), compared with traditional 3GPP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' INTRODUCTION With the exponential growth of the number of intercon- nected devices, the sixth generation (6G) is expected to achieve intelligent connections of everything, anywhere, anytime [1], which demands Tbit/s wireless data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To fulfill the de- mand, Terahertz (THz) communications gain increasing atten- tion as a vital technology of 6G systems, thanks to the ultra- broad bandwidth ranging from tens of GHz to hundreds of GHz [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The THz band is promising to address the spectrum scarcity and capacity limitations of current wireless systems, and realize long-awaited applications, extending from wireless cognition, localization/positioning, to integrated sensing and communication [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To design reliable THz wireless systems, one fundamental challenge lies in developing an accurate channel model to por- tray the propagation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Due to the high frequencies, new characteristics occur in the THz band, such as frequency- selective absorption loss and rough-surface scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' At- tribute to these new characteristics, THz channel modeling is required to capture these characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' However, traditional statistical channel modeling methods suffer from the prob- lem of low accuracy with the assumed certain distributions and empirical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' For example, a geometric based stochastic channel model (GSCM) assumes that the positions of scatters follow certain statistical distributions, such as the uniform distribution within a circle around the transmitters and receivers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' However, the positions of scatters are hard to characterize by certain statistical distributions, making the GSCM not accurate for utilization in the THz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, it is time-consuming and costly to acquire extensive channel measurement for THz channel modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To this end, an accurate channel modeling method with limited measurement data for the THz band is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Recently, deep learning (DL) is popular and widely applied in wireless communications, such as channel estimation [5], [6] and channel state information (CSI) feedback [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Among different kinds of DL methods, the generative adversarial network (GAN) has the advantage of modeling complex dis- tribution accurately without any statistical assumptions, based on which GAN can be utilized to develop channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The authors in [8] train GAN to approximate the probability distri- bution functions (PDFs) of stochastic channel response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In [9], GAN is applied to generate synthetic channel samples close to the distribution of real channel samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The researchers in [10] model the channel with GAN through channel input- output measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In [11], a model-driven GAN-based channel modeling method is developed in intelligent reflecting surface (IRS) aided communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' These methods achieve good performance in modeling the channel and prove high consistency between the target channel distribution and the generated channel distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' However, the GAN based channel modeling method has not been exploited in the THz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, it is a challenge to train GAN for channel modeling with the scarce THz channel measurement dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In this paper, a transfer GAN (T-GAN)-based THz channel modeling method is proposed, which can learn the distribution of power delay profile (PDP) of the THz channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, to tackle the challenge of limited channel measurement in the THz band, the transfer learning technique is introduced in T-GAN, which reduces the size requirement of channel dataset for training and enhances the performance of channel modeling, through transferring the knowledge stored in a pre- trained model to a new model [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Furthermore, the performance of T-GAN in modeling the channel distribution is validated by real measurements [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The contributions of this paper are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' We propose a T-GAN based THz channel modeling arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='00981v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='SP] 3 Jan 2023 method, in which a GAN is designed to capture the distribution of PDPs of the THz channel, by training on the dataset of PDP samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To tackle the challenge of limited measurement data for THz channel modeling, transfer learning is further exploited by T-GAN, which reduces the size requirement of training dataset, and enhances the performance of GAN, through transferring the knowledge stored in a pre- trained model to a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The rest of the sections are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' II details the proposed T-GAN based channel modeling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' III demonstrates the performance of the proposed T-GAN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The paper is concluded in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Notation: a is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' a denotes a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' A represents a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' E{·} describes the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' ∇ denotes the gradient operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' ∥·∥ represent the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' IN defines an N dimen- sional identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' N denotes the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' TRANSFER GAN (T-GAN) BASED CHANNEL MODELING In this section, the channel modeling problem is first for- mulated into a channel distribution learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Then, the proposed GAN in T-GAN method is elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Finally, T-GAN is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Problem Formulation The channel impulse response (CIR) can be represented as h(τ) = L−1 � l=0 αlejφlδ(τ − τl), (1) where τl denotes the delay of the lth multi-path components (MPCs), L denotes the number of MPC, αl refers to the path gain and φl represents the phase of the corresponding MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To characterize the channel, PDP is an important feature, which indicates the dispersion of power over the time delay, specifically, the received power with respect to the delay in a multi-path channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It can be extracted from the channel impulse response by P(τ) = |h(τ)|2, (2) Then, the channel modeling problem is exploited by learning the distribution of PDPs denoted by pr, which is difficult to be analytically represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Instead, the distribution pr can be captured by generating fake PDP samples with distribution pg, such that the generated distribution pg of PDPs can match the actual distribution pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Proposed GAN The GAN can be utilized to learn the distribution of PDPs denoted by pr, with the framework depicted in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The GAN consists of two sub-networks, namely, generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The generator is aimed at generating fake samples G(z) to fool the discriminator, in which z is the noise sample, by mapping the input noise distribution pz(z) to the generated distribution pg = p(G(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The discriminator tries to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Framework of GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' distinguish between real samples x from pr and fake samples G(z) from pg, and the output of the discriminator D(x) and D(G(z)) can be treated as the probability of being a real sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The two networks are trained in an adversarial manner, which can be considered as a two-player zero-sum minimax game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Specifically, the training objective can be represented by min G max D Ex∼pr[log D(x)] + Ez∼pz[log(1 − D(G(z)))], (3) where the generator minimizes the probability (1 − D(G(z)) that the generated sample is detected as fake by the dis- criminator, while the discriminator maximizes this probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Therefore, the generator and discriminator compete against each other with the opposite objectives in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Through the adversarial training, the Nash equilibrium can be achieved, such that the generator and discriminator cannot improve their objectives by changing only their own network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the global optimum of the training objective can be achieved in the equilibrium when pg = pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' However, training with the objective function in (3) is unstable, since the training objective is potentially not continuous with respect to the generator’s parameters [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Therefore, the improved version of GAN, namely, Wasserstein GAN with gradient penalty (WGAN-GP) [15] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The modified objective function is expressed as min G max D Ex∼pr[D(x)]+Ez∼pz[(1 − D(G(z)))] +λE˜x[(∥∇˜xD(˜x)∥ − 1)2)], (4) where the last term is the gradient penalty term to enforce Lipschitz constraint that the gradient of the GAN network is upper-bounded by a maximum value, the symbol ˜x is the uniformly sampled point between the points of x and G(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the parameter λ is the penalty coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' After introducing the framework of GAN, the detailed architecture of proposed GAN network is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The structures of generator G and discriminator D are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 2, where the number in the bracket denotes the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The input to the generator is a noise vector z with dimension nz = 100, which is sampled from the probability density function N(0, σ2Inz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The generator consists of five dense layers, and the numbers of neurons in the dense layers are 128, 128, 128, 128, 401, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It is worth noting that the size of the output layer is equal to the size of PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Structure of generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' activation function of the first four dense layers is LeakyReLU function, which can speed up the convergence and avoid the gradient vanishing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The formula of the LeakyReLU function is expressed as f(x) = � x, if x ≥ 0 αx, if x < 0 , (5) where α is the slope coefficient when the value of neuron x is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In addition to the LeakyReLU function, a Sigmoid function is utilized in the last layer, which maps the output to the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The Sigmoid function is defined as f(x) = 1 1 + e−x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' (6) After going through the dense layers and activation functions in the generator, the input noise vectors are transformed into the generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Then, the generated samples together with real samples are passed to the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The discriminator is designed to distinguish between gen- erated samples and real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The numbers of neurons for the five dense layers in the discriminator are 512, 256, 128, 64, 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The activation function chosen for the first 4 layers is the LeakyReLU function introduced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The activation function for the output layer is linear activation function, which is decided by the objective function of WGAN-GP introduced in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Proposed T-GAN The framework for the proposed T-GAN is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 3, in which the transfer learning is conducted between the measurement and 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='901 model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The measured PDPs denote the PDPs extracted from measurement with a small size, while the simulated PDPs refer to the PDPs simulated by the 3GPP model, which is implemented with the extracted statistics from measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Then, the proposed GAN and T-GAN with the same network structure, are trained on the simulated PDPs and measured PDPs, respectively, to capture the distribution of PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Since the size of measured PDPs is quite small for the training of T-GAN, which can cause the difficulty of converging or the over-fitting problem, the transfer learning is exploited to tackle these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Framework for T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To describe the transfer learning formally, a domain denoted by D consists of a feature space X and a marginal probability distribution P(X) defined on X = {x1, x2, · · · , xN} ∈ X, where N is the number of feature vectors in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 3, the target domain Dt and source domain Ds are defined on measurement and 3GPP model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The feature spaces for the two domains are both constructed by PDPs, with different marginal probability distributions defined on measured PDPs Xt and simulated PDPs Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, given a domain D(X, P(X)), a task denoted by T is defined by a label space L and a predictive function f(·), and the predictive function is learned from the pairs (xn, ln) with xn ∈ X and ln ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In the target domain Dt and source domain Ds, the tasks are the same to capture the distribution of PDPs, and the label space is L = {0, 1} representing whether the PDP sample is generated by the proposed GAN or from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The T-GAN and GAN serve as the predictive functions ft and fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Then, transfer learning is aimed at learning the function ft in target domain Dt with the knowledge of Ts in source domain Ds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', transferring the knowledge stored in GAN trained on simulated PDPs to T-GAN trained on the measured PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The method of fine-tuning [13] is adopted for the transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The T-GAN is initialized with the weights of the GAN trained on the simulated PDPs, and is then fine-tuned on the measured PDPs with small size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It is worth noting that both the generator and discriminator in the GAN are trans- ferred, which can yield the better performance in generating high quality samples and fast convergence, compared with transferring only the generator or the discriminator [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' With transfer learning, the performance of T-GAN can be largely enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Specifically, the channel statistics extracted for 3GPP method are captured by the proposed GAN trained on simulated PDPs, which are further transferred to T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, T-GAN can learn the features of PDPs that are not captured by 3GPP method, directly from measurement, which further improves the performance of T-GAN in modeling the distribution of PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Target Domain Source DomainFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Measurement layout in the indoor corridor scenario [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' EXPERIMENT AND PERFORMANCE EVALUATION In this section, the experiment settings are elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the performance of the T-GAN are evaluated by comparing the generated distribution of PDPs with measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Dataset and Setup The dataset is collected from the measurement campaign in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' which is conducted in an indoor corridor scenario at 306-321 GHz with 400 ns maximum delay, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' With the measurement data, the PDPs can be extracted to characterize the channel in the 21 receiver points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Since the sample frequency interval is relatively small, as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='5 MHz, the measured PDPs are very long, including 6001 sample points, which results in extraordinary computation and time consumption to train the GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To address this problem, we only use the measured channel transfer functions in the frequency band from 314 to 315 GHz, based on which the PDPs can be shorten to 401 sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The PDPs of the 21 measured channels make up the mea- sured dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In addition to the measured dataset, the dataset of simulated PDPs can be generated by 3GPP model with the extracted statistics from the measurement, which consists of 10000 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Compared to the measured dataset, the simulated dataset has larger data size with the channel statistics embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the PDPs in two datasets are normalized into the range of [0, 1] by the min-max normalization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The training procedure of the GAN network is explained in detail as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Firstly, the input noise vector z of size 100 is generated by the multivariate normal distribution, which can provide the capabilities to transform into the desired distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The gradient penalty parameter λ in (4) is set as 10, which works well in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the stochastic gradient descent (SGD) optimizer is applied for the generator network, and the adaptive moment estimation (Adam) optimizer is chosen for the discriminator network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In addition, the learning rates of the two optimizers are both set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='0002 to stabilize the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' All the experimental results are implemented on a PC with AMD Ryzen Threadripper 3990X @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='19 GHz and four Nvidia GeForce RTX 3090 Ti GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In addition, the training of GAN network is carried out in the Pytorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 0 2000 4000 6000 8000 10000 Epoch 3 2 1 0 1 2 3 4 Loss G_loss (simulated dataset) D_loss(simulated dataset) G_loss (measured dataset) D_loss (measured dataset) TG_loss (measured dataset) TD_loss (measured dataset) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Loss of the generator and discriminator in the GAN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Convergence The proposed GAN is first trained on the simulated dataset, and is then fine-tuned on the measured dataset with transfer learning to develop the T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The numbers of epochs for training the proposed GAN and T-GAN are both set as 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' A epoch is defined as a complete training cycle through the training dataset, in which the generator and discriminator are iteratively trained for once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' To demonstrate the benefits of transfer learning, the GAN is also trained on the measured dataset without transfer learning for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The loss of generator denoted by G loss and loss of discriminator denoted by D loss are shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 5, in which the TG loss and TD loss correspond to the losses for T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' For the simu- lated dataset, it is clear that the generator and discriminator reach the equilibrium in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' For the measured dataset, the loss of T-GAN is close to the loss for the simulated dataset except for some small fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The fluctuations are due to the small size of the measured dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' By comparison, the training is not stable for the GAN network without transfer leaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' There is large fluctuation in the discriminator loss, and the absolute values of G loss and D loss are quite large compared to the losses for the simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The comparison demonstrates the benefits of the transfer learning in the training of GAN network, which enables T-GAN to converge with a small training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, it takes only 4000 epochs for T-GAN to converge, compared to 6000 epochs for GAN trained on the simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The training time of T-GAN on the measured dataset is also small, which is only 114 seconds compared to 7 hours for GAN trained on the simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' From these results, it is clear that the transfer learning technique can improve the convergence rate of T-GAN, and reduce the training overhead with the knowledge from the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' nDoor - Wooden wall Glass wall Concrete wall Metal pillars a g ka D 业下 不业 Rx 1 ～ Rx 15 Rx 16 ～ Rx 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='86 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='26 m N TX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='93 m 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='88 m F 5 m 3 19 m S 31 m 58 m D buD ID C 9000 100 200 300 400 [ns] 85 80 75 70 65 60 55 50 Power [dB] Measurement 3GPP GAN T-GAN (a) Samples of PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 0 100 200 300 400 [ns] 82 80 78 76 74 72 70 68 66 Power [dB] Measurement 3GPP GAN T-GAN (b) Average PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Plot of PDPs generated by measurement, 3GPP, the proposed GAN and T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='9 SSIM 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='8 1 Culmultative probability function 3GPP GAN T-GAN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' SSIM of PDP for 3GPP, the proposed GAN and T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Power Delay Profile In the experiment, the samples of PDP from measurement, 3GPP method, the proposed GAN and T-GAN are compared as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It is clear that the PDPs are similar to each other, which proves that the proposed GAN and T-GAN can learn the features of PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, it is observed that PDP of measurement is more complex than PDP of 3GPP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' There are more peaks and fluctuations in the temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' This shows that 3GPP cannot well capture the channel effects embedded in PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Comparing PDPs generated by the proposed GAN and T-GAN, the PDP generated by T-GAN is close to measurement, while the PDP generated by the proposed GAN is similar to the 3GPP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' This is reasonable, since the T-GAN can capture the features of PDP from measurement 0 20 40 60 80 100 120 140 Delay spread [ns] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='8 1 Culmulative probability function Measurement 3GPP GAN T-GAN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Delay spread for 3GPP, the proposed GAN and T-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' through transfer learning, while the propose GAN can only learn the features of the simulated PDPs by 3GPP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' In addition, the average PDPs for these method are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It is clear that T-GAN shows good agreement with measurement, while 3GPP and GAN have large deviations from measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The deviations can be measured by root- mean-square error (RMSE), calculated as RMSE = � 1 Nτ � (Pm(i∆τ) − Pg(i∆τ))2, (7) where Nτ denotes the number of sampling points in PDP, i indexs temporal sample points of PDPs, Nτ represents the number of sampling points and ∆τ is the sampling interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, Pm(i∆τ) and Pg(i∆τ) are the average power in the ith sample point of measured PDPs and generated PDPs, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The results of RMSE for 3GPP, the proposed GAN and T-GAN are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='29 dB, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='12 dB and -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='82 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The T-GAN improves the performance of RMSE by about 9 dB, compared with other methods, which demonstrates that the T-GAN outperforms the other methods in terms of modeling the average power of PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' This is attributed to the powerful capability of GAN in modeling the complex distribution, and the benefits of transfer learning in better utilizing the small measurement dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, to measure the similarity quantitatively, Structure Similarity Index Measure (SSIM) is introduced, which is widely applied to evaluate the quality and similarity of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The range of SSIM is from 0 to 1, and the value of SSIM is larger when the similarity between images is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The PDPs generated by 3GPP method, the proposed GAN and T-GAN are compared with measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' The cumulative probability functions (CDFs) of SSIM for these method are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It can be observed that the proposed T-GAN can achieve higher SSIM values compared with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' More than 40% of SSIM values are higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='6 for T- GAN, compared to only 20% for 3GPP and the proposed GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' This further demonstrates the better performance of T- GAN in modeling the PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Delay Spread Delay spread characterizes the power dispersion of multi- path components in the temporal domain, which can be calculated as the second central moment of PDPs, by ¯τ = �Nτ i=0 i∆τP(i∆τ)∆τ �Nτ i=0 P(i∆τ)∆τ , τrms = � � � � �Nτ i=0(i∆τ − ¯τ)2P(i∆τ)∆τ �Nτ i=0 P(i∆τ)∆τ , (8) where ¯τ denotes the mean delay weighted by the power, τrms refers to the root-mean-square (RMS) delay spread, and P(i∆τ) are the power in the ith sample point of PDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Then, the CDF plot of delay spread for measurement, 3GPP, the proposed GAN and T-GAN is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' It can be observed that the CDFs of delay spread for 3GPP, the proposed GAN and T-GAN match the measurement well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a T-GAN based THz channel modeling method, which can capture the distribution of PDPs for the THz channel with the designed GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Moreover, the transfer learning is exploited in T-GAN to reduce the size requirement of training dataset and enhance the performance of GAN, through transferring the knowledge stored in the pre-trained GAN on the simulated dataset to the target T- GAN trained on limited measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Finally, we validate the performance of T-GAN with measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' T-GAN can generate the PDPs that have good agreement with measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Compared with conventional methods, T-GAN has better performance in modeling the distribution of PDPs, with 9 dB improved RMSE and higher SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' More than 40% of SSIM values are higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='6 for T-GAN, compared to only 20% for 3GPP and the proposed GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' REFERENCES [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Akyildiz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Nie, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Jornet, “Terahertz band communication: An old problem revisited and research directions for the next decade (invited paper),” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 4250–4285, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', “Terahertz wireless communications for 2030 and beyond: A cutting-edge frontier,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 66– 72, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wang, “THz ISAC: A Physical- Layer Perspective of Terahertz Integrated Sensing and Communication,” arXiv preprint:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='03145, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Petrus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Reed, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Rappaport, “Geometrical-based statistical macrocell channel model for mobile environments,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 495–502, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Yan, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Han, “Hybrid spherical- and planar-wave modeling and DCNN-powered estimation of Terahertz Ultra-Massive MIMO channels,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 7063– 7076, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Han, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Tao, “Millidegree-level direction- of-arrival estimation and tracking for terahertz ultra-massive mimo systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 869–883, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Jin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Li, “Deep learning-based CSI feedback approach for time-varying massive MIMO channels,” IEEE Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 416–419, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' O’Shea, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Roy, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' West, “Approximating the void: Learning stochastic channel models from observation with variational generative adversarial networks,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 681–686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Xiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Tian, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Shen, “ChannelGan: Deep learning- based channel modeling and generating,” IEEE Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 650–654, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Orekondy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Behboodi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Soriaga, “MIMO-GAN: Generative MIMO channel modeling,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 5322–5328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Zhao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Zhao, “Channel distribution learning: Model-driven gan-based channel modeling for irs-aided wireless com- munication,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 4482–4497, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Huynh and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Li, “Transfer learning for signal detection in wireless networks,” IEEE Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 1–1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Herranz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' van de Weijer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Gonzalez-Garcia, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Raducanu, “Transferring gans: generating images from limited data,” in ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Yu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Han, “Channel measurement and analysis in an indoor corridor scenario at 300 ghz,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 2888–2893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Gulrajani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Ahmed, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Arjovsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Dumoulin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Courville, “Improved training of Wasserstein GANs,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Neural Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=', 2017, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' 5769–5779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content=' [16] Study on Channel Model for Frequencies From 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='5 to 100 GHz (Release 15), document TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} +page_content='901, 3GPP, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfDvpk/content/2301.00981v1.pdf'} diff --git a/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf b/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..34c0c01527114200421ce9a393f3e57bf062dffc --- /dev/null +++ b/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3a73294f6d766dcc1c27105c28d2820cd67c5f9fda070e130b89f37c7cf0c5d0 +size 1495493 diff --git a/2tE1T4oBgHgl3EQflgTe/vector_store/index.faiss b/2tE1T4oBgHgl3EQflgTe/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..55cc73e33e677ade176dfeacd3d40bb1c06db773 --- /dev/null +++ b/2tE1T4oBgHgl3EQflgTe/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9b71792098cdc558dcd56cfe7f8d1861931ff682dfc02000e053aa44d1df814e +size 3866669 diff --git a/2tE1T4oBgHgl3EQflgTe/vector_store/index.pkl b/2tE1T4oBgHgl3EQflgTe/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a57c1766b56e6f549500be9678ae6b3909146c69 --- /dev/null +++ b/2tE1T4oBgHgl3EQflgTe/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a30a0d4868b0d08a647582eff8cc825a3ef9659c3d97ab4e9a24f6eaffad5b09 +size 157027 diff --git a/3NE3T4oBgHgl3EQfoQom/content/tmp_files/2301.04631v1.pdf.txt b/3NE3T4oBgHgl3EQfoQom/content/tmp_files/2301.04631v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8be2e5860c5942db6469f57424982d6b4c9ee5fb --- /dev/null +++ b/3NE3T4oBgHgl3EQfoQom/content/tmp_files/2301.04631v1.pdf.txt @@ -0,0 +1,1043 @@ +Deep Residual Axial Networks +Nazmul Shahadat, Anthony S. Maida +University of Louisiana at Lafayette +Lafayette LA 70504, USA +nazmul.ruet@gmail.com, maida@louisiana.edu +Abstract +While residual networks (ResNets) demonstrate out- +standing performance on computer vision tasks, their +computational cost still remains high. Here, we focus +on reducing this cost by proposing a new network archi- +tecture, axial ResNet, which replaces spatial 2D convo- +lution operations with two consecutive 1D convolution +operations. Convergence of very deep axial ResNets has +faced degradation problems which prevent the networks +from performing efficiently. To mitigate this, we apply a +residual connection to each 1D convolutional operation +and propose our final novel architecture namely residual +axial networks (RANs). Extensive benchmark evaluation +shows that RANs outperform with about 49% fewer pa- +rameters than ResNets on CIFAR benchmarks, SVHN, +and Tiny ImageNet image classification datasets. More- +over, our proposed RANs show significant improvement +in validation performance in comparison to the wide +ResNets on CIFAR benchmarks and the deep recursive +residual networks on image super resolution dataset. +1. Introduction +Deep convolutional neural network (CNN) based ar- +chitectures, specifically ResNets [11], have achieved +significant success for image processing tasks, includ- +ing classification [10,11,21], object detection [6,22] and +image super-resolution [17, 18, 28]. The performance +of deep ResNets and wide ResNets has improved in re- +cent years. Along with the increasing depth or widening +of ResNets, the computational cost of the networks also +rises. Moreover, training these deeper or wider networks +has faced exploding or vanishing gradient and degrada- +tion problems. Different initialization, optimization, and +normalization techniques [9,10,16,25,27,30], skip con- +nections [10], and transfer learning [5] have been used +used to mitigate these problems. The rising computa- +tional cost and/or trainable parameter is still unexplored +which is the main purpose of this paper. +However, the computational cost of these deeper and +wider ResNets has not been analyzed yet. +Deep or +wide ResNets gain popularity and impressive perfor- +mance due to their simple but effective architectures +[4, 8, 14, 31, 33]. Deep ResNets can be factored as en- +sembles of shallow networks [1] and represent func- +tions more efficiently for complex tasks than shallow +networks [2]. However, constructing deeper ResNets is +not as simple as adding more residual layers. The de- +sign of deeper ResNets demands better optimization and +initialization schemes, and proper use of identity con- +nections. Deeper ResNets have great success in image +classification and object detection tasks [10, 11]. How- +ever, the computational cost increases linearly with the +number of layers [12]. +Wide ResNets use a shallow network with wide (high +channel count) architecture to attain better performance +than the deeper networks [4, 31, 33]. For example, [33] +represented their wide residual network as WRN-n-k +where n is the number of convolutional layers and k rep- +resents the widening factor. They have shown that their +WRN-28-10, wide ResNet that adopts 28 convolutional +layers with k = 10 widening factor, outperforms the +deep ResNet-1001 network (1001 layers). However, the +computational cost is quadratic with a widening factor +of k. +This work revisits the designs of deep and wide +ResNets to boost their performance further, reduce the +above-mentioned high computational costs, and im- +prove the model inference speed. +To get these, we +propose our novel architecture, residual axial networks +(RANs), obtained using axial operations, height or +width-axis, instead of spatial operations in the residual +block. Here, we split 2D spatial (3 × 3) convolution +operation into two consecutive 1D convolution opera- +tions. These 1D convolution operations are mapped to +the height-axis (3 × 1) and width-axis (1 × 3). As axial +1D convolution operations propagate information along +one axis at a time, this modification reduces cost sig- +nificantly. To capture global information, we use these +layers in consecutive pairs. +arXiv:2301.04631v1 [cs.CV] 11 Jan 2023 + +Figure 1. Residual block architectures. “bn” stands for batch +normalization. (Left) ResNet basic block and (Right) ResNet +bottleneck blocks are depicted. +A simple axial architecture reduces cost but does not +improve performance. The reason is that forward in- +formation flows across the axial blocks degrades (di- +minishing feature reuse [15]). To address this, we add +residual connections to span the axial blocks. By using +both modifications, we made a novel, effective residual +axial architecture (RAN). The effectiveness of our pro- +posed model is demonstrated experimentally on four im- +age classification datasets and an image super-resolution +dataset. Our assessments are based on parameter counts, +FLOPS counts (number of multiply-add operations), la- +tency to process one image after training, and validation +accuracy. +2. Background and Related Work +2.1. Convolutional Neural Networks +In a convolutional layer, the core building block is +a convolution operation using one kernel W applied to +small neighborhoods to find input correlations. For an +input image X with height h, width w, and channel +count din, the convolution operation operates on region +(a, b) ∈ X centered at pixel (i, j) with spatial extend k. +The output for this operation Co where o = (i, j) is [26], +Co = +� +(a,b)∈Nk×k(i,j) +W (m) +i−a,j−b, xa,b +(1) +where Nk×k is the neighborhood of pixel (i, j) with spa- +tial extent k, and W is the shared weights to calculate the +output for all pixel positions (i, j). +2.2. Residual Networks +Residual networks (ResNets) are constructed using +convolutional layers linked by additive identity connec- +tions [12]. They were introduced to address the problem +of vanishing gradients found in standard deep CNNs. +Although, the vanishing gradient problem may be ad- +dressed by using normalized inputs and normalization +layers which help to make networks till ten layers. In +this situation, when more layers were stacked, the net- +work depth increases but accuracy gets saturated and +then degrades rapidly. The degradation (of training ac- +curacy) indicates that not all systems are similarly opti- +mized. To address these problems, He et al. proposed +residual networks by adding identity mapping among +the layers [12]. As a result, the subsequent deeper lay- +ers are shared inputs from the learned shallower model. +This helps to address all of the problems. +The key architectural feature of ResNets is the resid- +ual block with identity mapping to tackle the degrada- +tion problem. Two kinds of residual blocks are used +in residual networks, the basic block and the bottleneck +block, both depicted in Figure 1. Figure 1 (left) is known +as the basic architecture of ResNet which is constructed +with two k × k convolution layers where k is the size of +the kernel and an identity shortcut connection is added +to the end of these two layers to address vanishing gradi- +ents. These operations can be expressed mathematically +as, +y = F(Ckm,kn(x, W)) + x +(2) +where F, x, y, W, and Ckm,kn represent residual func- +tion, input vector, output vector, weight parameters, and +output of two convolution layers with kernels Km and +Kn respectively. Figure 1 (right) is a bottleneck archi- +tecture that is constructed using 1 × 1, k × k, and 1 × 1 +convolution layers, where the 1 × 1 layers reduce and +then increase the number of channels, and the 3×3 layer +performs feature extraction. The identity shortcuts (ex- +plained in equation 3) are very important for this block +as it leads to more efficient designs [11]. These can be +expressed mathematically as, +y = F(Ck1,km,k1(x, W)) + x +(3) +where F, x, y, W, and Ck1,km,k1 represent residual +function, input vector, output vector, weight parame- +ters, and output of three convolution layers with kernels +1 × 1, Km × Km and 1 × 1 respectively. Its perfor- +mance surpasses the learning speed, number of learning +parameters, way of layer-wise representation, difficult +optimization property, and memory mechanisms. +2.3. Wide Residual Networks +Wide ResNets [4, 33] use fewer layers compared to +standard ResNets but use high channel counts (wide ar- +chitectures) which compensate for the shallower archi- +tecture. The comparison between shallow and deep net- +works has been revealed in circuit complexity theory + +X +X +3x3 conv2d +1x1 conv2d +bn +bn + relu +★ relu +3x3 conv2d +3x3 conv2d +bn +bn + relu +1x1 conv2d +relu +bn +reluwhere shallow circuits require more components than +the deeper circuit. Inspired by this observation, [11] pro- +posed deeper networks with thinner architecture where +a gradient goes through the layers. +But the problem +such networks face is that the residual block weights +do not flow through the network layers. For this, the +network may be forced to avoid learning during train- +ing. To address these issues, [33] proposed shallow but +wide network architectures and showed that widening +the residual blocks improves the performance of resid- +ual networks compared to increasing their depth. For +example, a 16-layer wide ResNet has similar accuracy +performance to a 1000-layer thinner network. +2.4. Recursive Residual Networks +Image super-resolution (SR) is the process of gen- +erating a high-resolution (HR) image from a low- +resolution (LR) image. +It is also known as single +image super-resolution (SISR). A list of convolution- +based models has shown promising results on SISR +[7, 17, 18, 28]. These 2D convolutional networks learn +a nonlinear mapping from an LR to an HR image in an +end-to-end manner. Convolution-based recursive neural +networks have been used on SISR, where recursive net- +works learn detailed and structured information about an +image. As image SR requires more image details, pool- +ing is not used in deep models for SISR. Convolution- +based SR [7] has shown that the convolution-based LR- +HR mapping significantly improves performance for +classical shallow methods. Kim et al., introduce two +deep CNNs for SR by stacking weight layers [17, 18]. +Among them, [18] uses a chain structure recursive layer +along with skip-connections to control the model pa- +rameters and improve the performance. Deep SR mod- +els [17,18,23] demand large parameter counts and more +storage. +To address these issues, deep recursive residual net- +works (DRRNs) were proposed as a very deep network +structure, which achieves better performance with fewer +parameters [28]. It includes both local and global resid- +ual learning, where global residual learning (GRL) is be- +ing used in the identity branch to estimate the residual +image from the input and output of the network. GRL +might face degradation problems for deeper networks. +To handle this problem, local residual learning (LRL) +has been used which carries rich image details to deeper +layers and helps gradient flow. The DRRN also used +recursive learning of residual units to keep the model +more compact. Several recursive blocks (B) has been +stacked, followed by a CNN layer which is used to re- +construct the residual between the LR and HR images. +Each of these residual blocks decomposed into a num- +ber of residual units (U). The number of recursive block +B, and the number of residual units U are responsible +for defining network depth. The depth of DRRN d is +calculated as, +d = (1 + 2 × U) × B + 1 +(4) +Recursive block definition, DRRN formulation, and the +loss function of DRRN are defined in [28]. +3. Proposed Residual Axial Networks +The convolution-based residual basic and bottleneck +blocks [11, 12] have demonstrated significant perfor- +mance with the help of several state-of-the-art archi- +tectures like, ResNets [11], wide ResNets [31], scal- +ing wide ResNets [33], and deep recursive residual net- +works (DRRNs) [28] on image classification and im- +age super-resolution datasets. Although the bottleneck +residual block makes the networks thinner still the ba- +sic and bottleneck blocks are not cost-effective and/or +parameter efficient. The 2D convolutional operation of +these blocks is consuming O(N 2) resources, where N is +the flattened pixels of an image, and N = hw (for a 2D +image of height h, width w, and h = w). So the cost for +a 2D convolutional operation, for an image with height +h, and width w, is O((hw)2) = O(h2w2) = O(h4) +[13, 29]. To reduce this impractical computational cost, +we are proposing a novel architectural design, residual +axial networks (RANs). +Due to high computational expenses, we replace all +spatial 2D convolution operations (conv2D) of the resid- +ual basic blocks, and the only spatial 2D convolution op- +eration of the residual bottleneck block by using two 1D +convolutional operations. Also, each 1D convolutional +operation has a residual connection to reduce vanishing +gradients. Although this axial technique was introduced +in [13] for auto-regressive transformer models, we pro- +pose novel architectures by factorizing 2D convolution +into two consecutive 1D convolutions. Figures 2, and 3 +show our novel proposed residual blocks. +For each location, o = (i, j), a local input kernel k×k +is extracted from an input image X with height h, width +w, and channel count din to serve convolutional opera- +tion. Residual units, used by [12], are defined as, +Yo = R(Xo) + F(Xo, Wo) +(5) +where, Xo and Yo are input and output for the location +o = (i, j), R(Xo) is the original input or identity map- +ping, and F is the residual function. This residual func- +tion is defined using convolutional operation for vision +tasks. The structure of this residual function depends on +the residual block, we use. Two spatial 2D convolutional +operations are used for residual basic block, and a spa- +tial (kernel k > 1) 2D convolution operation is used in +between two convolutional operations (kernel k = 1) for + +Figure 2. RAN basic block used in our proposed networks. “bn” stands for batch normalization. +Figure 3. RAN bottleneck block used in our proposed networks. “bn” stands for batch normalization. +bottleneck block. These spatial 2D convolutional oper- +ations for kernel k > 1 and o = (i, j) can be defined +as [26], +Co = +� +(a,b)∈Nk×k(o) +Wi−a,j−b, xa,b +(6) +where, Nk ∈ Rk×k×din is the neighborhood of pixel +(i, j) with the spatial square region k × k and W ∈ +Rk×k×dout×din is the shared weights that are for cal- +culating output for all pixel positions centered by (i, j). +The computational cost is O(hwk2) which is high. +To reduce this computation cost and make parame- +ter efficient architecture, we propose to adopt the axial +concept and replace 2D convolution using two 1D con- +volutions with residual connections. These two 1D con- +volutions are performing convolution along the height +axis and the width axis. The 1D convolution along the +height axis is defined as follows. +Ch = +� +(a,b)∈Nk×1(i,j) +Wi−a,j−b, xa,b +(7) +where, Nk ∈ Rk×1×din is the neighborhood of pixel +(i, j) with spatial extent k × 1 and W ∈ Rk×1×dout×din +is the shared weights that are for calculating output for +all pixel positions (i, j). And, for width axis is as fol- +lows. +Cw = +� +(a,b)∈N1×k(i,j) +Wi−a,j−b, xa,b +(8) +where, Nk ∈ R1×k×din is the neighborhood of pixel +(i, j) with spatial extent 1 × k and W ∈ R1×k×dout×din +is the shared weights that are for calculating output for +all pixel positions (i, j). +To construct our basic and bottleneck blocks, we re- +place each 2D convolution layer from the original resid- +ual blocks with a pair of consecutive 1D convolution +layers. When we did this but omitted a residual connec- +tion, the network faced the vanishing gradient problem. +To handle this, we added a residual connection along +each 1D convolution operation. Each 2D convolution in +Equation 6 is equivalent to our proposed method defined +as, +Yh = Ch(Wh, Xo) + Xo +(9) +Yo = Cw(Ww, Yh) + Yh +(10) +where, Ch, and Cw are the height and width outputs of +Equations 7 and 8, respectively, Wh, and Ww is the con- +volutional weights for height, and width axis 1D convo- +lutional operations, respectively. Equation 10 describes +the residual basic and bottleneck blocks. As two 1D op- +erations equal one 2D operation, the use of these two + +.... +1x3 Conv1D +3x1 Conv1D +Height-Axis +Width-Axis +1x3 Conv1D +Width-Axis +I. +relu +X D +D +.S + Conv1D +Vidth-Axis +D +Conv1l +Axi +2 +2 +E +Conv +bn +n +0 +C +c +3 +X +X +H +X +X +3layers does not increase the layer count. The RAN ba- +sic and bottleneck blocks are shown in Figures 2 and 3. +These blocks are used to construct our proposed residual +axial networks (RANs). The output Yo from Equation 10 +is applied to other 2D convolution-based networks, for +example, wide residual networks (to make our proposed +wide RANs) and deep recursive residual networks (to +make RARNets), to check the effectiveness of our pro- +posed method. +4. Experimental Analysis +We present experimental results on four image classi- +fication datasets and one image super-resolution dataset. +Our experiments evaluate the proposed residual axial +networks, the original ResNets, the wide ResNets, wide +RANs, the deep recursive residual networks (DRRNs), +and RARNets. We compare our proposed network’s per- +formance with the corresponding original ResNets, as +these original networks used 2D spatial convolutional +layers. Our comparisons use parameter counts, FLOPS, +latency, and validation performance. +4.1. Method: Residual Networks +To explore scalability, we compare our proposed +RANs and baseline models on four datasets: CIFAR- +10 and CIFAR-100 benchmarks [19], Street View House +Number (SVHN) [24], and Tiny ImageNet datasets +[20]. +The CIFAR benchmarks have 10 and 100 dis- +tinct classes, and 60,000 color images (split into 50,000 +training and 10,000 testing images) of size 32 × 32. We +perform data normalization using per-channel mean and +standard deviation. In preprocessing, we do a horizontal +flip and randomly crop after padding with four pixels on +each side of the image. The SVHN and Tiny ImageNet +datasets contain 600,000 images of size 32 × 32 with +ten classes and 110,000 images of 200 distinct classes +downsized to 64 × 64 colored images, respectively. Our +only preprocessing is mean/std normalization for both +datasets. +All the models (baselines and proposed RANs) were +trained using similar architectures (same hyperparame- +ters and the same number of output channels). As our +main concern was to reduce the parameter counts of the +bottleneck residual block, we implemented all network +architecture, baselines, and proposed, using only bottle- +neck blocks. The numbers of output channels of bottle- +neck groups are 120, 240, 480,, and 960 for all networks. +This experiment analyzes 26, 35, 50, 101, and 152- +layer architectures with the bottleneck block multipliers +“[1, 2, 4, 1]”, “[2, 3, 4, 2]”, “[3, 4, 6, 3]”, “[3, 4, 23, 3]”, +and “[3, 8, 36, 3]”, respectively. All models were run us- +ing the stochastic gradient descent optimizer, and using +linearly warmed-up learning for 10 epochs from zero +to 0.1 and then used cosine learning scheduling from +epochs 11 to 150. All models were trained using batch +sizes of 128 for all datasets, we used except the 101, and +152-layer architectures of the Tiny ImageNet dataset. +We used a batch size of 64 for these two architectures +on Tiny ImageNet. +4.2. Results: Residual Networks +Table 1 summarizes the classification results of the +original ResNets and our proposed RANs on the four +datasets. We tested shallow and deeper networks by im- +plementing 26, 35, 50, 101, and 152-layer architectures. +These architectures compare performance to check the +effectiveness of our proposed methods for shallow and +deep networks. Our proposed method is compared with +original ResNets in terms of parameter count, FLOPS +count (number of multiply-add operations), inference +time or latency (time used to test one image after train- +ing), and the percentage accuracy of validation results +on the four datasets. +The 26, 35, 50, 101, and 152-layer architectures re- +duce by 48.6%, 46.5%, 44.8%, 43.2%, and 42.6% the +trainable parameters respectively needed in comparison +to the baseline networks. In addition to parameter re- +duction, our proposed method requires 15 to 36 percent +fewer FLOPS for all analyzed architectures. Also, the +validation performance improvement is significantly no- +ticeable for all datasets. The latency to process one im- +age after training our proposed models is comparatively +high as the RANs use two convolution layers sequen- +tially. It is also shown that the deeper networks perform +better than the shallow networks and it has demonstrated +“the deeper, the better” in classification. +4.3. Method: Wide Residual Networks +The previous experiment did not assess wide +ResNets. +To assess the widening factor on our pro- +posed RANs, we increase the width of our RANs by +factorizing the number of output channels for shallow +networks like [33]. +Like the original wide residual +networks (WRNs) [33], we analyzed our proposed 26- +layer bottleneck block of RANs with a widening factor, +k = 2, 4, 6, 8, and 10. We multiplied the number of out- +put channels of RANs with k to obtain wide RANs. We +performed training with the same optimizer and hyper- +parameters used in 4.1. +4.4. Results: Wide Residual Networks +Table 1 shows “the deeper the better” in vision classi- +fication for our proposed methods. To compare our pro- +posed RANs with the original wide ResNets (WRNs), +we analyze our proposed method for different widening +factors. Table 2 shows an overall comparison among the +original WRN-28-10 (28-layers with a widening factor + +Dataset +Model Name +Layers +Params +FLOPs +Latency +Accuracy +CIFAR-10 +ResNet [11] +26 +40.9M +0.66G +0.66ms +94.68 +RAN (Ours) +21M +0.56G +0.73ms +96.08 +ResNet [11] +35 +57.8M +0.86G +0.82ms +94.95 +RAN (Ours) +30.9M +0.68G +0.91ms +96.15 +ResNet [11] +50 +82.5M +1.18G +1.02ms +95.08 +RAN (Ours) +45.5M +0.87G +1.17ms +96.25 +ResNet [11] +101 +149.2M +2.29G +1.68ms +95.36 +RAN (Ours) +84.7M +1.52G +1.86ms +96.27 +ResNet [11] +152 +204.1M +3.41G +2.39ms +95.36 +RAN (Ours) +117.1M +2.18G +2.55ms +96.37 +CIFAR-100 +ResNet [11] +26 +41.2M +0.66G +0.66ms +78.21 +RAN (Ours) +21.1M +0.56G +0.74ms +79.66 +ResNet [11] +35 +58.1M +0.86G +0.80ms +78.72 +RAN (Ours) +31.1M +0.68G +0.91ms +80.38 +ResNet [11] +50 +82.9M +1.18G +1.11ms +78.95 +RAN (Ours) +45.7M +0.87G +1.17ms +80.84 +ResNet [11] +101 +149.5M +2.29G +1.72ms +78.80 +RAN (Ours) +84.9M +1.52G +1.86ms +80.88 +ResNet [11] +152 +204.5M +3.41G +2.36ms +79.85 +RAN (Ours) +117.2M +2.18G +2.55ms +80.94 +SVHN +ResNet [11] +26 +40.9M +0.66G +0.64ms +96.04 +RAN (Ours) +21M +0.56G +0.73ms +97.60 +ResNet [11] +35 +57.8M +0.86G +0.79ms +95.74 +RAN (Ours) +30.9M +0.68G +0.90ms +97.50 +ResNet [11] +50 +82.5M +1.18G +1.05ms +95.76 +RAN (Ours) +45.5M +0.87G +1.11ms +97.32 +ResNet [11] +101 +149.2M +2.29G +1.64ms +96.29 +RAN (Ours) +84.7M +1.52G +1.80ms +97.29 +ResNet [11] +152 +204.1M +3.41G +2.28ms +96.35 +RAN (Ours) +117.1M +2.18G +2.5ms +97.38 +ImageNet-Tiny +ResNet [11] +26 +41.6M +0.66G +2.31ms +57.21 +RAN (Ours) +21.3M +0.56G +2.58ms +62.28 +ResNet [11] +35 +58.5M +0.86G +2.85ms +57.80 +RAN (Ours) +31.3M +0.68G +3.0ms +59.31 +ResNet [11] +50 +82.6M +1.18G +3.75ms +59.06 +RAN (Ours) +45.8M +0.87G +4.02ms +62.40 +ResNet [11] +101 +149.3M +2.29G +6.86ms +60.62 +RAN (Ours) +85.1M +1.52G +7.19ms +64.18 +ResNet [11] +152 +204.2M +3.41G +9.29ms +61.57 +RAN (Ours) +117.4M +2.18G +9.72ms +66.16 +Table 1. Image classification performance on the CIFAR benchmarks, SVHN, and Tiny ImageNet datasets for 26, 35, 50, 101, and +152-layer architectures. +of 10), and our proposed 26-layer networks with differ- +ent widening factors (k = 2, 4, 6, 8, and 10). Our pro- +posed wide RANs show significant accuracy improve- +ment over the original WRN-28-10. +This table also +demonstrates “the wider the better” for our proposed +wide RANs. +4.5. Method: Recursive Networks +This experiment compares the cost and performance +of our novel RARnet with the DRRN on the super- +resolution tasks. The RARnet is built by replacing the +residual unit U with a RAN layer described in Equation +10. These modifications form a new architecture, recur- +sive axial residual network (RARNet) whose depth d is + +Figure 4. Recursive axial residual network (RARNet) architec- +ture with B = 4 and U = 3. Here, “RB” layer, and RAN refer +to a recursive block, and residual axial block, respectively. +defined as, +d = (1 + URAN) × B + 1 +(11) +As two 1D layers are equivalent to one 2D layer and we +replace each residual unit by a RAN unit (see Equation +10). Hence, we rewrite Equation 4 to Equation 11 by +removing the multiplier for the residual unit. The pro- +posed RARNet with four RB blocks is shown on the left +in Figure 4. An RB block is expanded on the right. +We trained our proposed RARNet using 291 images +dataset [32] and tested using the Set5 dataset [3]. We +also use different scales (×2, ×3, and ×4) in training +and testing images. +We used similar data augmenta- +tion, training hyperparameters, and implementation de- +tails like [28]. +4.6. Results: Recursive Networks +Table 3 shows the Peak Signal-to-Noise Ratio +(PSNR) results of several CNN models including +DRRN, and our proposed RARNet on the Set5 dataset. +The comparison between DRRN and RARNet is our +main focus as it directly indicates the effectiveness of us- +ing our proposed RAN block. DRRN19 and DRRN125 +are constructed using B = 1, U = 9, and B = 1, U = +25, respectively. For fair comparison, we also construct +similar architecture like RARNet19 (B = 1, URAN = +9) and RARNet125 (B = 1, URAN = 25). Our pro- +posed models outperform all CNN models in Table 3 +on the Set5 dataset and for all scaling factors. As we +are trying to propose a parameter-efficient architecture, +parameter comparison is very essential along with the +testing performance. Our proposed model for B = 1, +and URAN = 9 takes 213,254 parameters compared to +297,216 parameters of DRRN (B = 1, and U = 9). +RARNet, which is constructed using RAN blocks, re- +duces by 28.2% the trainable parameters compared to +Dataset +Model Name +Accuracy +CIFAR-10 +WRN-28-10 [33] +94.68 +RAN-26-2 (Ours) +96.32 +RAN-26-4 (Ours) +96.68 +RAN-26-6 (Ours) +96.77 +RAN-26-8 (Ours) +96.83 +RAN-26-10 (Ours) +96.87 +CIFAR-100 +WRN-28-10 [33] +79.57 +RAN-26-2 (Ours) +83.54 +RAN-26-4 (Ours) +83.75 +RAN-26-6 (Ours) +83.78 +RAN-26-8 (Ours) +83.82 +RAN-26-10 (Ours) +83.92 +Table 2. Image classification performance comparison on the +CIFAR benchmarks for 26-layer RAN architectures with dif- +ferent widening factors. +the DRRN. +5. Discussion and Conclusions +This work introduces a new residual block that can +be used as a replacement for the ResNet basic and bot- +tleneck blocks. This RAN block replaced the 2D convo- +lution from the original ResNet blocks with two sequen- +tial 1D convolutions along with a residual connection. +These modifications help to reduce trainable parame- +ters as well as improve validation performance on vision +classification. But the latency of our proposed model is +comparatively high. The proposed model’s performance +and parameter reduction outweigh this latency time limi- +tation. We also checked this proposed block for widened +ResNets and showed that the wide RANs obtain better +accuracy performance than the WRNs. We also checked +the effectiveness of RANs on the SISR task. Specifi- +cally, we applied our proposed RAN block on am image +restoration dataset and found that our proposed recursive +axial ResNets (RARNets) improve image resolution and +reduce trainable parameters more than the other CNN- +based super-resolution models. Extensive experiments +and analysis show that RANs can be deep, and wide +and these are parameter-efficient and superior models +for image classification and SISR. We have shown that +our proposed model is a viable replacement for ResNets +on the tasks that were tested. Further work is required +to determine the range of applications for which RANs +may offer advantages. +References +[1] Serge Belongie, Michael Wilber, and Andreas Veit. +Residual networks behave like ensembles of relatively +shallow networks. 2016. 1 + +X1 +X +个 +RAN +RB +RAN +RB +RB +4 +RAN +RB +conv +RAN +★ +-- +Y +X2Dataset +Scale +SRCNN [7] +VDSR [17] +DRCN [18] +DRRN19 +DRRN125 +RARNet19 +RARNet25 +Set5 +x2 +36.66 +37.53 +37.63 +37.66 +37.74 +37.73 +37.84 +x3 +32.75 +33.66 +33.82 +33.93 +34.03 +33.99 +34.11 +x4 +30.48 +31.35 +31.53 +31.58 +31.68 +31.63 +31.84 +Table 3. Benchmark testing PSNR results for scaling factors ×2, ×3, and ×4 on Set5 dataset. +[2] Yoshua Bengio, Aaron Courville, and Pascal Vincent. +Representation learning: A review and new perspectives. +IEEE transactions on pattern analysis and machine in- +telligence, 35(8):1798–1828, 2013. 1 +[3] Marco Bevilacqua, Aline Roumy, Christine Guillemot, +and Marie Line Alberi-Morel. Low-complexity single- +image super-resolution based on nonnegative neighbor +embedding. 2012. 7 +[4] Liang-Chieh Chen, Huiyu Wang, and Siyuan Qiao. Scal- +ing wide residual networks for panoptic segmentation. +arXiv preprint arXiv:2011.11675, 2020. 1, 2 +[5] Tianqi Chen, Ian Goodfellow, and Jonathon Shlens. +Net2net: Accelerating learning via knowledge transfer. +arXiv preprint arXiv:1511.05641, 2015. 1 +[6] Jifeng Dai, Yi Li, Kaiming He, and Jian Sun. R-fcn: Ob- +ject detection via region-based fully convolutional net- +works. Advances in neural information processing sys- +tems, 29, 2016. 1 +[7] Chao Dong, Chen Change Loy, Kaiming He, and Xiaoou +Tang. Image super-resolution using deep convolutional +networks. IEEE transactions on pattern analysis and ma- +chine intelligence, 38(2):295–307, 2015. 3, 8 +[8] Shang-Hua Gao, Ming-Ming Cheng, Kai Zhao, Xin-Yu +Zhang, Ming-Hsuan Yang, and Philip Torr. +Res2net: +A new multi-scale backbone architecture. IEEE trans- +actions on pattern analysis and machine intelligence, +43(2):652–662, 2019. 1 +[9] Xavier Glorot and Yoshua Bengio. Understanding the +difficulty of training deep feedforward neural networks. +In Proceedings of the thirteenth international conference +on artificial intelligence and statistics, pages 249–256. +JMLR Workshop and Conference Proceedings, 2010. 1 +[10] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian +Sun. Delving deep into rectifiers: Surpassing human- +level performance on imagenet classification. +In Pro- +ceedings of the IEEE international conference on com- +puter vision, pages 1026–1034, 2015. 1 +[11] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian +Sun. Deep residual learning for image recognition. In +Proceedings of the IEEE conference on computer vision +and pattern recognition, pages 770–778, 2016. 1, 2, 3, 6 +[12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian +Sun. Identity mappings in deep residual networks. In Eu- +ropean conference on computer vision, pages 630–645. +Springer, 2016. 1, 2, 3 +[13] Jonathan Ho, Nal Kalchbrenner, Dirk Weissenborn, and +Tim Salimans. Axial attention in multidimensional trans- +formers. arXiv preprint arXiv:1912.12180, 2019. 3 +[14] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation +networks. +In Proceedings of the IEEE conference on +computer vision and pattern recognition, pages 7132– +7141, 2018. 1 +[15] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and +Kilian Q. Weinberger. +Deep networks with stochastic +depth. In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max +Welling, editors, Computer Vision – ECCV 2016, pages +646–661, Cham, 2016. Springer International Publish- +ing. 2 +[16] Sergey Ioffe and Christian Szegedy. Batch normaliza- +tion: Accelerating deep network training by reducing in- +ternal covariate shift. In International conference on ma- +chine learning, pages 448–456. PMLR, 2015. 1 +[17] Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee. Ac- +curate image super-resolution using very deep convolu- +tional networks. In Proceedings of the IEEE conference +on computer vision and pattern recognition, pages 1646– +1654, 2016. 1, 3, 8 +[18] Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee. +Deeply-recursive convolutional network for image super- +resolution. In Proceedings of the IEEE conference on +computer vision and pattern recognition, pages 1637– +1645, 2016. 1, 3, 8 +[19] Alex Krizhevsky, Geoffrey Hinton, et al. Learning mul- +tiple layers of features from tiny images. 2009. 5 +[20] Ya Le and Xuan S. Yang. Tiny imagenet visual recogni- +tion challenge. 2015. 5 +[21] Xiu Li and Zuoying Cui. +Deep residual networks for +plankton classification. +In OCEANS 2016 MTS/IEEE +Monterey, pages 1–4, 2016. 1 +[22] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He, +Bharath Hariharan, and Serge Belongie. Feature pyra- +mid networks for object detection. In Proceedings of the +IEEE conference on computer vision and pattern recog- +nition, pages 2117–2125, 2017. 1 +[23] Xiaojiao Mao, Chunhua Shen, and Yu-Bin Yang. Im- +age restoration using very deep convolutional encoder- +decoder networks with symmetric skip connections. Ad- +vances in neural information processing systems, 29, +2016. 3 +[24] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bis- +sacco, Bo Wu, and Andrew Y Ng. Reading digits in nat- +ural images with unsupervised feature learning. 2011. +5 +[25] Siyuan Qiao, Huiyu Wang, Chenxi Liu, Wei Shen, and +Alan Yuille. +Micro-batch training with batch-channel +normalization and weight standardization. arXiv preprint +arXiv:1903.10520, 2019. 1 +[26] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Ir- +wan Bello, Anselm Levskaya, and Jonathon Shlens. +Stand-alone self-attention in vision models. +arXiv +preprint arXiv:1906.05909, 2019. 2, 4 + +[27] Ilya Sutskever, James Martens, George Dahl, and Geof- +frey Hinton. On the importance of initialization and mo- +mentum in deep learning. In International conference on +machine learning, pages 1139–1147. PMLR, 2013. 1 +[28] Ying Tai, Jian Yang, and Xiaoming Liu. Image super- +resolution via deep recursive residual network. In Pro- +ceedings of the IEEE conference on computer vision and +pattern recognition, pages 3147–3155, 2017. 1, 3, 7 +[29] Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig +Adam, Alan Yuille, and Liang-Chieh Chen. +Axial- +deeplab: Stand-alone axial-attention for panoptic seg- +mentation. In European Conference on Computer Vision, +pages 108–126. Springer, 2020. 3 +[30] Yuxin Wu and Kaiming He. Group normalization. In +Proceedings of the European conference on computer vi- +sion (ECCV), pages 3–19, 2018. 1 +[31] Zifeng Wu, Chunhua Shen, and Anton Van Den Hengel. +Wider or deeper: Revisiting the resnet model for visual +recognition. Pattern Recognition, 90:119–133, 2019. 1, +3 +[32] Jianchao Yang, John Wright, Thomas S Huang, and Yi +Ma. +Image super-resolution via sparse representation. +IEEE transactions on image processing, 19(11):2861– +2873, 2010. 7 +[33] Sergey Zagoruyko and Nikos Komodakis. Wide residual +networks. arXiv preprint arXiv:1605.07146, 2016. 1, 2, +3, 5, 7 + diff --git a/3NE3T4oBgHgl3EQfoQom/content/tmp_files/load_file.txt b/3NE3T4oBgHgl3EQfoQom/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a08cd0dc945d9ddd38750e94cdc7a89fb8cc5f02 --- /dev/null +++ b/3NE3T4oBgHgl3EQfoQom/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf,len=544 +page_content='Deep Residual Axial Networks Nazmul Shahadat, Anthony S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Maida University of Louisiana at Lafayette Lafayette LA 70504, USA nazmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='ruet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='com, maida@louisiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='edu Abstract While residual networks (ResNets) demonstrate out- standing performance on computer vision tasks, their computational cost still remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Here, we focus on reducing this cost by proposing a new network archi- tecture, axial ResNet, which replaces spatial 2D convo- lution operations with two consecutive 1D convolution operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Convergence of very deep axial ResNets has faced degradation problems which prevent the networks from performing efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To mitigate this, we apply a residual connection to each 1D convolutional operation and propose our final novel architecture namely residual axial networks (RANs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Extensive benchmark evaluation shows that RANs outperform with about 49% fewer pa- rameters than ResNets on CIFAR benchmarks, SVHN, and Tiny ImageNet image classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' More- over, our proposed RANs show significant improvement in validation performance in comparison to the wide ResNets on CIFAR benchmarks and the deep recursive residual networks on image super resolution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Introduction Deep convolutional neural network (CNN) based ar- chitectures, specifically ResNets [11], have achieved significant success for image processing tasks, includ- ing classification [10,11,21], object detection [6,22] and image super-resolution [17, 18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The performance of deep ResNets and wide ResNets has improved in re- cent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Along with the increasing depth or widening of ResNets, the computational cost of the networks also rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Moreover, training these deeper or wider networks has faced exploding or vanishing gradient and degrada- tion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Different initialization, optimization, and normalization techniques [9,10,16,25,27,30], skip con- nections [10], and transfer learning [5] have been used used to mitigate these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The rising computa- tional cost and/or trainable parameter is still unexplored which is the main purpose of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' However, the computational cost of these deeper and wider ResNets has not been analyzed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep or wide ResNets gain popularity and impressive perfor- mance due to their simple but effective architectures [4, 8, 14, 31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep ResNets can be factored as en- sembles of shallow networks [1] and represent func- tions more efficiently for complex tasks than shallow networks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' However, constructing deeper ResNets is not as simple as adding more residual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The de- sign of deeper ResNets demands better optimization and initialization schemes, and proper use of identity con- nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deeper ResNets have great success in image classification and object detection tasks [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' How- ever, the computational cost increases linearly with the number of layers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Wide ResNets use a shallow network with wide (high channel count) architecture to attain better performance than the deeper networks [4, 31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For example, [33] represented their wide residual network as WRN-n-k where n is the number of convolutional layers and k rep- resents the widening factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' They have shown that their WRN-28-10, wide ResNet that adopts 28 convolutional layers with k = 10 widening factor, outperforms the deep ResNet-1001 network (1001 layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' However, the computational cost is quadratic with a widening factor of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This work revisits the designs of deep and wide ResNets to boost their performance further, reduce the above-mentioned high computational costs, and im- prove the model inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To get these, we propose our novel architecture, residual axial networks (RANs), obtained using axial operations, height or width-axis, instead of spatial operations in the residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Here, we split 2D spatial (3 × 3) convolution operation into two consecutive 1D convolution opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These 1D convolution operations are mapped to the height-axis (3 × 1) and width-axis (1 × 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As axial 1D convolution operations propagate information along one axis at a time, this modification reduces cost sig- nificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To capture global information, we use these layers in consecutive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='04631v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='CV] 11 Jan 2023 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Residual block architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' “bn” stands for batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' (Left) ResNet basic block and (Right) ResNet bottleneck blocks are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' A simple axial architecture reduces cost but does not improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The reason is that forward in- formation flows across the axial blocks degrades (di- minishing feature reuse [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To address this, we add residual connections to span the axial blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' By using both modifications, we made a novel, effective residual axial architecture (RAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The effectiveness of our pro- posed model is demonstrated experimentally on four im- age classification datasets and an image super-resolution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our assessments are based on parameter counts, FLOPS counts (number of multiply-add operations), la- tency to process one image after training, and validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Background and Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Convolutional Neural Networks In a convolutional layer, the core building block is a convolution operation using one kernel W applied to small neighborhoods to find input correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For an input image X with height h, width w, and channel count din, the convolution operation operates on region (a, b) ∈ X centered at pixel (i, j) with spatial extend k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The output for this operation Co where o = (i, j) is [26], Co = � (a,b)∈Nk×k(i,j) W (m) i−a,j−b, xa,b (1) where Nk×k is the neighborhood of pixel (i, j) with spa- tial extent k, and W is the shared weights to calculate the output for all pixel positions (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Residual Networks Residual networks (ResNets) are constructed using convolutional layers linked by additive identity connec- tions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' They were introduced to address the problem of vanishing gradients found in standard deep CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Although, the vanishing gradient problem may be ad- dressed by using normalized inputs and normalization layers which help to make networks till ten layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In this situation, when more layers were stacked, the net- work depth increases but accuracy gets saturated and then degrades rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The degradation (of training ac- curacy) indicates that not all systems are similarly opti- mized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To address these problems, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' proposed residual networks by adding identity mapping among the layers [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As a result, the subsequent deeper lay- ers are shared inputs from the learned shallower model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This helps to address all of the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The key architectural feature of ResNets is the resid- ual block with identity mapping to tackle the degrada- tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Two kinds of residual blocks are used in residual networks, the basic block and the bottleneck block, both depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Figure 1 (left) is known as the basic architecture of ResNet which is constructed with two k × k convolution layers where k is the size of the kernel and an identity shortcut connection is added to the end of these two layers to address vanishing gradi- ents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These operations can be expressed mathematically as, y = F(Ckm,kn(x, W)) + x (2) where F, x, y, W, and Ckm,kn represent residual func- tion, input vector, output vector, weight parameters, and output of two convolution layers with kernels Km and Kn respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Figure 1 (right) is a bottleneck archi- tecture that is constructed using 1 × 1, k × k, and 1 × 1 convolution layers, where the 1 × 1 layers reduce and then increase the number of channels, and the 3×3 layer performs feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The identity shortcuts (ex- plained in equation 3) are very important for this block as it leads to more efficient designs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These can be expressed mathematically as, y = F(Ck1,km,k1(x, W)) + x (3) where F, x, y, W, and Ck1,km,k1 represent residual function, input vector, output vector, weight parame- ters, and output of three convolution layers with kernels 1 × 1, Km × Km and 1 × 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Its perfor- mance surpasses the learning speed, number of learning parameters, way of layer-wise representation, difficult optimization property, and memory mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Wide Residual Networks Wide ResNets [4, 33] use fewer layers compared to standard ResNets but use high channel counts (wide ar- chitectures) which compensate for the shallower archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The comparison between shallow and deep net- works has been revealed in circuit complexity theory X X 3x3 conv2d 1x1 conv2d bn bn relu ★ relu 3x3 conv2d 3x3 conv2d bn bn relu 1x1 conv2d relu bn reluwhere shallow circuits require more components than the deeper circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Inspired by this observation, [11] pro- posed deeper networks with thinner architecture where a gradient goes through the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' But the problem such networks face is that the residual block weights do not flow through the network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For this, the network may be forced to avoid learning during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To address these issues, [33] proposed shallow but wide network architectures and showed that widening the residual blocks improves the performance of resid- ual networks compared to increasing their depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For example, a 16-layer wide ResNet has similar accuracy performance to a 1000-layer thinner network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Recursive Residual Networks Image super-resolution (SR) is the process of gen- erating a high-resolution (HR) image from a low- resolution (LR) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' It is also known as single image super-resolution (SISR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' A list of convolution- based models has shown promising results on SISR [7, 17, 18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These 2D convolutional networks learn a nonlinear mapping from an LR to an HR image in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Convolution-based recursive neural networks have been used on SISR, where recursive net- works learn detailed and structured information about an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As image SR requires more image details, pool- ing is not used in deep models for SISR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Convolution- based SR [7] has shown that the convolution-based LR- HR mapping significantly improves performance for classical shallow methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=', introduce two deep CNNs for SR by stacking weight layers [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Among them, [18] uses a chain structure recursive layer along with skip-connections to control the model pa- rameters and improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep SR mod- els [17,18,23] demand large parameter counts and more storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To address these issues, deep recursive residual net- works (DRRNs) were proposed as a very deep network structure, which achieves better performance with fewer parameters [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' It includes both local and global resid- ual learning, where global residual learning (GRL) is be- ing used in the identity branch to estimate the residual image from the input and output of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' GRL might face degradation problems for deeper networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To handle this problem, local residual learning (LRL) has been used which carries rich image details to deeper layers and helps gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The DRRN also used recursive learning of residual units to keep the model more compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Several recursive blocks (B) has been stacked, followed by a CNN layer which is used to re- construct the residual between the LR and HR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Each of these residual blocks decomposed into a num- ber of residual units (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The number of recursive block B, and the number of residual units U are responsible for defining network depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The depth of DRRN d is calculated as, d = (1 + 2 × U) × B + 1 (4) Recursive block definition, DRRN formulation, and the loss function of DRRN are defined in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Proposed Residual Axial Networks The convolution-based residual basic and bottleneck blocks [11, 12] have demonstrated significant perfor- mance with the help of several state-of-the-art archi- tectures like, ResNets [11], wide ResNets [31], scal- ing wide ResNets [33], and deep recursive residual net- works (DRRNs) [28] on image classification and im- age super-resolution datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Although the bottleneck residual block makes the networks thinner still the ba- sic and bottleneck blocks are not cost-effective and/or parameter efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The 2D convolutional operation of these blocks is consuming O(N 2) resources, where N is the flattened pixels of an image, and N = hw (for a 2D image of height h, width w, and h = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' So the cost for a 2D convolutional operation, for an image with height h, and width w, is O((hw)2) = O(h2w2) = O(h4) [13, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To reduce this impractical computational cost, we are proposing a novel architectural design, residual axial networks (RANs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Due to high computational expenses, we replace all spatial 2D convolution operations (conv2D) of the resid- ual basic blocks, and the only spatial 2D convolution op- eration of the residual bottleneck block by using two 1D convolutional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Also, each 1D convolutional operation has a residual connection to reduce vanishing gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Although this axial technique was introduced in [13] for auto-regressive transformer models, we pro- pose novel architectures by factorizing 2D convolution into two consecutive 1D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Figures 2, and 3 show our novel proposed residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For each location, o = (i, j), a local input kernel k×k is extracted from an input image X with height h, width w, and channel count din to serve convolutional opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Residual units, used by [12], are defined as, Yo = R(Xo) + F(Xo, Wo) (5) where, Xo and Yo are input and output for the location o = (i, j), R(Xo) is the original input or identity map- ping, and F is the residual function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This residual func- tion is defined using convolutional operation for vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The structure of this residual function depends on the residual block, we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Two spatial 2D convolutional operations are used for residual basic block, and a spa- tial (kernel k > 1) 2D convolution operation is used in between two convolutional operations (kernel k = 1) for Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' RAN basic block used in our proposed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' “bn” stands for batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' RAN bottleneck block used in our proposed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' “bn” stands for batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' bottleneck block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These spatial 2D convolutional oper- ations for kernel k > 1 and o = (i, j) can be defined as [26], Co = � (a,b)∈Nk×k(o) Wi−a,j−b, xa,b (6) where, Nk ∈ Rk×k×din is the neighborhood of pixel (i, j) with the spatial square region k × k and W ∈ Rk×k×dout×din is the shared weights that are for cal- culating output for all pixel positions centered by (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The computational cost is O(hwk2) which is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To reduce this computation cost and make parame- ter efficient architecture, we propose to adopt the axial concept and replace 2D convolution using two 1D con- volutions with residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These two 1D con- volutions are performing convolution along the height axis and the width axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The 1D convolution along the height axis is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Ch = � (a,b)∈Nk×1(i,j) Wi−a,j−b, xa,b (7) where, Nk ∈ Rk×1×din is the neighborhood of pixel (i, j) with spatial extent k × 1 and W ∈ Rk×1×dout×din is the shared weights that are for calculating output for all pixel positions (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' And, for width axis is as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Cw = � (a,b)∈N1×k(i,j) Wi−a,j−b, xa,b (8) where, Nk ∈ R1×k×din is the neighborhood of pixel (i, j) with spatial extent 1 × k and W ∈ R1×k×dout×din is the shared weights that are for calculating output for all pixel positions (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To construct our basic and bottleneck blocks, we re- place each 2D convolution layer from the original resid- ual blocks with a pair of consecutive 1D convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' When we did this but omitted a residual connec- tion, the network faced the vanishing gradient problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To handle this, we added a residual connection along each 1D convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Each 2D convolution in Equation 6 is equivalent to our proposed method defined as, Yh = Ch(Wh, Xo) + Xo (9) Yo = Cw(Ww, Yh) + Yh (10) where, Ch, and Cw are the height and width outputs of Equations 7 and 8, respectively, Wh, and Ww is the con- volutional weights for height, and width axis 1D convo- lutional operations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Equation 10 describes the residual basic and bottleneck blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As two 1D op- erations equal one 2D operation, the use of these two .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='. 1x3 Conv1D 3x1 Conv1D Height-Axis Width-Axis 1x3 Conv1D Width-Axis I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' relu X D D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='S Conv1D Vidth-Axis D Conv1l Axi 2 2 E Conv bn n 0 C c 3 X X H X X 3layers does not increase the layer count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The RAN ba- sic and bottleneck blocks are shown in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These blocks are used to construct our proposed residual axial networks (RANs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The output Yo from Equation 10 is applied to other 2D convolution-based networks, for example, wide residual networks (to make our proposed wide RANs) and deep recursive residual networks (to make RARNets), to check the effectiveness of our pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Experimental Analysis We present experimental results on four image classi- fication datasets and one image super-resolution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our experiments evaluate the proposed residual axial networks, the original ResNets, the wide ResNets, wide RANs, the deep recursive residual networks (DRRNs), and RARNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We compare our proposed network’s per- formance with the corresponding original ResNets, as these original networks used 2D spatial convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our comparisons use parameter counts, FLOPS, latency, and validation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Method: Residual Networks To explore scalability, we compare our proposed RANs and baseline models on four datasets: CIFAR- 10 and CIFAR-100 benchmarks [19], Street View House Number (SVHN) [24], and Tiny ImageNet datasets [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The CIFAR benchmarks have 10 and 100 dis- tinct classes, and 60,000 color images (split into 50,000 training and 10,000 testing images) of size 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We perform data normalization using per-channel mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In preprocessing, we do a horizontal flip and randomly crop after padding with four pixels on each side of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The SVHN and Tiny ImageNet datasets contain 600,000 images of size 32 × 32 with ten classes and 110,000 images of 200 distinct classes downsized to 64 × 64 colored images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our only preprocessing is mean/std normalization for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' All the models (baselines and proposed RANs) were trained using similar architectures (same hyperparame- ters and the same number of output channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As our main concern was to reduce the parameter counts of the bottleneck residual block, we implemented all network architecture, baselines, and proposed, using only bottle- neck blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The numbers of output channels of bottle- neck groups are 120, 240, 480,, and 960 for all networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This experiment analyzes 26, 35, 50, 101, and 152- layer architectures with the bottleneck block multipliers “[1, 2, 4, 1]”, “[2, 3, 4, 2]”, “[3, 4, 6, 3]”, “[3, 4, 23, 3]”, and “[3, 8, 36, 3]”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' All models were run us- ing the stochastic gradient descent optimizer, and using linearly warmed-up learning for 10 epochs from zero to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1 and then used cosine learning scheduling from epochs 11 to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' All models were trained using batch sizes of 128 for all datasets, we used except the 101, and 152-layer architectures of the Tiny ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We used a batch size of 64 for these two architectures on Tiny ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Results: Residual Networks Table 1 summarizes the classification results of the original ResNets and our proposed RANs on the four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We tested shallow and deeper networks by im- plementing 26, 35, 50, 101, and 152-layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These architectures compare performance to check the effectiveness of our proposed methods for shallow and deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our proposed method is compared with original ResNets in terms of parameter count, FLOPS count (number of multiply-add operations), inference time or latency (time used to test one image after train- ing), and the percentage accuracy of validation results on the four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The 26, 35, 50, 101, and 152-layer architectures re- duce by 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='6%, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5%, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='8%, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2%, and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='6% the trainable parameters respectively needed in comparison to the baseline networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In addition to parameter re- duction, our proposed method requires 15 to 36 percent fewer FLOPS for all analyzed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Also, the validation performance improvement is significantly no- ticeable for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The latency to process one im- age after training our proposed models is comparatively high as the RANs use two convolution layers sequen- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' It is also shown that the deeper networks perform better than the shallow networks and it has demonstrated “the deeper, the better” in classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Method: Wide Residual Networks The previous experiment did not assess wide ResNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To assess the widening factor on our pro- posed RANs, we increase the width of our RANs by factorizing the number of output channels for shallow networks like [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Like the original wide residual networks (WRNs) [33], we analyzed our proposed 26- layer bottleneck block of RANs with a widening factor, k = 2, 4, 6, 8, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We multiplied the number of out- put channels of RANs with k to obtain wide RANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We performed training with the same optimizer and hyper- parameters used in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Results: Wide Residual Networks Table 1 shows “the deeper the better” in vision classi- fication for our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' To compare our pro- posed RANs with the original wide ResNets (WRNs), we analyze our proposed method for different widening factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Table 2 shows an overall comparison among the original WRN-28-10 (28-layers with a widening factor Dataset Model Name Layers Params FLOPs Latency Accuracy CIFAR-10 ResNet [11] 26 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66ms 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68 RAN (Ours) 21M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='56G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='73ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='08 ResNet [11] 35 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='8M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='82ms 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='95 RAN (Ours) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='91ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='15 ResNet [11] 50 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='02ms 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='08 RAN (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='87G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='17ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='25 ResNet [11] 101 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68ms 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='36 RAN (Ours) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='7M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='52G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='27 ResNet [11] 152 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='41G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='39ms 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='36 RAN (Ours) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='55ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='37 CIFAR-100 ResNet [11] 26 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66ms 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='21 RAN (Ours) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='56G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='74ms 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66 ResNet [11] 35 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='80ms 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='72 RAN (Ours) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='91ms 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='38 ResNet [11] 50 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='11ms 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='95 RAN (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='7M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='87G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='17ms 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='84 ResNet [11] 101 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='72ms 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='80 RAN (Ours) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='52G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86ms 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='88 ResNet [11] 152 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='41G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='36ms 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='85 RAN (Ours) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='55ms 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='94 SVHN ResNet [11] 26 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='64ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='04 RAN (Ours) 21M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='56G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='73ms 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='60 ResNet [11] 35 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='8M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='79ms 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='74 RAN (Ours) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='9M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='90ms 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='50 ResNet [11] 50 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='05ms 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='76 RAN (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='87G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='11ms 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='32 ResNet [11] 101 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='64ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29 RAN (Ours) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='7M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='52G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='80ms 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29 ResNet [11] 152 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='41G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='28ms 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='35 RAN (Ours) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5ms 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='38 ImageNet-Tiny ResNet [11] 26 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='6M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='31ms 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='21 RAN (Ours) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='3M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='56G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='58ms 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='28 ResNet [11] 35 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86G 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='85ms 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='80 RAN (Ours) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='3M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68G 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='0ms 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='31 ResNet [11] 50 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='6M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='75ms 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='06 RAN (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='8M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='87G 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='02ms 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='40 ResNet [11] 101 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='3M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29G 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='86ms 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='62 RAN (Ours) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='1M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='52G 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='19ms 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18 ResNet [11] 152 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2M 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='41G 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='29ms 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='57 RAN (Ours) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='4M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='18G 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='72ms 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='16 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Image classification performance on the CIFAR benchmarks, SVHN, and Tiny ImageNet datasets for 26, 35, 50, 101, and 152-layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' of 10), and our proposed 26-layer networks with differ- ent widening factors (k = 2, 4, 6, 8, and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our pro- posed wide RANs show significant accuracy improve- ment over the original WRN-28-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This table also demonstrates “the wider the better” for our proposed wide RANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Method: Recursive Networks This experiment compares the cost and performance of our novel RARnet with the DRRN on the super- resolution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The RARnet is built by replacing the residual unit U with a RAN layer described in Equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These modifications form a new architecture, recur- sive axial residual network (RARNet) whose depth d is Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Recursive axial residual network (RARNet) architec- ture with B = 4 and U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Here, “RB” layer, and RAN refer to a recursive block, and residual axial block, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' defined as, d = (1 + URAN) × B + 1 (11) As two 1D layers are equivalent to one 2D layer and we replace each residual unit by a RAN unit (see Equation 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Hence, we rewrite Equation 4 to Equation 11 by removing the multiplier for the residual unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The pro- posed RARNet with four RB blocks is shown on the left in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' An RB block is expanded on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We trained our proposed RARNet using 291 images dataset [32] and tested using the Set5 dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We also use different scales (×2, ×3, and ×4) in training and testing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We used similar data augmenta- tion, training hyperparameters, and implementation de- tails like [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Results: Recursive Networks Table 3 shows the Peak Signal-to-Noise Ratio (PSNR) results of several CNN models including DRRN, and our proposed RARNet on the Set5 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The comparison between DRRN and RARNet is our main focus as it directly indicates the effectiveness of us- ing our proposed RAN block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' DRRN19 and DRRN125 are constructed using B = 1, U = 9, and B = 1, U = 25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' For fair comparison, we also construct similar architecture like RARNet19 (B = 1, URAN = 9) and RARNet125 (B = 1, URAN = 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our pro- posed models outperform all CNN models in Table 3 on the Set5 dataset and for all scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' As we are trying to propose a parameter-efficient architecture, parameter comparison is very essential along with the testing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Our proposed model for B = 1, and URAN = 9 takes 213,254 parameters compared to 297,216 parameters of DRRN (B = 1, and U = 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' RARNet, which is constructed using RAN blocks, re- duces by 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='2% the trainable parameters compared to Dataset Model Name Accuracy CIFAR-10 WRN-28-10 [33] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68 RAN-26-2 (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='32 RAN-26-4 (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68 RAN-26-6 (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='77 RAN-26-8 (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='83 RAN-26-10 (Ours) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='87 CIFAR-100 WRN-28-10 [33] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='57 RAN-26-2 (Ours) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='54 RAN-26-4 (Ours) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='75 RAN-26-6 (Ours) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='78 RAN-26-8 (Ours) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='82 RAN-26-10 (Ours) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='92 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Image classification performance comparison on the CIFAR benchmarks for 26-layer RAN architectures with dif- ferent widening factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' the DRRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Discussion and Conclusions This work introduces a new residual block that can be used as a replacement for the ResNet basic and bot- tleneck blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' This RAN block replaced the 2D convo- lution from the original ResNet blocks with two sequen- tial 1D convolutions along with a residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' These modifications help to reduce trainable parame- ters as well as improve validation performance on vision classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' But the latency of our proposed model is comparatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' The proposed model’s performance and parameter reduction outweigh this latency time limi- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We also checked this proposed block for widened ResNets and showed that the wide RANs obtain better accuracy performance than the WRNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We also checked the effectiveness of RANs on the SISR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Specifi- cally, we applied our proposed RAN block on am image restoration dataset and found that our proposed recursive axial ResNets (RARNets) improve image resolution and reduce trainable parameters more than the other CNN- based super-resolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Extensive experiments and analysis show that RANs can be deep, and wide and these are parameter-efficient and superior models for image classification and SISR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' We have shown that our proposed model is a viable replacement for ResNets on the tasks that were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Further work is required to determine the range of applications for which RANs may offer advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' References [1] Serge Belongie, Michael Wilber, and Andreas Veit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Residual networks behave like ensembles of relatively shallow networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 X1 X 个 RAN RB RAN RB RB 4 RAN RB conv RAN ★ -- Y X2Dataset Scale SRCNN [7] VDSR [17] DRCN [18] DRRN19 DRRN125 RARNet19 RARNet25 Set5 x2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='53 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='63 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='74 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='73 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='84 x3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='66 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='82 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='93 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='03 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='99 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='11 x4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='48 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='35 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='53 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='58 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='68 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='63 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='84 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Benchmark testing PSNR results for scaling factors ×2, ×3, and ×4 on Set5 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' [2] Yoshua Bengio, Aaron Courville, and Pascal Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Representation learning: A review and new perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine in- telligence, 35(8):1798–1828, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [3] Marco Bevilacqua, Aline Roumy, Christine Guillemot, and Marie Line Alberi-Morel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Low-complexity single- image super-resolution based on nonnegative neighbor embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 7 [4] Liang-Chieh Chen, Huiyu Wang, and Siyuan Qiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Scal- ing wide residual networks for panoptic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='11675, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 2 [5] Tianqi Chen, Ian Goodfellow, and Jonathon Shlens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Net2net: Accelerating learning via knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='05641, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [6] Jifeng Dai, Yi Li, Kaiming He, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' R-fcn: Ob- ject detection via region-based fully convolutional net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Advances in neural information processing sys- tems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [7] Chao Dong, Chen Change Loy, Kaiming He, and Xiaoou Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Image super-resolution using deep convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' IEEE transactions on pattern analysis and ma- chine intelligence, 38(2):295–307, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 3, 8 [8] Shang-Hua Gao, Ming-Ming Cheng, Kai Zhao, Xin-Yu Zhang, Ming-Hsuan Yang, and Philip Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Res2net: A new multi-scale backbone architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' IEEE trans- actions on pattern analysis and machine intelligence, 43(2):652–662, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [9] Xavier Glorot and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Understanding the difficulty of training deep feedforward neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pages 249–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' JMLR Workshop and Conference Proceedings, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [10] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Delving deep into rectifiers: Surpassing human- level performance on imagenet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Pro- ceedings of the IEEE international conference on com- puter vision, pages 1026–1034, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [11] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 2, 3, 6 [12] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Identity mappings in deep residual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Eu- ropean conference on computer vision, pages 630–645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 2, 3 [13] Jonathan Ho, Nal Kalchbrenner, Dirk Weissenborn, and Tim Salimans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Axial attention in multidimensional trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='12180, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 3 [14] Jie Hu, Li Shen, and Gang Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Squeeze-and-excitation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132– 7141, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [15] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep networks with stochastic depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, Computer Vision – ECCV 2016, pages 646–661, Cham, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Springer International Publish- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2 [16] Sergey Ioffe and Christian Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Batch normaliza- tion: Accelerating deep network training by reducing in- ternal covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In International conference on ma- chine learning, pages 448–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [17] Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Ac- curate image super-resolution using very deep convolu- tional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1646– 1654, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 3, 8 [18] Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deeply-recursive convolutional network for image super- resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1637– 1645, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 3, 8 [19] Alex Krizhevsky, Geoffrey Hinton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Learning mul- tiple layers of features from tiny images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 5 [20] Ya Le and Xuan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Tiny imagenet visual recogni- tion challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 5 [21] Xiu Li and Zuoying Cui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Deep residual networks for plankton classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In OCEANS 2016 MTS/IEEE Monterey, pages 1–4, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [22] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Feature pyra- mid networks for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recog- nition, pages 2117–2125, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [23] Xiaojiao Mao, Chunhua Shen, and Yu-Bin Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Im- age restoration using very deep convolutional encoder- decoder networks with symmetric skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Ad- vances in neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 3 [24] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bis- sacco, Bo Wu, and Andrew Y Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Reading digits in nat- ural images with unsupervised feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 5 [25] Siyuan Qiao, Huiyu Wang, Chenxi Liu, Wei Shen, and Alan Yuille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Micro-batch training with batch-channel normalization and weight standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='10520, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [26] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Ir- wan Bello, Anselm Levskaya, and Jonathon Shlens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Stand-alone self-attention in vision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='05909, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 2, 4 [27] Ilya Sutskever, James Martens, George Dahl, and Geof- frey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' On the importance of initialization and mo- mentum in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In International conference on machine learning, pages 1139–1147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [28] Ying Tai, Jian Yang, and Xiaoming Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Image super- resolution via deep recursive residual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Pro- ceedings of the IEEE conference on computer vision and pattern recognition, pages 3147–3155, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 3, 7 [29] Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Axial- deeplab: Stand-alone axial-attention for panoptic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In European Conference on Computer Vision, pages 108–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 3 [30] Yuxin Wu and Kaiming He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Group normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' In Proceedings of the European conference on computer vi- sion (ECCV), pages 3–19, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1 [31] Zifeng Wu, Chunhua Shen, and Anton Van Den Hengel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Wider or deeper: Revisiting the resnet model for visual recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Pattern Recognition, 90:119–133, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 3 [32] Jianchao Yang, John Wright, Thomas S Huang, and Yi Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Image super-resolution via sparse representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' IEEE transactions on image processing, 19(11):2861– 2873, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 7 [33] Sergey Zagoruyko and Nikos Komodakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' Wide residual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' arXiv preprint arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content='07146, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} +page_content=' 1, 2, 3, 5, 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE3T4oBgHgl3EQfoQom/content/2301.04631v1.pdf'} diff --git a/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf b/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..69ab601d2587ca9c6a70a648a1e7fc17ec4ac9b3 --- /dev/null +++ b/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:385b6ab986485c74f7a331c13ae5b2617a1cff457b7daeaa7fe679b7d74e6439 +size 1316514 diff --git a/4dAzT4oBgHgl3EQfuv2o/vector_store/index.pkl b/4dAzT4oBgHgl3EQfuv2o/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ad6e2371489981d047419620bf4fdba8a08c969a --- /dev/null +++ b/4dAzT4oBgHgl3EQfuv2o/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c5398059a290b5f17e26040908607a4830ab1c11df0663c84292d9a4287f6e33 +size 353784 diff --git a/5NE4T4oBgHgl3EQfBQty/vector_store/index.pkl b/5NE4T4oBgHgl3EQfBQty/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a63e8ecab63ad696e0d467549969195842df824e --- /dev/null +++ b/5NE4T4oBgHgl3EQfBQty/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e99c2eedeaba5eabd8abcaca8e1a215a8d692db7f15ce5bbea7f9b45cb92b301 +size 138488 diff --git a/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf b/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6fd65555c44b5a3e53580a84bddcf9a2ca12b6ef --- /dev/null +++ b/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:799a3af2a9c0381442f98ac482a7a6961f8916cbbfb6e8440395d906e0f5e7b3 +size 1485842 diff --git a/99AzT4oBgHgl3EQf_P4t/vector_store/index.pkl b/99AzT4oBgHgl3EQf_P4t/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c065509268444dee0236511e96216366befba0d8 --- /dev/null +++ b/99AzT4oBgHgl3EQf_P4t/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b75388a2d8276d8f67e689eb7a084440149d9a52cc2ae09acdae658b0e1b07da +size 332426 diff --git a/9NFST4oBgHgl3EQfaziV/content/tmp_files/load_file.txt b/9NFST4oBgHgl3EQfaziV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d04b7a54a8e2c6e3a9681e4e7a58539eafda048c --- /dev/null +++ b/9NFST4oBgHgl3EQfaziV/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf,len=673 +page_content='STOCHASTIC APPROACHES: MODELING THE PROBABILITY OF ENCOUNTERS BETWEEN H2-MOLECULES AND METALLIC ATOMIC CLUSTERS IN A CUBIC BOX Maximiliano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Leandro Andrini Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas Departamento de Química, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' de Ciencias Exactas (INIFTA/ UNLP-CONICET) Departamento de Matemática, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' de Ciencias Exactas, UNLP La Plata, Argentina mriddick@mate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='unlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='ar Enrique E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Álvarez Instituto de Cálculo, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón II, UBA (CABA) Departamento de Fisicomatemática, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' de Ingeniería, UNLP Ciudad Autónoma de Buenos Aires and La Plata, Argentina Félix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Requejo Instituto de Investigaciones Fisicoquimicas Teóricas y Aplicadas Departamento de Química, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' de Ciencias Exactas (INIFTA/ UNLP-CONICET) Departamento de Física, Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' de Ciencias Exactas, UNLP La Plata, Argentina ABSTRACT In recent years the advance of chemical synthesis has made it possible to obtain “naked”clusters of different transition metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' It is well known that cluster experiments allow studying the fundamental reactive behavior of catalytic materials in an environment that avoids the complications present in extended solid-phase research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In physicochemical terms, the question that arises is the chemical reduction of metallic clusters could be affected by the presence of H2 molecules, that is, by the probability of encounter that these small metal atomic agglomerates can have with these reducing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Therefore, we consider the stochastic movement of N molecules of hydrogen in a cubic box containing M metallic atomic clusters in a confined region of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We use a Wiener process to simulate the stochastic process, with σ given by the Maxwell-Boltzmann relationships, which enabled us to obtain an analytical expression for the probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This expression is an exact expression, obtained under an original proposal outlined in this work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' obtained from considerations of mathematical rebounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' On this basis, we obtained the probability of encounter for three different volumes, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 3 m 3, at three different temperatures in each case, 293, 373 and 473 K, for 10 1 ≤ N ≤ 10 10, comparing the results with those obtained considering the distribution of the position as a Truncated Normal Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Finally, we observe that the probability is significantly affected by the number N of molecules and by the size of the box, not by the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Keywords Wiener Process · Probability of encounters · Molecular Collisions · Atomic-Clusters · Mathematical Rebounds arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='13797v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='mtrl-sci] 10 Jan 2023 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 1 Introduction In the last two decades there has been an important development in clusters chemistry, and consequently new questions arise on the basis of these developments [1, 2, 3, 4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This interest is due to an atomic clusters containing up to a few dozen atoms exhibit features that are very different from the corresponding bulk properties and that can depend very sensitively on cluster size [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In particular, many of these transition metal clusters are used in the field of catalysis [1, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' One of the basic principles of catalysis is that when the smaller the metal particles, the larger the fraction of the metal atoms that are exposed at surfaces, where they are accessible to reactant molecules and available for catalysis [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' It is well known in chemistry that the encounter between two molecules can give rise to a chemical reaction, and from the mathematical aspect there are two fundamental ways to represent these types of situations as continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by stochastic processes whose variables are the number of molecules [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Without loss of generality, it can be considered that the molecular chemisorption is due to the encounter between a molecule and a surface (or a cluster in this case) with the energy necessary for the phenomenon of adsorption to occur [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Besides, the kinetics of hydrogen chemisorption by neutral gas-phase metal clusters exhibits a complex dependence on both cluster size and metal type [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' For different chemical purposes, for example, in the case of copper clusters (Cun) is very important to have control of the chemisorption of hydrogen on these clusters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' the formation of Cun-H2 species [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' From a reductionist point of view, the molecular chemisorption is a problem of encounter between bodies: metal clusters and reactant molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In our first approximation (mathematical reduction) we will consider the problem as a problem of encounter or collisions between bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We are interested in proposing this strategy because we are focused to answer what is the probability of meeting between N hydrogen molecules (N-H2) and a fixed M metallic clusters (M-Men), for a given time t, where the H2 move freely in a bounded volume V of R3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Under this assumption, we are going to consider H2-molecules and Men-clusters as rigid spheres of radii r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, it is considered that there will be a collision whenever the center-to-center distance between an H2-molecule and a Men-cluster is equal to r12 = r1 + r2 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Also, in this context we propose the H2-molecules follow a Brownian motion, namely: (a) it has continuous trajectories (sample paths) and (b) the increments of the paths in disjoint time intervals are independent zero mean Gaussian random variables with variance proportional to the duration of the time interval [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The pioneering work of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Gillespie [16, 18, 19] have given rise to a large number of works that are proposed different algorithms for the calculation for numerically simulating the time evolution of a well-stirred chemically reacting system, although despite recent major improvements in the efficiency of the stochastic simulation algorithm, its drawback remains the great amount of computer time that is often required to simulate a desired amount of system time [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' While our method is a simple reduction to collisions of molecules, allows to calculate the probability of encounter (scheduled in R) for a large number of molecules (≈ 106) and clusters (≈ 1020) with advantages regarding the cost of calculation, and the effects of first approximation can provide statistical support to the design of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This calculation is possible using a stochastic model (Wiener process) in the context of considerations from the Maxwell-Boltzmann theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2 A first theoretical approaching As we announced in the introduction, we will assume that hydrogen molecules have a random movement, whence let H(t) = (X(t), Y (t), Z(t)) the random variable which specify the space point where the H2 hydrogen molecule is at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Trivially, H(t) depends on an initially point H(0) = (x0, y0, z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Thus, when the initial starting point is undefined, H(t) = H(t, x0, y0, z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Our interest is in how probably is that the distance between H(t) and a fixed point (a, b, c) is smaller than ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The fixed point (a, b, c) are the coordinates for Men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Let’s consider the random variable D(t) as the variable that measures the distance between H(t) and the fixed point (a, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Following the classical Pythagorean relationship, D(t) = � (X(t) − a)2 + (Y (t) − b)2 + (Z(t) − c)2, and in general D(t) = D(t, x0, y0, z0, a, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Now, given a time window [0, τ], let Rτ := � � � 1 if min t∈[0,τ) D(t) ≤ ϵ, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' (1) So, for a fixed t0 > 0, we define G(t0) = P(D(t0) ≤ ϵ) = � ϵ 0 D(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, P(Rτ = 1) = � τ 0 G(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Thus, given τ > 0, Rτ depends only on the initial values (x0, y0, z0, a, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Now, if we have M-Men, the probability that the H2 molecule does not meet with any of the clusters is P(Rτ1 = 0, Rτ2 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=', RτM = 0) = pA, where Rτi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=', M}, follows the definition given in the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' If N-H2 molecules are in the environment, let Aj the event “the j-th hydrogen molecule meet with a metallic cluster”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Under random starting points, we are interested in P(AC 1 ∩ AC 2 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ∩ AC N) = pN A according to the independence among the hydrogen molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 Adaptation to our context Next, we proceed to realize the analysis according to the Brownian Motion Theory [17], in which the movement of the particle is independent among different axis, and we are going to assume that it follows a Wiener process [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, X(t) = x0 + WX(t) Y (t) = y0 + WY (t) Z(t) = z0 + WZ(t) And we will say that WX(t), WY (t) and WZ(t) are following a Wiener processes with σ = � kbT m , where kb is the Boltzmann’s constant, T is the absolute temperature in Kelvin (K) and m is the H2’s mass in kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' That is, we are imposing a physical behavior that obeys Maxwell-Boltzmann’s considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' According this: X(t) ∼ N(x0, σ2t) Y (t) ∼ N(y0, σ2t) Z(t) ∼ N(z0, σ2t) With density function fX(x, t|x0), fY (y, t|y0) and fZ(z, t|z0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Under these assumptions: fX(x, t|x0) = 1 √ 2πσ2t exp � −1 2 �x − x0 σ √ t �2� fY (y, t|y0) = 1 √ 2πσ2t exp � −1 2 �y − y0 σ √ t �2� fZ(z, t|z0) = 1 √ 2πσ2t exp � −1 2 �z − z0 σ √ t �2� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 Unbounded conditions Under unbounded conditions, as it is well known, the density of the particle position in the space for a fixed t follows the expression: fXY Z(x, y, z, t|x0, y0, z0) = fX(x, t|x0) · fY (y, t|y0) · fZ(z, t|z0) = = 1 ( � 2πσ2t)3 exp � −1 2 �(x − x0)2 + (y − y0)2 + (z − z0)2 σ2t �� This function is continuous in the variables x, y, z, t, then is also integrable in a measurable context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Because of this fact, Fubini’s theorem is aplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Now, calling ν = � (x−x0)2+(y−y0)2+(z−z0)2 2σ2 , and integrating over the variable t by substitution, results: 3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' fXY Z(x, y, z, τ|x0, y0, z0) = 1 −ν( √ 2πσ2)3 � τ 0 −ν ( √ t)3 exp � − � ν √ t �2� dt = 1 ν( √ 2πσ2)3 � ∞ √ ν τ e−u2du Remembering that the erfc function [23] is defined by: erfc(z) = 2 √π � ∞ z e−t2dt we conclude: fXY Z(x, y, z, τ|x0, y0, z0) = 1 2πν( √ 2σ2)3 erfc ��ν τ � From the physical-experimental perspective that the problem is lays out, the unbounded system lacks interest, so we will proceed to study the case of the bounded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 Bounded conditions We assume that the experiment takes place into a cubic recipe centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This implies that X(t), Y (t) and Z(t) ∈ [−L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L], for a fixed volume V = L3 in R3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In a similar issue the traditional way of approaching is by “truncation" [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' A drawback of this approach is the fact that the truncation does not represent precisely the reflection on the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' An illustrative and motivational argument is given by the following example: suppose a random walk of N = 4 steps, with starting point at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, the walker moves 1 step at right or left (with equal probability) at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, after four steps, the resultant probabilities of the walker position are: 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' with probability 3/8, −2 or 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' with probability 2/8, −4 or 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' with probability 1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The probability values (under truncation) in the closed interval [−2, 2] for the values (−2, −1, 0, 1, 2) are, respectively: (2/7, 0, 3/7, 0, 2/7) With fixed boundaries, considering reflections at [−2, 2], we can construct the following Markov transition matrix P: P = 0 1 0 0 0 1/2 0 1/2 0 0 0 1/2 0 1/2 0 0 0 1/2 0 1/2 0 0 0 1 0 At the fourth step, after some algebra, we obtain the respectively mass point probability for the position of the walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This is provided by the stochastic vector (1/4, 0, 1/2, 0, 1/4) (given by the third file of P 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' : with starting point at the origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' At this point, is clearly the difference between truncation and “rebounds" (considering reflection on the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We must modify the density of the position H(t) according to the particle rebounds (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' It is important to note that the rebounds indicated in the figure in gray colour do not correspond to the physical rebounds of the particles in the cubic box, but to the contributions of the displaced distribution considering an infinite behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Figure 1: In red colour an arbitrary normal distribution, N(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We observe in gray colour the folding of the normal distribution at the edge of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We could see that A + B = 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Inside the box, the derived density fB according to the variable X(t) ∼ N(x0, σ2t) with density function fX of the particle position (for each dimension, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 1) follows the expression: fB(x) = [fX(x) + fB+(x) + fB−(x)] × I[−L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L](x) where: fB+(x) = f(x + 2A) + f(x + 2A + 2B) + f(x + 2A + 2B + 2A) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' = = f(x + 2(L − x)) + f(x + 2(L − x) + 2(x − (−L)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' = f(−x + 2L) + f(x + 4L) + f(−x + 6L) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' = ∞ � k=1 f((−1)kx + 2kL) = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �((−1)kx + 2kL) − x0 σ √ t �2� and fB−(x) = f(x − 2B) + f(x − 2B − 2A) + f(x − 2B − 2A − 2B) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' = = f(x − 2(x − (−L))) + f(x − 2(x − (−L)) − 2(L − x)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' = f(−x − 2L) + f(x − 4L) + f(−x − 6L) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=" = ∞ � k=1 f((−1)kx − 2kL) = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �((−1)kx − 2kL) − x0 σ √ t �2� 5 B B' B 0 XM." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The proof that fB is a density function is straightforward its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Trivially, fB > 0, and by construction: � ∞ −∞ fB(t)dt = � L −L fB(t)dt = � ∞ −∞ fX(t)dt = 1 For practical purposes, we now try to find an upper bound to this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Looking at in the model proposed, the next constraint is straightforward |(−1)kx − x0| ≤ 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Following these constraints: fB+(x) = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �((−1)kx + 2kL) − x0 σ √ t �2� ≤ ∞ � k=1 1 √ 2πσ2t exp � −1 2 �−2L + 2kL σ √ t �2� = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �2(k − 1)L σ √ t �2� = ∞ � k=0 1 √ 2πσ2t exp � −1 2 � 2kL σ √ t �2� = 1 √ 2πσ2t ∞ � k=0 exp � −1 2 �4L2 σ2t ��k2 It is known that ∞ � k=0 rk2 = 1 2 + 1 2ΘE[3, 0, r] where ΘE is the Jacobi theta elliptic function [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' So: fB+(x) ≤ 1 √ 2πσ2t ∞ � k=0 exp � −1 2 �4L2 σ2t ��k2 = 1 √ 2πσ2t �1 2 + 1 2ΘE � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' exp � −1 2 �4L2 σ2t ���� and fB−(x) = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �((−1)kx − 2kL) − x0 σ √ t �2� ≤ ∞ � k=1 1 √ 2πσ2t exp � −1 2 �−2L − 2kL σ √ t �2� = ∞ � k=1 1 √ 2πσ2t exp � −1 2 �−2(k + 1)L σ √ t �2� = ∞ � k=2 1 √ 2πσ2t exp � −1 2 �−2kL σ √ t �2� = 1 √ 2πσ2t � ∞ � k=0 exp � −1 2 �4L2 σ2t ��k2 − 1 − exp � −1 2 �4L2 σ2t ��� = 1 √ 2πσ2t �1 2 + 1 2ΘE � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' exp � −1 2 �4L2 σ2t ��� − 1 − exp � −1 2 �4L2 σ2t ��� 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, fB+(x) + fB−(x) ≤ 1 √ 2πσ2t � ΘE � 3, 0, exp � −1 2 �4L2 σ2t ��� − exp � −1 2 �4L2 σ2t ��� = CB For each x ∈ [−L, L], fB(x) ≤ fX(x) + CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Besides CB does not depends on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Consequently, we have a maximum for the density fB which is equal to f(x0) + CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Calling PB = (f(x0) + CB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2ϵ, we can conclude that: P(X(t) ∈ (a0 − ϵ, a0 + ϵ)) ≤ PB, for any a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Analogous, CB is the same for the variables Y (t) and Z(t), and we know that f(x0) = f(y0) = f(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, the same result is available for the variables Y (t) and Z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' According the bounded CB, it is straightforward the uniform convergence of the series fB+ and fB− (by the M Weierstrass criteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' An important fact to remark is that PB is not even a probability, but in the case in we are interested, we know that is a real number bigger than the probability desired, and then, under certain conditions, we can work with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' For practical purposes, the error through the CB implementation can be minimized, since the first S terms are available, and the tail can be compared with S−1 � k=0 rk2 ≤ ∞ � k=0 rk2 = S−1 � k=0 rk2 + ∞ � k=S rk2 And, ∞ � k=S rk2 = ∞ � k=0 r(k+S)2 = ∞ � k=0 rk2+2kS+S2 = rS2 ∞ � k=0 rk2r2kS ≤ rS2 ∞ � k=0 rk2 Then, ∞ � k=S rk2 ≤ rS2 �1 2 + 1 2ΘE[3, 0, r] � Controlling the value of S controls the value of the error made by truncating the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' As we said, CB does not depend on x, thus, the desired probability can be estimated with any degree of accuracy, according the computational cost necessary to this development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Taking into consideration the Brownian Motion Theory, in the time lapse of 1 second, the particle position under unbounded conditions follows a N(x0, σ2) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' To discretize the problem, if we partitioned the time axis of τ seconds in τ intervals of 1 second each one, then: P(H(t) ∈ Bϵ(a, b, c)) ≤ P(H(t) ∈ Qϵ(a, b, c)) where Qϵ(a, b, c) denotes the cube centered in (a, b, c) with side size 2 × ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' And, considering the independence between X(t), Y (t) and Z(t), with X(t) ∈ (a − ϵ, a + ϵ), Y (t) ∈ (b − ϵ, b + ϵ) and Z(t) ∈ (c − ϵ, c + ϵ), P(H(t) ∈ Qϵ(a, b, c)) = P(H ∈ Qϵ) is P(H ∈ Qϵ) = P(X(t)) × P(Y (t)) × P(Z(t)) ≤ PB × PB × PB = P 3 B For each second τj for τj ∈ {1 : τ}, P(H(t) ∈ Qϵ(a, b, c)) ≤ P 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, under the Wiener process formulation, H(τj) ⊥ H(τk|τj) if j ̸= k, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 7 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' P(H(τj) ∈ Qϵ(a, b, c)) ≤ P 3 B, ∀τj ∈ {1 : τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Calling F :=“# of τj ∈ {1 : T} in which H(τj) ∈ Qϵ(a, b, c)", we are interesting in the event F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' According its nature, F is a Binomial random variable B(τ, P 3 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Consequently, the non-collision probability is pNC = P(F = 0) ≤ (1 − P 3 B)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' At this point, we only can conclude that the probability of the encounter between a hydrogen molecule and a Men cluster in a time τ is less than p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We proceed to analyze what happens when the number of hydrogen molecules and metallic clusters increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' We emphasize that the H2 molecules have a random movement while the clusters are confined in a fixed region of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Since p is the probability that a random hydrogen molecule meets in the cube Qϵ in which a Men cluster is, the most unfavorable case with M clusters is when there is no intersection among the cubes that contain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In this case: pA = P(Rτ1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=', RτM = 0) = 1 − M � i=1 P(Rτi = 1) ≥ 1 − � M � i=1 P(Rτi = 1) � = 1 − M × p In view of this analysis, we can conclude that the non-collision probability is higher than pNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In regular conditions, when this approach is used, the values of pNC and N outcomes into a several numerical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In this case, the small value of pNC and the large value of N place us in conditions to use the Poisson approach to the Binomial distribution (with parameter λ = N × p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Then, P(X = 0) ≈ exp(−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Even in the cases when the probability is still unavailable, the expected number of collisions is presented according a time window, and then we can estimate the probability of collisions in a time window T using the relationship between the Poisson and Exponential distributions[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Next, we present the results of the analysis whit different box dimensions (in meters) and number of hydrogen molecules (N), according to M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9 × 10 20 Cu20-clusters [27], where the Cu20-clusters have been considered as spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 3 Results and analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 Obtaining non-collision probability values The situation we consider is approximately a “realistic”situation, with M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9 × 10 20 Cu20-clusters in a cubic box according to the standard dimensions of reaction chambers (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 3 m 3), and a variable N-H2-molecules “contamination”(10 1 ≤ N ≤ 10 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' It worked with three temperatures, T, 293, 373 and 473 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The choice of T is arbitrary, conditioned by the possible reaction temperatures [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2 we observe the results obtained for the simulations, considering the maximum sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' That is, take S = 10 6, perform the sum, and add the maximum level for the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Clearly, a greater probability of non-collision, pNC, is observed depending on the increase in volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' For a detailed study, we proceed as follows: we model the data obtained through a non-linear graphic fitting considering a Boltzmann decrease function, g(x) = A2 + A1−A2 1+exp � x−x0 dx � (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 we show the statistical results for each parameter in each data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Under these considerations, we can calculate the critical value (criticality)[29, 30] of hydrogen molecules, that is “what is the value of N for which the non-collision probability is greater than 1 2”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' the value of the exponent for which 1 2 < pNC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' It should be clarified that, in the strict physical sense, there is no abrupt phase transition to consider “criticality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' As we assumed in the introduction, we consider that there is a chemical reaction if there is an encounter between two molecules, and under this assumption we are considering as critical the level of presence of hydrogen for a chemical reaction to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In any case, it can be demonstrated that there is an “abrupt”transition behavior, for a well defined interval in the number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 3 we can observe this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Figure 2: Results for the non-collision probability, pNC, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ln(N) for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05m (V1), L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1m (V2) and L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2m (V3), at T = 293 K (blue square), 373 K (black star) and 493 K (red triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Figure 3: Data (blue square) modeling using a non-linear Boltzmann decrease function (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 9 V1 V2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0 - 4 GD △ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8 0 ★ 口 293 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6 V 373 K 0 473 K 文 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 - ★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0- 支立文支安 123456789101234567891012345678910 Ln(N)293 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0 - Boltzmann Fit 293 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0 0 3 4 5 6 7 8 10 Ln(N)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Table 1: Critical values obtained from the decrease model for each box and each temperature, for mathematical robounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L [m] 293 K 373 K 473 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='96 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='96 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='96 Figure 4: Results for the non-collision probability, pNC, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ln(N) for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05m (V1), at T = 293 K, 373 K and 493 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In Table 1 we can see the critical values obtained from the decrease model for each box and each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' For the smallest volumes, V1 and V2, it is observed that the critical value of N depends more strongly on the temperature than in the case of the larger volume (V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Although it is remarkable the fact of dependence with the size of the box, it can be seen directly from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In this way, and under these simplified assumptions, we can obtain control of contaminant molecules in relation to the volume and temperature parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Linear behavior is evident from the values obtained (Table 1, N vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Moreover, as the volume increases the slope increases from negative values to null value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 4 Conclusion By way of conclusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' it can be indicated that considering a Wiener stochastic process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' for thermodynamic-statistical movements of a gas confined in a box,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' and considering mathematical rebounds bounded by the physical-geometric contour of the problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' the analytical expression could be obtained for the probability density function of encounters between two differentiated species of molecules (one of the species fixed in the box -solid or liquid- and the other species is a gas whose molecules move stochastically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In addition, the function obtained can be calculated numerically or can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The bounded process allows to reduce the computational cost, and to limit the error from cutting the Table 2: Critical values obtained from the decrease model for each box and each temperature, for truncated normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L [m] 293 K 373 K 473 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='96 10 293 Trunc ☆373 Trunc 473 Trunc 口 293 K 373 K 473 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0 0 0 ★ Non-collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0+ 口口OO口 1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910 Ln(N)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Figure 5: Results for the non-collision probability, pNC, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ln(N) for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1m (V1), at T = 293 K, 373 K and 493 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Figure 6: Results for the non-collision probability, pNC, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ln(N) for L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2m (V1), at T = 293 K, 373 K and 493 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Comparison between models:“xxx Trunc”correspond to the truncated normal model and “xxx K”to the mathematical rebound model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 11 293 Trunc ☆373 Trunc → 473 Trunc 口 293 ★373 473 ★★★ 口 Non-collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0 - OOOOG 1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910 Ln(N)293 Trunc ☆373 Trunc → 473 Trunc 口 293 K 373 K 473K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0- Non-collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0- 123456789101234567891012345678910 Ln(N)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' sum in a finite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In particular, there is an error control that can be made, and it is possible to refine the process according to the precision required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' From the physical-chemical point of view, it is observed that both the number of gas molecules and the dimensions of the box affect the probability of encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' For this model, temperature is a parameter that has a lower incidence on the values of the probability of encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' At this point some considerations have to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The first is that in a strict sense a chemical reaction is more than the encounter of two chemical entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The second is the exceptional chemical nature of metal clusters, which make them highly reactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Despite the simplicity of the model we are proposing, this model can account in an experiment design about the collision probability between two chemical entities (and this collision can lead to a chemical reaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' From the point of view of computation, it is a system that requires less computational cost (time + memory) than the algorithmic systems developed for this type of problems, so it contributes as a test method in the design of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The comparison with an established method (truncated normal model) was optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In the method of mathematical rebounds the number of molecules needed for a reaction is less than the number obtained by the truncated normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' This is an advantage when strict contamination control is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' On the other hand, in terms of obtaining the density function, mathematical results can be generalized for volumes of rectangular prisms of uneven sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In addition, it remains to calculate the first and second order moments of the density function obtained, work that exceeded the purposes of present communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Acknowledgments This was was supported in part by PICT-2019-0784, PICT-2017-3944, PICT-2017-1220, PICT-2017-3150 (PICT, Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación) and PPID-I231 (PPID, Universiad Nacional de La Plata).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' References [1] Bruce C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Supported metal clusters: synthesis, structure, and catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Chemical reviews, 95(3):511–522, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [2] M Arturo López-Quintela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Synthesis of nanomaterials in microemulsions: formation mechanisms and growth control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Current Opinion in Colloid & Interface Science, 8(2):137–144, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [3] Puru Jena and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Welford Castleman Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Clusters: A bridge across the disciplines of physics and chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 103(28):10560–10569, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [4] Shahana Huseyinova, Joseé Blanco, Feélix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Requejo, Joseé M Ramallo-López, M Carmen Blanco, David Buceta, and M Arturo Loópez-Quintela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Synthesis of highly stable surfactant-free cu5 clusters in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The Journal of Physical Chemistry C, 120(29):15902–15908, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [5] Lichen Liu and Avelino Corma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Confining isolated atoms and clusters in crystalline porous materials for catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Nature Reviews Materials, 6(3):244–263, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [6] Huixia Luo, Peifeng Yu, Guowei Li, and Kai Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Topological quantum materials for energy conversion and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Nature Reviews Physics, 4(9):611–624, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [7] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Is there evidence for exponential quantum advantage in quantum chemistry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02199, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [8] Mingli Yang, Koblar A Jackson, Christof Koehler, Thomas Frauenheim, and Julius Jellinek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Structure and shape variations in intermediate-size copper clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The Journal of chemical physics, 124(2):024308, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [9] Manfred T Reetz and Wolfgang Helbig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Size-selective synthesis of nanostructured transition metal clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of the American Chemical Society, 116(16):7401–7402, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [10] John D Aiken III and Richard G Finke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' A review of modern transition-metal nanoclusters: their synthesis, characterization, and applications in catalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of Molecular Catalysis A: Chemical, 145(1-2):1–44, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [11] Gareth S Parkinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Unravelling single atom catalysis: The surface science approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='09473, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [12] Michael A Gibson and Jehoshua Bruck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Efficient exact stochastic simulation of chemical systems with many species and many channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The journal of physical chemistry A, 104(9):1876–1889, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [13] David E Brown, Douglas J Moffatt, and Robert A Wolkow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Isolation of an intrinsic precursor to molecular chemisorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Science, 279(5350):542–544, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [14] MR Zakin, RO Brickman, DM Cox, and A Kaldor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Dependence of metal cluster reaction kinetics on charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' chemisorption of hydrogen by neutral and positively charged iron clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The Journal of chemical physics, 88(10):6605–6610, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [15] Xiang-Jun Kuang, Xin-Qiang Wang, and Gao-Bin Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' A density functional study on the adsorption of hydrogen molecule onto small copper clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of Chemical Sciences, 123(5):743–754, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [16] Daniel T Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Exact stochastic simulation of coupled chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The journal of physical chemistry, 81(25):2340–2361, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [17] Zeev Schuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Theory and applications of stochastic processes: an analytical approach, volume 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Springer Science & Business Media, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [18] Daniel T Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of computational physics, 22(4):403–434, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [19] Daniel T Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Concerning the validity of the stochastic approach to chemical kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of Statistical Physics, 16(3):311–318, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [20] Daniel T Gillespie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Approximate accelerated stochastic simulation of chemically reacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The Journal of chemical physics, 115(4):1716–1733, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [21] Ben Leimkuhler and Charles Matthews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Interdisciplinary applied mathematics, 36, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [22] Ben Leimkuhler and Charles Matthews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Numerical methods for stochastic molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In Molecular Dynamics, pages 261–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [23] Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Formulas and theorems for the special functions of mathematical physics, volume 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [24] James J Heckman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' In Annals of economic and social measurement, volume 5, number 4, pages 475–492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' NBER, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [25] Charles M Stein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Estimation of the mean of a multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' The annals of Statistics, pages 1135–1151, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [26] Jeroen Gerritsen and J Rudi Strickler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Encounter probabilities and community structure in zooplankton: a mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of the Fisheries Board of Canada, 34(1):73–82, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [27] Leandro Andrini, Germán J Soldano, Marcelo M Mariscal, Félix G Requejo, and Yves Joly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Structure stability of free copper nanoclusters: Fsa-dft cu-building and fdm-xanes study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of Electron Spectroscopy and Related Phenomena, 235:1–7, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [28] Avelino Corma, Patricia Concepción, Mercedes Boronat, María J Sabater, Javier Navas, Miguel José Yacaman, Eduardo Larios, Álvaro Posadas, M Arturo López-Quintela, David Buceta, Ernest Mendoza, Gemma Guilera, and Álvaro Mayoral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Exceptional oxidation activity with size-controlled supported gold clusters of low atomicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Nature Chemistry, 5(9):775–781, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [29] Per Bak and Maya Paczuski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Complexity, contingency, and criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 92(15):6689–6696, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' [30] Terrie M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Criticality in stochastic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Journal of the Operational Research Society, 43(4):353–357, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 Errors in the Boltzmann model for the probability calculated according to mathematical rebounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Program used: Origin 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 In all cases, number of points is 10, and degrees of freedon is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' 13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='009 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0008 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='66387 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0016 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99888 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='006 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='003 x0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 Reduced Chi-Sqr 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='93743 × 10 −5 Residual Sum of Squares: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='76246 × 10 −4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99957 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='008 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0009 x0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='30175 × 10 −4 Residual Sum of Squares: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='8105 × 10 −4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99927 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='009 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0011 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='66387 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0016 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99888 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='008 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='002 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 Reduced Chi-Sqr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='88888 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00113 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9992 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='005 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0019 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 Reduced Chi-Sqr 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='76599 × 10 −5 Residual Sum of Squares: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='65959 × 10 −4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99967 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='003 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='64593 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00159 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9989 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='008 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='64587 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00159 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9989 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='008 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='64587 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00159 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9989 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 Errors in the Boltzmann model for the probability calculated according to the truncated normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='006 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0001 x0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 Reduced Chi-Sqr 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='6611 × 10 −5 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='000399 15 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99965 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='006 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='003 x0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 Reduced Chi-Sqr 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='69947 × 10 −5 Residual Sum of Squares: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0196 × 10 −4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99960 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='005 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0003 x0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 Reduced Chi-Sqr 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='69508 × 10 −4 Residual Sum of Squares: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01705 × 10 −4 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99964 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='003 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0009 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='07 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='5392 × 10 −5 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00033 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99977 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='006 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 Reduced Chi-Sqr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='20178 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='000721 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9995 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='1 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='005 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='05 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 Reduced Chi-Sqr 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='70051 × 10 −5 Residual Sum of Squares: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2203 × 10 −4 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Riddick, Stochastic approaches: modeling the probability of encounters, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99963 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 293 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='003 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0065 x0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='75184 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00165 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99885 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 373 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='007 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0075 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='07471 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00124 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='99913 L=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='2 m, T = 473 K Parameter Value Standard Error A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='008 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='0025 x0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='02 dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='03 Reduced Chi-Sqr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='64587 × 10 −4 Residual Sum of Squares: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='00159 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content=' R-Square: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} +page_content='9989 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFST4oBgHgl3EQfaziV/content/2301.13797v1.pdf'} diff --git a/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf b/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9599555efb4b69638d120ea1bbae81cf9da5a964 --- /dev/null +++ b/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:21d0fddaa7cc80733690766b7ee0c9e8ce487d1f4f57374dc909985d767dd683 +size 1468140 diff --git a/A9AyT4oBgHgl3EQfRvd3/vector_store/index.faiss b/A9AyT4oBgHgl3EQfRvd3/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..64d8f495d3254e1ef57e5b9c6dc200ea084c3d92 --- /dev/null +++ b/A9AyT4oBgHgl3EQfRvd3/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a588373e0c70eccf27aa2e166d5b319b0ee7256663bb390b78c21058a37fcfa8 +size 4915245 diff --git a/A9AyT4oBgHgl3EQfRvd3/vector_store/index.pkl b/A9AyT4oBgHgl3EQfRvd3/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2e1301733684e640f90398b517ee8c76ba4407a5 --- /dev/null +++ b/A9AyT4oBgHgl3EQfRvd3/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:86b11c9855600226696bea4f38fa60a776e77df86e5bf7757a15db205d7c02d9 +size 197048 diff --git a/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf b/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f04cb612634d81e14731cd625c210584ce9f272 --- /dev/null +++ b/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c193a78ca62e816c5dd0eada8a4c24ba608b6931f38aca740235c53c72bec300 +size 516482 diff --git a/AdE2T4oBgHgl3EQfRQcf/vector_store/index.pkl b/AdE2T4oBgHgl3EQfRQcf/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2f09b869a74d4f0669c599b62b08a2783e6ed617 --- /dev/null +++ b/AdE2T4oBgHgl3EQfRQcf/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:feff0c7f76c17afc2559f1a1364484fd773fa5e02e1c2e81d35c8e5aa5c33e8d +size 98656 diff --git a/AtAzT4oBgHgl3EQf__8r/vector_store/index.pkl b/AtAzT4oBgHgl3EQf__8r/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..098d8146377fb5cca15f2a2d48897afcd3b0f979 --- /dev/null +++ b/AtAzT4oBgHgl3EQf__8r/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1ed8159929f9780ef64d657108dd8876575b0feeaebba6c5a831b95e9341266a +size 135517 diff --git a/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/2301.04669v1.pdf.txt b/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/2301.04669v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0669c99f57752d86b781f68f27eb618aaeafcef2 --- /dev/null +++ b/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/2301.04669v1.pdf.txt @@ -0,0 +1,22028 @@ +Kondo Lattice Model of Magic-Angle Twisted-Bilayer Graphene: Hund’s Rule, Local-Moment +Fluctuations, and Low-Energy Effective Theory +Haoyu Hu,1 B. Andrei Bernevig,2, 1, 3, ∗ and Alexei M. Tsvelik4 +1Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain +2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA +3IKERBASQUE, Basque Foundation for Science, Bilbao, Spain +4Division of Condensed Matter Physics and Materials Science, +Brookhaven National Laboratory, Upton, NY 11973-5000, USA +We apply a generalized Schrieffer–Wolff transformation to the extended Anderson-like topological heavy +fermion (THF) model for the magic-angle (θ = 1.05◦) twisted bilayer graphene (MATBLG) (Phys. Rev. Lett. +129, 047601 (2022)), to obtain its Kondo Lattice limit. In this limit localized f-electrons on a triangular lattice +interact with topological conduction c-electrons. By solving the exact limit of the THF model, we show that +the integer fillings ν = 0, ±1, ±2 are controlled by the heavy f-electrons, while ν = ±3 is at the border of a +phase transition between two f-electron fillings. For ν = 0, ±1, ±2, we then calculate the RKKY interactions +between the f-moments in the full model and analytically prove the SU(4) Hund’s rule for the ground state +which maintains that two f-electrons fill the same valley-spin flavor. Our (ferromagnetic interactions in the) spin +model dramatically differ from the usual Heisenberg antiferromagnetic interactions expected at strong coupling. +We show the ground state in some limits can be found exactly by employing a positive semidefinite ”bond- +operators” method. We then compute the excitation spectrum of the f-moments in the ordered ground state, +prove the stability of the ground state favored by RKKY interactions, and discuss the properties of the Goldstone +modes, the (reason for the accidental) degeneracy of (some of) the excitation modes, and the physics of their +phase stiffness. We develop a low-energy effective theory for the f-moments and obtain analytic expressions for +the dispersion of the collective modes. We discuss the relevance of our results to the spin-entropy experiments +in twisted bilayer graphene. +Introduction— The discovery of the correlated insulating +phase [1] and superconductivity [2] in the MATBLG [3] has +driven considerable theoretical [4–20] and experimental ef- +forts [21–44] to understand its topology [45–61] and corre- +lation physics [45, 60–76]. Theoretically, correlated insula- +tors [77–88], ferromagnetic order [85, 89–92], superconduc- +tivity [81, 93–108], and other exotic quantum phases [109– +116] have been identified and systematically studied which +all point to rich physics [60] of the MATBLG. The recent +experiments [28, 64, 117, 118] have provided evidence for +fluctuating local moments and Hubbard-like physics. Mean- +while the theoretical understanding is challenging since the +stable topology [52, 57] of the flat bands obstructs the sym- +metric real-space description. A real-space extended Hubbard +model [68, 70, 119–121] can still be constructed, but a certain +symmetry (C2zT or P) becomes non-local. To address this +problem the authors of Ref. [122] have introduced an exact +mapping of MATBG to a THF model. THF is a version of +the extended Anderson lattice model describing localized f- +electrons interacting with topological conduction c-electrons. +The f-electrons have zero kinetic energy and strong Hubbard +interactions; they admit a description in terms of localized +Wannier orbitals centered at the AA-stacking region. +The +topological flat bands can be recovered from the hybridization +between the f- and the c-electrons [122]. +In this letter, we map the THF model to a Kondo lattice +model using a generalized Schrieffer-Wolff (SW) transforma- +tion which takes into account the density-density interaction +term between f- and c-electrons. In this limit, the dynamics +∗ bernevig@princeton.edu +of the localized orbitals becomes the one of the f-moments. +By solving exactly a particular limit of the THF model, we +show that the integer fillings ν = 0, ±1, ±2 are controlled +by the f-electrons, while the situation is drastically different +for ν = ±3 which sits at the phase transition between two +f-electrons fillings. In the Kondo lattice model of MATBLG, +the local moments formed by the fully-localized f-electrons +interact with topological conduction electron bands via Kondo +superexchange and direct ferromagnetic exchange interaction. +These two types of exchange interactions induce an RKKY in- +teraction. At ν = 0, −1, −2, the RKKY interaction dominates +the physics and stabilizes the ferromagnetic ground states that +obey the Hund’s rule. This result provides an analytic deriva- +tion of the Hund’s rule found numerically in the Hartree-Fock +calculations of THF model [122]. We then proceed to inves- +tigate fluctuations of the f-moments in the symmetry broken +ground states by developing the low-energy effective theory +and calculating the excitation spectrum. +Schrieffer–Wolff transformation and Kondo lattice model— +The single-particle Hamiltonian of the THF model contains +the kinetic term ˆHc describing the topological conduction c- +electron bands (SM [123], Sec. I), and the hybridization be- +tween the f- and the c-electrons ˆHfc [122, 123]: +ˆHfc = +� +|k|<Λc,R +i,ξ,ξ′ +�eik·R− |k|2λ2 +2 +˜H(fc) +ξξ′ (k) +√NM +ψf,ξ,† +R,i ψc′,ξ′ +k,i ++ h.c. +� +˜H(fc)(k) = +� +γ +v′ +⋆(kx − iky) +v′ +⋆(kx + iky) +γ +� +. +(1) +where ψc′,ξ,† +k,i +creates Γ3 “conduction” c-electron with mo- +arXiv:2301.04669v1 [cond-mat.str-el] 11 Jan 2023 + +2 +mentum k, valley-spin flavor i ∈ {1, 2, 3, 4} (with (1, 2, 3, 4) +corresponding to (+ ↑, − ↑, + ↓, − ↓)), and ”orbital” index +ξ = (−1)a+1η (with a = 1, 2 are original orbital indices and +η = ± are valley indices defined in Ref. [122]). ψf,ξ,† +R,i creates +f-electron at moir´e unit cell R with valley-spin flavor i, and +orbital index ξ = (−1)a+1η, where a = 1, 2, η = ± [122]. +Λc is the momentum cutoff, NM is the total number of moir´e +unit cells and λ is the damping factor [122]. In the hybridiza- +tion matrix ˜H(fc)(k), we only keep the first two terms (γ and +v′ +⋆) in the expansion in powers of k. The flat band limit is re- +alized by setting M = 0, where M is taken as a parameter of +ˆHc. +The interaction Hamiltonian of the THF model is ˆHI = +ˆHU + ˆHJ + ˆHV + ˆHW , where ˆHU, ˆHJ describe respec- +tively the on-site Hubbard interaction of the f-electrons (U = +57.95meV), and the ferromagnetic exchange between the f- +and the c-electrons (J = 16.38meV), ˆHV , ˆHW describe re- +spectively the repulsion between the c-electrons (∼ 48meV) +and the repulsion between the f- and the c-electrons (W = +47meV) [122, 124]. The full Hamiltonian is ˆHc + ˆHfc + ˆHI. +The model possesses the U(4)×U(4) symmetry in the chiral- +flat limit (M = 0, v′ +⋆ = 0), a flat U(4) symmetry in the +nonchiral-flat limit (M = 0, v′ +⋆ ̸= 0), a chiral U(4) sym- +metry in the chiral-nonflat limit (M ̸= 0, v′ +⋆ = 0) and a +U(2) × U(2) symmetry in the nonchiral-nonflat limit (M ̸= +0, v′ +⋆ ̸= 0) [4, 85, 119, 122, 123, 125, 126]. +At sufficiently strong on-site Coulomb interaction U, the f- +electrons are fully localized and give rise to local f-moments +which are defined as +ˆΣ(f,ξξ′) +µν +(R) = +� +i,j +1 +2T µν +ij ψf,ξ,† +R,i ψf,ξ′ +R,j . +(2) +{T µν +ij } with µ, ν ∈ {0, x, y, z} are given in SM [123], Sec. I. +Eq. 2 are the generators of the U(8) group (8 = 2(orbital) × +2(valley) × 2(spin)) where the U(1) charge component can be +gauged away for the fully-localized f-electrons. +We first analyze the zero-hybridization limit of the model +(γ = 0, v′ +⋆ = 0). v′ +⋆ is small [122] while γ changes rapidly +and goes through zero at the value of w0/w1 = 0.9 close to +the actual MATBLG value w0/w1 = 0.8 [124]. Hence this +limit can be thought as a meaningful approximation close to +the MATBLG. The zero-hybridization model is exactly solv- +able at zero Coulomb repulsion between c-electrons. Here, we +treat ˆHV in the mean-field approximation ( ˆHMF +V +) and drop +the ˆHJ which is relatively weak. We then solve the model +under the assumption that each site is filled with νf + 4 f- +electrons with an integer νf (SM. [123], Sec. II). We use νc +to denote the filling of the c-electrons and use ν = νf + νc +to denote the total fillings. In Fig. 1 (a), we plot νf and νc of +the ground state as a function of ν. At ν = 0, −1, −2, −3, the +ground state has νf = ν and νc = 0. ν = −3 is close to the +transition point between νf = −3 state and νf = −2 states. +Thus, at ν = −3, our assumption of uniform charge distri- +bution may be violated [115], and a Kondo model description +fails. This is consistent with the special place that ν = −3 has +in the TBG physics [115]. ++ ↑ ++ ↓ +− ↑ +− ↓ +ξ = + 1 +ξ = − 1 +ξ = + 1 +ξ = − 1 +ξ = + 1 +ξ = − 1 +ν = 0 +ν = − 1 +ν = − 2 +(a) +(b) +(c) +(d) +FIG. 1. (a) Filling of f electrons(νf) and c-electrons(νc) as a func- +tion of total filling(ν) in the zero hybridization model. (b), (c), (d) +Illustrations of ground states at ν = 0, −1, −2. The red dot means +the filling of one f-electron. +At ν = 0, −1, −2, we fix the filling of the f-electrons (ac- +cording to Fig. 1 (a)) and perform a generalized SW transfor- +mation (SM. [123], Sec. IV), which leads to the following +Kondo lattice Hamiltonian: +ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ + ˆHMF +V ++ ˆHW +(3) +where ˆHK and ˆHcc are the Kondo interaction and one-body +scattering term generated by the SW transformation respec- +tively [123, 127]. The Kondo interaction ˆHK takes the form +of +ˆHK = +� +R +� +|k|<Λc,|k+q|<Λc +� +µνξξ′ +e−iq·Re− |k|2+|k+q|2 +2 +λ2 +NM +� +γ2 +Dνc,νf +: ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′ξ) +µν +(k, q) : + +(4) +�v′ +⋆γ(kx − iξky) +Dνc,νf +: ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′−ξ) +µν +(k, q) : +h.c. +�� +where the colon symbols represent the normal ordering and +Σ(c′,ξ′ξ) +µν +(k, q) (SM [123], Sec. I) is the U(8) moment of +c-electrons defined in the manner similar to Σ(f,ξ′ξ) +µν +(R) in +Eq. 2. The parameter Dνc,νf at ν = νf = 0, −1, −2 is defined +as +1 +Dνc=0,νf += − +1 +(U − W)νf − U +2 ++ +1 +(U − W)νf + U +2 +. (5) +The distinct feature of this expression is the presence W +absent in the standard Kondo Hamiltonians. +In addition, +we perform a k-expansion in the square bracket of Eq. 4, +and keep only the zeroth and the linear order terms in k; +as was done in the expression for the hybridization matrix +˜H(fc,η)(k) (Eq. S195). +The zeroth order Kondo coupling +has strength γ2/Dνc=0,νf = 42.3meV, 49.3meV, 98.6meV +at ν = 0, −1, −2 respectively. +The RKKY interactions and the Hund’s rule— By integrat- +ing out the c-electrons in the Kondo Hamiltonian(Eq. 3), one + +3 +(a) +(b) +(c) +FIG. 2. Excitation spectrum at ν = 0, −1, −2. Blue, orange, red and green denote the fluctuations in the full-empty sector, half-empty sector, +full-half sector and half-half sector respectively. +induces an RKKY interaction between the f-moments [128– +130] . We restrict ourselves to deriving the RKKY interac- +tions [123] in the leading order in ˆHK, ˆHJ at integer fillings +ν = 0, −1, −2. In the chiral-flat limit (v′ +⋆ = 0, M = 0), +the RKKY interactions can be described by the following +U(4) × U(4) symmetric Hamiltonian +ˆHv′ +⋆=0,M=0 +RKKY += +� +R,R′ +µν,ξξ′ +[JRKKY +0 +(R′) + JRKKY +1 +(R′)δξ,ξ′] +: Σ(f,ξξ′) +µν +(R) :: Σ(f,ξ′ξ) +µν +(R + R′) : , +(6) +where both JRKKY +0 +(R′)(≤ 0) and JRKKY +1 +(R′)(≤ 0) can +be analytically obtained and are ferromagnetic (SM [123], +Sec.V). Using bond operators Aξ,ξ′ +R,R′ += � +i ψf,ξ,† +R,i ψf,ξ′ +R′,i, +we show that ˆHv′ +⋆=0,M=0 +RKKY +is a Positive Semidefinite Hamil- +tonian [4, 68], where the exact ground state can be ob- +tained [4, 68, 85, 123]. The grounds states are ferromagnetic +with the form of (SM [123], Sec. VI) +� +R +� ν+ +f +2 +� +n=1 +ψf,+,† +R,in +νf +4 +� +m=ν+ +f +3 +ψf,−,† +R,im +� +|0⟩ . +(7) +where ν+ +f denotes the filling of the f-electrons with index ξ = +1, and ν+ +f = 0 at ν = 0, ν+ +f = −1, 0 at ν = −1, and ν+ +f = −1 +at ν = −2. {in} are chosen arbitrarily and |0⟩ is the vacuum +with ψf,ξ +R,i|0⟩ = 0. We note that our ground states form a +subset of the ground states in Ref. [85], due to the additional +non-zero kinetic energy generated by f-c hybridizations in our +model. +We then consider the nonchiral-flat limit (v′ +⋆ ̸= 0, M = 0), +where the RKKY interactions are flat-U(4) symmetric. They +lift the ground-state degeneracy in Eq. 7. +We obtain the +ground states by treating v′ +⋆-induced RKKY interactions as +perturbations (SM [123], Sec. VI) , and the true ground state +is selected by the following RKKY interactions +� +R,R′ +µν,ξ +JRKKY +2 +(R′) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R + R′) : (8) +where JRKKY +2 +(R)(≤ 0) is analytically obtained and ferro- +magnetic (SM [123], Sec. V). JRKKY +2 +(R2 − R) tends to +align two f-moments : ˆΣ(f,++) +µν +(R) : and : ˆΣ(f,−−) +µν +(R2) : +and stabilize the ground states obeying the Hund’s rule (see +Fig. 1). The corresponding ground states are consistent with +Ref. [68, 85] (nonchiral-flat limit). +Moreover, we also derive the RKKY interactions in the +zero-hybridization limit (γ = 0, v′ +⋆ = 0) with non-zero +J(̸= 0) (SM [123], Sec. III), where the corresponding ground +states are consistent with Ref. [85] (chiral-nonflat limit). +Fluctuations of the f-moments— We check the stability of +the ferromagnetic ground state derived from RKKY Hamil- +tonian by studying small fluctuations. We restrict ourselves +to the flat-U(4) nonchiral-flat limit M += 0, v′ +⋆ ̸= 0 at +ν = 0, −1, −2. To describe the fluctuations we introduce for +each site a 8 × 8 traceless Hermitian matrix uiξ,jξ′(R), where +i, j ∈ {1, 2, 3, 4} are valley-spin flavors and ξ, ξ′ ∈ {+, −}. +The f-moments in Eq. 2 can then be written as (SM [123], +Sec. VIII). +ˆΣ(f,ξξ′) +µν +(R) = +� +ij +T µν +ij [eiu(R)Λe−iu(R)]iξ,jξ′ +(9) +where Λ is an 8 × 8 matrix defined as Λiξ,jξ′ = ⟨ψ0| : +ψf,ξ,† +R,i ψf,ξ′ +R,j : |ψ0⟩/2 and |ψ0⟩ is the ground state. A non- +zero uiξ,jξ′(R) will generate fluctuations by rotating the f- +moments from their ground-state expectation values. +The ferromagnetic order in the ground state opens a gap in +the single-particle spectrum of the c-electrons which allows us +to safely integrate them out [123] and to develop an effective +theory for small fluctuations by expanding the action to the +second order in uiξ,jξ′(R, τ), where τ is the imaginary time +(SM [123], Sec. VIII). The Lagrangian of the effective theory +is provided in SM [123], Sec. VIII, where we find the diagonal +components uiξ,iξ(R, τ) only contribute a total derivative and +we will focus on the off-diagonal components: uiξ,jξ′(R, τ) +with iξ ̸= jξ′. +We then introduce two sets Sfill and Semp to character- +ize the ground state, where Sfill and Semp denote the sets of +iξ indices that are filled with one and zero number of the f- +electrons at each site, respectively. A fluctuation generated by +uiξ,jξ′(R) (iξ ̸= jξ′) is described by the procedure of mov- +ing one f-electron from jξ′ flavor at site R to iξ flavor at + +4 +the same site (SM [123], Sec. VIII). This procedure can only +be valid when iξ ∈ Semp, jξ′ ∈ Sfill. Consequently, only +uiξ,jξ′(R, τ) with jξ′ ∈ Sfill, iξ ∈ Semp and also its com- +plex conjugate ujξ′,iξ(R, τ) that describes the reverse proce- +dure appear in our effective Lagrangian. We diagonalize the +Lagrangian and plot the excitation spectrum in Fig. 2. +We first analyze the spectrum at ν = 0, −2. +The La- +grangian density in the long-wavelength limit and in mo- +mentum (k) and frequency (ω) spaces has the form of L = +LGoldstone + Lgapped: +LGoldstone = +� +jξ∈Sfill +iξ∈Semp +u† +j,i(k, ω) +� +ω − k2 +2m0 +� +uj,i(k, ω), +Lgapped = +� +jξ∈Sfill +iξ∈Semp +U † +j,i(k, ω)[ω ˆI − ˆH(k)]Uj,i(k, ω), +H(k) = +� +� +� +� +k+k− +2m1 + ∆1 +V k+ +−V k− +V k− +k+k− +2m2 + ∆2 +k2 +− +2m3 +−V k+ +k2 ++ +2m3 +k+k− +2m2 + ∆2 +� +� +� +� (10) +where uj,i += +(u(j+,i+) + u(j−,i−))/ +√ +2, and U T +j,i += +((u(j+,i+) − u(j−,i−))/ +√ +2, u(j+,i−), u(j−,i+)) and k± += +kx ± iky, k = |k|. m0, m1, m2, V, ∆1, ∆2, V are analyti- +cally defined constants that depend on the parameters of the +original THF Hamiltonian (SM [123], Sec. VIII). The Gold- +stone modes with quadratic dispersion decouple from the rest +(gapped modes). Their stiffness is 1/m0 = 5.9meV · a2 +M (at +ν = 0), 13.2meV · a2 +M (at ν = −2), where aM is the moir´e +lattice constant. +The gaps at k = 0 are ∆1 and ∆2, with ∆2 correspond- +ing to two-fold degenerate modes. +This feature is repro- +duced in Fig. 2 (c). +The exceptional case when all three +modes are degenerate (∆1 = ∆2) is depicted in Fig. 2 +(a) and the condition for the degeneracy is α = [−0.37 + +� +0.14 + 0.23(v′⋆)2/(γλ)2]γ2/(JDνcνf ) = 1. Using the re- +alistic values of parameters, we find α = 1.07 ≈ 1. By +directly evaluating the Lagrangian, we also observe ”roton” +minima in the gapped modes (most obviously at ν = −2) at +|k| ∼ 0.3|bM1| with bM1 the moir´e reciprocal lattice vector. +The roton mode has small anisotropy with the minimum lo- +cated along the ΓM-MM line. +We next discuss the number of the Goldstone modes. Each +combination of (i, j) that satisfies i+, i− ∈ Semp, j+, j− ∈ +Sfill produce a Goldstone mode [123]. This leads to four +Goldstone modes at ν = 0 and three Goldstone modes at +ν = −2 [86]. Furthermore, all excitation modes depicted in +Fig. 2 are four-fold degenerate at ν = 0 and three-fold degen- +erate at ν = −2, due to the remaining U(2) × U(2) symme- +tries at ν = 0 and the remaining U(1) × U(3) symmetries at +ν = −2. +We now analyze the spectrum at ν = −1. Unlike ν = 0, −2 +where each valley-spin flavor is filled with either two or zero +f- electrons, at ν = −1, there is one valley-spin flavor (de- +noted by i = 1) filled with two f-electrons and one valley-spin +flavor (denoted by i = 2) filled with one f-electron and two +empty valley-spin flavors (denoted by i = 3, 4) as shown in +Fig. 1 (c). This allows us to classify uiξ,jξ′(R, τ) at ν = −1 +into four sectors: (1) full-empty sector with i = 3, 4, j = 1 +or i = 1, j = 3, 4; (2) half-empty sector with i = 2, j = 3, 4 +or i = 3, 4, j = 2 ; (3) full-half sector with i = 1, j = 2 +or j = 2, i = 1; (4) half-half sector with i = 2, j = 2. In +Fig. 2, we label the excitation in different sectors with differ- +ent colors. Due to the remaining U(1) × U(1) × U(2) sym- +metry of the ν = −1 ground state , each mode is two-fold de- +generate in the full-empty and half-empty sectors and is non- +degenerate in the full-half and half-half sectors. We find 2 de- +generate Goldstone modes with stiffness 6.7meV · a2 +M in the +full-empty sector, 2 degenerate Goldstone modes with stiff- +ness 1.3meV · a2 +M in the half-empty sector, and 1 Goldstone +mode with stiffness 1.8meV · a2 +M in the full-half sector [86]. +Several remarks are in order. Firstly, some of the gapped +modes at ν = 0, −1, −2 are relatively flat as shown in Fig. 2. +Secondly, at ν = −1, one of the flat modes (green curve +in Fig. 2 (b)) with eigenfunction u2−,2+(k) has a tiny gap +0.12meV at ΓM point and a very narrow bandwidth 1.5meV. +Thirdly, the dispersion along MM to KM is also relatively +flat which is related to the approximate C∞ symmetry of the +excitation modes. +Summary and discussions— We have constructed and stud- +ied a Kondo lattice model for MATBLG. Its distinct feature +is the Dirac character of the c-electron spectrum: at integer +fillings, there is no Fermi surface. Hence the Kondo screening +is irrelevant and the low energy physics is dominated by the +RKKY interactions, which is also responsible for the Hund’s +rule and ferromagnetism of the ground states. We have devel- +oped an effective theory describing small fluctuations of the +local moments and found their excitation spectrum. We have +also discussed the properties of the Goldstone and gapped +modes and their degeneracies. We believe that our work pro- +vides insight into the correlated ground states and the local- +moment fluctuations of the MATBLG. +We also comment on the connection with previous +works [86, 131]. Some features, including the soft modes and +accidental degeneracy of gapped modes at ΓM, also appear +in the projected Coulomb model [86], but the roton modes +do not. We note that the projected Coulomb model [86] has +ignored the effect of remote bands. Roton modes have also +been seen in Ref. [131], where the remote bands have been +included. However, our model gives a larger bandwidth of the +gapped modes than Ref. [131]. We point out that, we take +a different set of parameter values (including dielectric con- +stant, and gate distance). Besides, in our model, all Goldstone +modes have quadratic dispersions, in contrast to Ref. [131] +which found a linear one. The difference in the results has +its origin in the different symmetry properties. We consider +the flat limit of the model, and the quadratic dispersion comes +from the broken flat U(4) symmetry. Ref. [131] takes non-flat +bands, and the linear Goldstone modes come from the broken +U(1) valley symmetry. +We conclude the paper with a discussion of the relevance of +our results to the recent entropy experiments [117, 118]. Ex- +perimentally, the entropy in TBG near ν = −1 has been found +to be of the order of Boltzmann’s constant and is suppressed +by the applications of magnetic fields. This large entropy can + +5 +be explained by the presence of soft mode at ν = −1, which +will be suppressed by the magnetic field. Finally, we point out +that the fluctuations of the f-moments could potentially gen- +erate attractive interactions between c-electrons and drive the +system to the superconducting phase. Thus, our work also es- +tablishes a platform for understanding the superconductivity. +Note added— After finishing this work, we have learned +that related, but not identical, results had recently been ob- +tained by the S. Das Sarma’s [132], P. Coleman’s [133] and Z. +Song’s groups [134]. +Acknowledgements— We thank Z. Song for discussions. +We also thank P. Coleman and S. Das Sarma for discussions. +B. A. B.’s work was primarily supported by the DOE Grant +No. DE-SC0016239. H. H. was supported by the European +Research Council (ERC) under the European Union’s Horizon +2020 research and innovation program (Grant Agreement No. +101020833). Further support was provided by the Gordon and +Betty Moore Foundation through the EPiQS Initiative, Grant +GBMF11070 and the European Research Council (ERC) un- +der the European Union’s Horizon 2020 research and inno- +vation program (Grant Agreement No. 101020833) A. M. T. +was supported by the Office of Basic Energy Sciences, Ma- +terial Sciences and Engineering Division, U.S. Department of +Energy (DOE) under Contract No. DE-SC0012704. +[1] Yuan Cao, +Valla Fatemi, +Ahmet Demir, +Shiang Fang, +Spencer L. Tomarken, Jason Y. Luo, Javier D. Sanchez- +Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios +Kaxiras, Ray C. Ashoori, +and Pablo Jarillo-Herrero, “Cor- +related insulator behaviour at half-filling in magic-angle +graphene superlattices,” Nature 556, 80–84 (2018). +[2] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, +Takashi Taniguchi, Efthimios Kaxiras, +and Pablo Jarillo- +Herrero, “Unconventional superconductivity in magic-angle +graphene superlattices,” Nature 556, 43–50 (2018). +[3] Rafi Bistritzer and Allan H MacDonald, “Moir´e bands in +twisted double-layer graphene,” Proceedings of the National +Academy of Sciences 108, 12233–12237 (2011). +[4] B. Andrei Bernevig, Zhi-Da Song, Nicolas Regnault, +and +Biao Lian, “Twisted bilayer graphene. iii. interacting hamilto- +nian and exact symmetries,” Phys. Rev. B 103, 205413 (2021). +[5] Kasra Hejazi, Chunxiao Liu, Hassan Shapourian, Xiao Chen, +and Leon Balents, “Multiple topological transitions in twisted +bilayer graphene near the first magic angle,” Phys. Rev. B 99, +035111 (2019). +[6] Bikash Padhi, Apoorv Tiwari, Titus Neupert, +and Shin- +sei Ryu, “Transport across twist angle domains in moir\’e +graphene,” Phys. Rev. Research 2, 033458 (2020). +[7] J. F. Dodaro, S. A. Kivelson, Y. Schattner, X. Q. Sun, and +C. Wang, “Phases of a phenomenological model of twisted +bilayer graphene,” Phys. Rev. B 98, 075154 (2018). +[8] Tomohiro Soejima, Daniel E. Parker, Nick Bultinck, Johannes +Hauschild, and Michael P. Zaletel, “Efficient simulation of +moir\’e materials using the density matrix renormalization +group,” Phys. Rev. B 102, 205111 (2020). +[9] Paul Eugenio and Ceren Dag, “DMRG study of strongly in- +teracting $\mathbb{Z} 2$ flatbands: A toy model inspired by +twisted bilayer graphene,” SciPost Physics Core 3, 015 (2020). +[10] Yixuan Huang, Pavan Hosur, +and Hridis K. Pal, “Quasi- +flat-band physics in a two-leg ladder model and its relation +to magic-angle twisted bilayer graphene,” Phys. Rev. B 102, +155429 (2020). +[11] Xu Zhang, Gaopei Pan, Yi Zhang, Jian Kang, and Zi Yang +Meng, “Momentum Space Quantum Monte Carlo on Twisted +Bilayer Graphene,” Chinese Physics Letters 38, 077305 +(2021), arXiv:2105.07010 [cond-mat.str-el]. +[12] Jian Kang, B. Andrei Bernevig, +and Oskar Vafek, “Cas- +cades between light and heavy fermions in the normal state +of magic angle twisted bilayer graphene,” arXiv:2104.01145 +[cond-mat] (2021), arXiv:2104.01145 [cond-mat]. +[13] Johannes S. Hofmann, Eslam Khalaf, Ashvin Vishwanath, +Erez Berg, +and Jong Yeon Lee, “Fermionic Monte Carlo +Study of a Realistic Model of Twisted Bilayer Graphene,” +Physical Review X 12, 011061 (2022), arXiv:2105.12112 +[cond-mat.str-el]. +[14] Ya-Hui Zhang, Dan Mao, Yuan Cao, Pablo Jarillo-Herrero, +and T. Senthil, “Nearly flat Chern bands in moir\’e superlat- +tices,” Phys. Rev. B 99, 075127 (2019). +[15] Xiao-Chuan Wu, Chao-Ming Jian, and Cenke Xu, “Coupled- +wire description of the correlated physics in twisted bilayer +graphene,” Phys. Rev. B 99, 161405 (2019). +[16] Rafael M. Fernandes and J¨orn W. F. Venderbos, “Nematicity +with a twist: Rotational symmetry breaking in a moir´e super- +lattice,” Science Advances 6, eaba8834 (2020). +[17] Justin H. Wilson, Yixing Fu, S. Das Sarma, and J. H. Pixley, +“Disorder in twisted bilayer graphene,” Phys. Rev. Research +2, 023325 (2020). +[18] Jie Wang, Yunqin Zheng, Andrew J. Millis, +and Jennifer +Cano, “Chiral approximation to twisted bilayer graphene: Ex- +act intravalley inversion symmetry, nodal structure, and im- +plications for higher magic angles,” Phys. Rev. Research 3, +023155 (2021). +[19] B. Andrei Bernevig, Zhi-Da Song, Nicolas Regnault, +and +Biao Lian, “Twisted bilayer graphene. I. Matrix elements, ap- +proximations, perturbation theory, and a k ˙p two-band model,” +Physical Review B 103, 205411 (2021), publisher: American +Physical Society. +[20] Jiabin Yu, Ming Xie, B. Andrei Bernevig, and Sankar Das +Sarma, to be published. +[21] Yuan Cao, Debanjan Chowdhury, Daniel Rodan-Legrain, +Oriol Rubies-Bigorda, Kenji Watanabe, Takashi Taniguchi, +T. Senthil, +and Pablo Jarillo-Herrero, “Strange metal in +magic-angle graphene with near planckian dissipation,” Phys. +Rev. Lett. 124, 076801 (2020). +[22] Yuan Cao, Daniel Rodan-Legrain, Jeong Min Park, Noah F. Q. +Yuan, Kenji Watanabe, Takashi Taniguchi, Rafael M. Fernan- +des, Liang Fu, +and Pablo Jarillo-Herrero, “Nematicity and +competing orders in superconducting magic-angle graphene,” +Science 372, 264–271 (2021). +[23] Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mo- +hammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji Watan- +abe, Takashi Taniguchi, Guangyu Zhang, Adrian Bachtold, +Allan H. MacDonald, and Dmitri K. Efetov, “Superconduc- +tors, orbital magnets and correlated states in magic-angle bi- +layer graphene,” Nature 574, 653–657 (2019). +[24] Matthew Yankowitz, Shaowen Chen, Hryhoriy Polshyn, Yux- +uan Zhang, K. Watanabe, T. Taniguchi, David Graf, Andrea F. + +6 +Young, +and Cory R. Dean, “Tuning superconductivity in +twisted bilayer graphene,” Science 363, 1059–1064 (2019). +[25] Petr Stepanov, Ipsita Das, Xiaobo Lu, Ali Fahimniya, Kenji +Watanabe, Takashi Taniguchi, Frank H. L. Koppens, Johannes +Lischner, Leonid Levitov, +and Dmitri K. Efetov, “Unty- +ing the insulating and superconducting orders in magic-angle +graphene,” Nature 583, 375–378 (2020). +[26] Ming Xie and A. H. MacDonald, “Weak-Field Hall Resistivity +and Spin-Valley Flavor Symmetry Breaking in Magic-Angle +Twisted Bilayer Graphene,” Phys. Rev. Lett. 127, 196401 +(2021). +[27] Yuhang Jiang, +Xinyuan Lai, +Kenji Watanabe, +Takashi +Taniguchi, Kristjan Haule, Jinhai Mao, and Eva Y. Andrei, +“Charge order and broken rotational symmetry in magic-angle +twisted bilayer graphene,” Nature 573, 91–95 (2019). +[28] Dillon Wong, Kevin P. Nuckolls, Myungchul Oh, Biao +Lian, Yonglong Xie, Sangjun Jeon, Kenji Watanabe, Takashi +Taniguchi, B. Andrei Bernevig, +and Ali Yazdani, “Cas- +cade of electronic transitions in magic-angle twisted bilayer +graphene,” Nature 582, 198–202 (2020). +[29] U. +Zondiner, +A. +Rozen, +D. +Rodan-Legrain, +Y. +Cao, +R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Op- +pen, Ady Stern, E. Berg, P. Jarillo-Herrero, and S. Ilani, “Cas- +cade of phase transitions and Dirac revivals in magic-angle +graphene,” Nature 582, 203–208 (2020). +[30] Ipsita Das, +Xiaobo Lu, +Jonah Herzog-Arbeitman, +Zhi- +Da Song, Kenji Watanabe, Takashi Taniguchi, B. Andrei +Bernevig, +and Dmitri K. Efetov, “Symmetry-broken Chern +insulators and Rashba-like Landau-level crossings in magic- +angle bilayer graphene,” Nat. Phys. 17, 710–714 (2021). +[31] Jeong Min Park, +Yuan Cao, +Kenji Watanabe, +Takashi +Taniguchi, and Pablo Jarillo-Herrero, “Flavour Hund’s cou- +pling, Chern gaps and charge diffusivity in moir´e graphene,” +Nature 592, 43–48 (2021). +[32] Xiaobo Lu, Biao Lian, Gaurav Chaudhary, Benjamin A. Piot, +Giulio Romagnoli, Kenji Watanabe, Takashi Taniguchi, Mar- +tino Poggio, Allan H. MacDonald, B. Andrei Bernevig, and +Dmitri K. Efetov, “Multiple flat bands and topological Hofs- +tadter butterfly in twisted bilayer graphene close to the second +magic angle,” PNAS 118 (2021), 10.1073/pnas.2100006118. +[33] Maximilian +Otteneder, +Stefan +Hubmann, +Xiaobo +Lu, +Dmitry A. Kozlov, Leonid E. Golub, Kenji Watanabe, Takashi +Taniguchi, Dmitri K. Efetov, +and Sergey D. Ganichev, +“Terahertz photogalvanics in twisted bilayer graphene close to +the second magic angle,” Nano Letters 20, 7152–7158 (2020). +[34] Simone Lisi, Xiaobo Lu, Tjerk Benschop, Tobias A. de Jong, +Petr Stepanov, Jose R. Duran, Florian Margot, Ir`ene Cuc- +chi, Edoardo Cappelli, Andrew Hunter, Anna Tamai, Vik- +tor Kandyba, Alessio Giampietri, Alexei Barinov, Johannes +Jobst, Vincent Stalman, Maarten Leeuwenhoek, Kenji Watan- +abe, Takashi Taniguchi, Louk Rademaker, Sense Jan van der +Molen, Milan P. Allan, Dmitri K. Efetov, +and Felix +Baumberger, “Observation of flat bands in twisted bilayer +graphene,” Nature Physics 17, 189–193 (2021). +[35] Tjerk Benschop, Tobias A. de Jong, Petr Stepanov, Xiaobo Lu, +Vincent Stalman, Sense Jan van der Molen, Dmitri K. Efetov, +and Milan P. Allan, “Measuring local moir´e lattice heterogene- +ity of twisted bilayer graphene,” Phys. Rev. Res. 3, 013153 +(2021). +[36] Paul Seifert, +Xiaobo Lu, +Petr Stepanov, +Jos´e Ram´on +Dur´an Retamal, John N. Moore, Kin-Chung Fong, Alessan- +dro Principi, +and Dmitri K. Efetov, “Magic-angle bi- +layer graphene nanocalorimeters: Toward broadband, energy- +resolving single photon detection,” Nano Letters 20, 3459– +3464 (2020). +[37] Niels C. H. Hesp, Iacopo Torre, Daniel Rodan-Legrain, Pietro +Novelli, Yuan Cao, Stephen Carr, Shiang Fang, Petr Stepanov, +David Barcons-Ruiz, Hanan Herzig Sheinfux, Kenji Watan- +abe, Takashi Taniguchi, Dmitri K. Efetov, Efthimios Kaxi- +ras, Pablo Jarillo-Herrero, Marco Polini, +and Frank H. L. +Koppens, “Observation of interband collective excitations in +twisted bilayer graphene,” Nature Physics 17, 1162–1168 +(2021). +[38] Jaime Diez-Merida, Andr´es D´ıez-Carl´on, SY Yang, Y-M Xie, +X-J Gao, Kenji Watanabe, Takashi Taniguchi, Xiaobo Lu, +Kam Tuen Law, and Dmitri K Efetov, “Magnetic josephson +junctions and superconducting diodes in magic angle twisted +bilayer graphene,” arXiv preprint arXiv:2110.01067 (2021). +[39] Petr Stepanov, Ming Xie, Takashi Taniguchi, Kenji Watanabe, +Xiaobo Lu, Allan H. MacDonald, B. Andrei Bernevig, and +Dmitri K. Efetov, “Competing zero-field chern insulators in +superconducting twisted bilayer graphene,” Phys. Rev. Lett. +127, 197701 (2021). +[40] S. Hubmann, P. Soul, G. Di Battista, M. Hild, K. Watanabe, +T. Taniguchi, D. K. Efetov, +and S. D. Ganichev, “Nonlin- +ear intensity dependence of photogalvanics and photoconduc- +tance induced by terahertz laser radiation in twisted bilayer +graphene close to magic angle,” Phys. Rev. Mater. 6, 024003 +(2022). +[41] Alexandre Jaoui, Ipsita Das, Giorgio Di Battista, Jaime D´ıez- +M´erida, Xiaobo Lu, Kenji Watanabe, Takashi Taniguchi, +Hiroaki Ishizuka, Leonid Levitov, +and Dmitri K. Efetov, +“Quantum critical behaviour in magic-angle twisted bilayer +graphene,” Nature Physics 18, 633–638 (2022). +[42] Ipsita Das, Cheng Shen, Alexandre Jaoui, Jonah Herzog- +Arbeitman, Aaron Chew, Chang-Woo Cho, Kenji Watanabe, +Takashi Taniguchi, Benjamin A. Piot, B. Andrei Bernevig, +and Dmitri K. Efetov, “Observation of reentrant correlated in- +sulators and interaction-driven fermi-surface reconstructions +at one magnetic flux quantum per moir´e unit cell in magic- +angle twisted bilayer graphene,” Phys. Rev. Lett. 128, 217701 +(2022). +[43] Sameer Grover, Matan Bocarsly, Aviram Uri, Petr Stepanov, +Giorgio Di Battista, Indranil Roy, Jiewen Xiao, Alexan- +der Y. Meltzer, Yuri Myasoedov, Keshav Pareek, Kenji Watan- +abe, Takashi Taniguchi, Binghai Yan, Ady Stern, Erez Berg, +Dmitri K. Efetov, +and Eli Zeldov, “Chern mosaic and +berry-curvature magnetism in magic-angle graphene,” Nature +Physics 18, 885–892 (2022). +[44] Giorgio Di Battista, Paul Seifert, Kenji Watanabe, Takashi +Taniguchi, Kin Chung Fong, Alessandro Principi, +and +Dmitri K. Efetov, “Revealing the thermal properties of super- +conducting magic-angle twisted bilayer graphene,” Nano Let- +ters 22, 6465–6470 (2022). +[45] Kevin P. Nuckolls, Myungchul Oh, Dillon Wong, Biao Lian, +Kenji Watanabe, Takashi Taniguchi, B. Andrei Bernevig, and +Ali Yazdani, “Strongly correlated Chern insulators in magic- +angle twisted bilayer graphene,” Nature 588, 610–615 (2020). +[46] Youngjoon Choi, Hyunjin Kim, Yang Peng, Alex Thom- +son, Cyprian Lewandowski, Robert Polski, Yiran Zhang, +Harpreet Singh Arora, Kenji Watanabe, Takashi Taniguchi, Ja- +son Alicea, and Stevan Nadj-Perge, “Correlation-driven topo- +logical phases in magic-angle twisted bilayer graphene,” Na- +ture 589, 536–541 (2021). +[47] Yu Saito, Jingyuan Ge, Louk Rademaker, Kenji Watanabe, +Takashi Taniguchi, Dmitry A. Abanin, and Andrea F. Young, +“Hofstadter subband ferromagnetism and symmetry-broken +Chern insulators in twisted bilayer graphene,” Nat. Phys. , 1–4 + +7 +(2021). +[48] Shuang Wu, Zhenyuan Zhang, K. Watanabe, T. Taniguchi, +and Eva Y. Andrei, “Chern insulators, van Hove singulari- +ties and topological flat bands in magic-angle twisted bilayer +graphene,” Nat. Mater. 20, 488–494 (2021). +[49] Liujun Zou, Hoi Chun Po, Ashvin Vishwanath, and T. Senthil, +“Band structure of twisted bilayer graphene: Emergent sym- +metries, commensurate approximants, and Wannier obstruc- +tions,” Phys. Rev. B 98, 085435 (2018). +[50] Dmitry K. Efimkin and Allan H. MacDonald, “Helical net- +work model for twisted bilayer graphene,” Phys. Rev. B 98, +035404 (2018). +[51] Jian Kang and Oskar Vafek, “Symmetry, Maximally Localized +Wannier States, and a Low-Energy Model for Twisted Bilayer +Graphene Narrow Bands,” Phys. Rev. X 8, 031088 (2018). +[52] Zhida Song, Zhijun Wang, Wujun Shi, Gang Li, Chen Fang, +and B. Andrei Bernevig, “All magic angles in twisted bi- +layer graphene are topological,” Phys. Rev. Lett. 123, 036401 +(2019). +[53] Hoi Chun Po, Liujun Zou, T. Senthil, and Ashvin Vishwanath, +“Faithful tight-binding models and fragile topology of magic- +angle bilayer graphene,” Phys. Rev. B 99, 195455 (2019). +[54] Bikash Padhi, Chandan Setty, and Philip W. Phillips, “Doped +Twisted Bilayer Graphene near Magic Angles: Proximity to +Wigner Crystallization, Not Mott Insulation,” Nano Letters 18, +6175–6180 (2018), arXiv:1804.01101 [cond-mat.str-el]. +[55] Kasra Hejazi, Xiao Chen, and Leon Balents, “Hybrid Wannier +Chern bands in magic angle twisted bilayer graphene and the +quantized anomalous Hall effect,” Physical Review Research +3, 013242 (2021), publisher: American Physical Society. +[56] Jian Kang and Oskar Vafek, “Non-Abelian Dirac node braid- +ing and near-degeneracy of correlated phases at odd integer +filling in magic-angle twisted bilayer graphene,” Phys. Rev. B +102, 035161 (2020). +[57] Zhi-Da Song, Biao Lian, Nicolas Regnault, +and B. An- +drei Bernevig, “Twisted bilayer graphene. II. Stable sym- +metry anomaly,” Physical Review B 103, 205412 (2021), +arXiv:2009.11872 [cond-mat]. +[58] Biao Lian, Fang Xie, and B. Andrei Bernevig, “Landau level +of fragile topology,” Phys. Rev. B 102, 041402 (2020). +[59] Kasra Hejazi, Chunxiao Liu, and Leon Balents, “Landau lev- +els in twisted bilayer graphene and semiclassical orbits,” Phys. +Rev. B 100, 035115 (2019). +[60] Leon Balents, Cory R. Dean, Dmitri K. Efetov, and Andrea F. +Young, “Superconductivity and strong correlations in moir´e +flat bands,” Nature Physics 16, 725–733 (2020). +[61] Eva Y. Andrei, Dmitri K. Efetov, Pablo Jarillo-Herrero, Al- +lan H. MacDonald, Kin Fai Mak, T. Senthil, Emanuel Tutuc, +Ali Yazdani, and Andrea F. Young, “The marvels of moir´e +materials,” Nature Reviews Materials 6, 201–206 (2021). +[62] Hryhoriy Polshyn, Matthew Yankowitz, Shaowen Chen, Yux- +uan Zhang, K. Watanabe, T. Taniguchi, Cory R. Dean, and +Andrea F. Young, “Large linear-in-temperature resistivity in +twisted bilayer graphene,” Nat. Phys. 15, 1011–1016 (2019). +[63] Xiaoxue Liu, Zhi Wang, K. Watanabe, T. Taniguchi, Oskar +Vafek, and J. I. A. Li, “Tuning electron correlation in magic- +angle twisted bilayer graphene using Coulomb screening,” +Science 371, 1261–1265 (2021). +[64] Yonglong Xie, Biao Lian, Berthold J¨ack, Xiaomeng Liu, +Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, B. An- +drei Bernevig, +and Ali Yazdani, “Spectroscopic signatures +of many-body correlations in magic-angle twisted bilayer +graphene,” Nature 572, 101–105 (2019). +[65] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex Thom- +son, Harpreet Arora, Robert Polski, Yiran Zhang, Hechen Ren, +Jason Alicea, Gil Refael, Felix von Oppen, Kenji Watanabe, +Takashi Taniguchi, and Stevan Nadj-Perge, “Electronic corre- +lations in twisted bilayer graphene near the magic angle,” Nat. +Phys. 15, 1174–1180 (2019). +[66] Alexander Kerelsky, Leo J. McGilly, Dante M. Kennes, Lede +Xian, Matthew Yankowitz, Shaowen Chen, K. Watanabe, +T. Taniguchi, James Hone, Cory Dean, Angel Rubio, +and +Abhay N. Pasupathy, “Maximized electron interactions at the +magic angle in twisted bilayer graphene,” Nature 572, 95–100 +(2019). +[67] Hoi Chun Po, Liujun Zou, Ashvin Vishwanath, and T. Senthil, +“Origin of mott insulating behavior and superconductivity in +twisted bilayer graphene,” Phys. Rev. X 8, 031089 (2018). +[68] Jian Kang and Oskar Vafek, “Strong Coupling Phases of Par- +tially Filled Twisted Bilayer Graphene Narrow Bands,” Phys. +Rev. Lett. 122, 246401 (2019). +[69] Fang Xie, Aditya Cowsik, Zhi-Da Song, Biao Lian, B. Andrei +Bernevig, and Nicolas Regnault, “Twisted bilayer graphene. +VI. An exact diagonalization study at nonzero integer filling,” +Physical Review B 103, 205416 (2021), publisher: American +Physical Society. +[70] J¨orn W. F. Venderbos and Rafael M. Fernandes, “Correlations +and electronic order in a two-orbital honeycomb lattice model +for twisted bilayer graphene,” Phys. Rev. B 98, 245103 (2018). +[71] Yi-Zhuang You and Ashvin Vishwanath, “Superconductivity +from valley fluctuations and approximate SO(4) symmetry in a +weak coupling theory of twisted bilayer graphene,” npj Quan- +tum Mater. 4, 1–12 (2019). +[72] Nick Bultinck, Shubhayu Chatterjee, and Michael P. Zaletel, +“Mechanism for Anomalous Hall Ferromagnetism in Twisted +Bilayer Graphene,” Phys. Rev. Lett. 124, 166601 (2020). +[73] Alex Thomson, Shubhayu Chatterjee, Subir Sachdev, +and +Mathias S. Scheurer, “Triangular antiferromagnetism on the +honeycomb lattice of twisted bilayer graphene,” Phys. Rev. B +98, 075109 (2018). +[74] Yuan Da Liao, Zi Yang Meng, and Xiao Yan Xu, “Valence +Bond Orders at Charge Neutrality in a Possible Two-Orbital +Extended Hubbard Model for Twisted Bilayer Graphene,” +Phys. Rev. Lett. 123, 157601 (2019). +[75] Yuan Da Liao, Jian Kang, Clara N. Breiø, Xiao Yan Xu, Han- +Qing Wu, Brian M. Andersen, Rafael M. Fernandes, +and +Zi Yang Meng, “Correlation-Induced Insulating Topological +Phases at Charge Neutrality in Twisted Bilayer Graphene,” +Phys. Rev. X 11, 011014 (2021). +[76] Noah F. Q. Yuan and Liang Fu, “Model for the metal-insulator +transition in graphene superlattices and beyond,” Phys. Rev. B +98, 045103 (2018). +[77] Ming Xie and A. H. MacDonald, “Nature of the Correlated +Insulator States in Twisted Bilayer Graphene,” Phys. Rev. Lett. +124, 097601 (2020). +[78] Masayuki Ochi, Mikito Koshino, and Kazuhiko Kuroki, “Pos- +sible correlated insulating states in magic-angle twisted bi- +layer graphene under strongly competing interactions,” Phys. +Rev. B 98, 081102 (2018). +[79] Jianpeng Liu, Junwei Liu, and Xi Dai, “Pseudo Landau level +representation of twisted bilayer graphene: Band topology and +implications on the correlated insulating phase,” Phys. Rev. B +99, 155415 (2019). +[80] Kangjun Seo, Valeri N. Kotov, and Bruno Uchoa, “Ferromag- +netic Mott state in Twisted Graphene Bilayers at the Magic +Angle,” Phys. Rev. Lett. 122, 246402 (2019). +[81] Maine +Christos, +Subir +Sachdev, +and +Mathias +S. +Scheurer, “Superconductivity, +correlated insulators, +and + +8 +Wess–Zumino–Witten terms in twisted bilayer graphene,” +PNAS 117, 29543–29554 (2020). +[82] Jianpeng Liu and Xi Dai, “Theories for the correlated insulat- +ing states and quantum anomalous Hall effect phenomena in +twisted bilayer graphene,” Phys. Rev. B 103, 035427 (2021). +[83] Tommaso Cea and Francisco Guinea, “Band structure and in- +sulating states driven by Coulomb interaction in twisted bi- +layer graphene,” Phys. Rev. B 102, 045107 (2020). +[84] Yi Zhang, Kun Jiang, Ziqiang Wang, +and Fuchun Zhang, +“Correlated insulating phases of twisted bilayer graphene at +commensurate filling fractions: A Hartree-Fock study,” Phys. +Rev. B 102, 035136 (2020). +[85] Biao Lian, Zhi-Da Song, Nicolas Regnault, Dmitri K. Efe- +tov, Ali Yazdani, and B. Andrei Bernevig, “Twisted bilayer +graphene. iv. exact insulator ground states and phase diagram,” +Phys. Rev. B 103, 205414 (2021). +[86] B. Andrei Bernevig, Biao Lian, Aditya Cowsik, Fang Xie, +Nicolas Regnault, +and Zhi-Da Song, “Twisted bilayer +graphene. v. exact analytic many-body excitations in coulomb +hamiltonians: Charge gap, goldstone modes, and absence of +cooper pairing,” Phys. Rev. B 103, 205415 (2021). +[87] Huaiming Guo, +Xingchuan Zhu, +Shiping Feng, +and +Richard T. Scalettar, “Pairing symmetry of interacting +fermions on a twisted bilayer graphene superlattice,” Phys. +Rev. B 97, 235453 (2018). +[88] Jonah Herzog-Arbeitman, Aaron Chew, Dmitri K. Efetov, +and B. Andrei Bernevig, “Reentrant correlated insulators in +twisted bilayer graphene at 25 t (2π flux),” Phys. Rev. Lett. +129, 076401 (2022). +[89] Jed H. Pixley and Eva Y. Andrei, “Ferromagnetism in magic- +angle graphene,” Science 365, 543–543 (2019). +[90] Fengcheng Wu and Sankar Das Sarma, “Collective Excitations +of Quantum Anomalous Hall Ferromagnets in Twisted Bilayer +Graphene,” Phys. Rev. Lett. 124, 046403 (2020). +[91] C´ecile Repellin, +Zhihuan Dong, +Ya-Hui Zhang, +and +T. Senthil, “Ferromagnetism in Narrow Bands of Moir\’e Su- +perlattices,” Phys. Rev. Lett. 124, 187601 (2020). +[92] Jianpeng Liu, Zhen Ma, Jinhua Gao, and Xi Dai, “Quantum +Valley Hall Effect, Orbital Magnetism, and Anomalous Hall +Effect in Twisted Multilayer Graphene Systems,” Phys. Rev. +X 9, 031021 (2019). +[93] Biao Lian, Zhijun Wang, and B. Andrei Bernevig, “Twisted +Bilayer Graphene: A Phonon-Driven Superconductor,” Phys. +Rev. Lett. 122, 257002 (2019). +[94] Fengcheng Wu, A. H. MacDonald, and Ivar Martin, “The- +ory of Phonon-Mediated Superconductivity in Twisted Bilayer +Graphene,” Phys. Rev. Lett. 121, 257001 (2018). +[95] Cenke Xu and Leon Balents, “Topological Superconductiv- +ity in Twisted Multilayer Graphene,” Phys. Rev. Lett. 121, +087001 (2018). +[96] Francisco Guinea and Niels R. Walet, “Electrostatic effects, +band distortions, and superconductivity in twisted graphene +bilayers,” PNAS 115, 13174–13179 (2018). +[97] Hiroki Isobe, Noah F. Q. Yuan, and Liang Fu, “Unconven- +tional Superconductivity and Density Waves in Twisted Bi- +layer Graphene,” Phys. Rev. X 8, 041041 (2018). +[98] Cheng-Cheng Liu, Li-Da Zhang, Wei-Qiang Chen, and Fan +Yang, “Chiral Spin Density Wave and $d+id$ Superconduc- +tivity in the Magic-Angle-Twisted Bilayer Graphene,” Phys. +Rev. Lett. 121, 217001 (2018). +[99] J. Gonz´alez and T. Stauber, “Kohn-Luttinger Superconductiv- +ity in Twisted Bilayer Graphene,” Phys. Rev. Lett. 122, 026801 +(2019). +[100] Eslam +Khalaf, +Shubhayu +Chatterjee, +Nick +Bultinck, +Michael P. Zaletel, +and Ashvin Vishwanath, “Charged +skyrmions and topological origin of superconductivity in +magic-angle graphene,” Science Advances 7, eabf5299 +(2021). +[101] Cyprian Lewandowski, Debanjan Chowdhury, and Jonathan +Ruhman, “Pairing in magic-angle twisted bilayer graphene: +Role of phonon and plasmon umklapp,” Phys. Rev. B 103, +235401 (2021). +[102] Shang Liu, Eslam Khalaf, Jong Yeon Lee, +and Ashvin +Vishwanath, “Nematic topological semimetal and insulator in +magic-angle bilayer graphene at charge neutrality,” Phys. Rev. +Research 3, 013033 (2021). +[103] Dante M. Kennes, Johannes Lischner, +and Christoph Kar- +rasch, “Strong correlations and $d+\mathit{id}$ supercon- +ductivity in twisted bilayer graphene,” Phys. Rev. B 98, +241407 (2018). +[104] Tongyun Huang, Lufeng Zhang, and Tianxing Ma, “Antifer- +romagnetically ordered Mott insulator and d+id superconduc- +tivity in twisted bilayer graphene: A quantum Monte Carlo +study,” Science Bulletin 64, 310–314 (2019). +[105] A. Julku, T. J. Peltonen, L. Liang, T. T. Heikkil¨a, +and +P. T¨orm¨a, “Superfluid weight and Berezinskii-Kosterlitz- +Thouless transition temperature of twisted bilayer graphene,” +Phys. Rev. B 101, 060505 (2020). +[106] Fang Xie, Zhida Song, Biao Lian, and B. Andrei Bernevig, +“Topology-Bounded Superfluid Weight in Twisted Bilayer +Graphene,” Phys. Rev. Lett. 124, 167002 (2020). +[107] E. J. K¨onig, Piers Coleman, and A. M. Tsvelik, “Spin mag- +netometry as a probe of stripe superconductivity in twisted bi- +layer graphene,” Phys. Rev. B 102, 104514 (2020). +[108] Dmitry V. Chichinadze, Laura Classen, +and Andrey V. +Chubukov, “Nematic superconductivity in twisted bilayer +graphene,” Phys. Rev. B 101, 224513 (2020). +[109] Ahmed Abouelkomsan, Zhao Liu, +and Emil J. Bergholtz, +“Particle-Hole Duality, Emergent Fermi Liquids, and Frac- +tional Chern Insulators in Moir\’e Flatbands,” Phys. Rev. Lett. +124, 106803 (2020). +[110] Yves H. Kwan, Glenn Wagner, Tomohiro Soejima, Michael P. +Zaletel, Steven H. Simon, Siddharth A. Parameswaran, and +Nick Bultinck, “Kekul\’e spiral order at all nonzero integer +fillings in twisted bilayer graphene,” arXiv:2105.05857 [cond- +mat] (2021), arXiv:2105.05857 [cond-mat]. +[111] C´ecile Repellin and T. Senthil, “Chern bands of twisted bi- +layer graphene: Fractional Chern insulators and spin phase +transition,” Phys. Rev. Research 2, 023238 (2020). +[112] Patrick J. Ledwith, Grigory Tarnopolsky, Eslam Khalaf, and +Ashvin Vishwanath, “Fractional Chern insulator states in +twisted bilayer graphene: An analytical approach,” Phys. Rev. +Research 2, 023237 (2020). +[113] Leon Balents, Cory R. Dean, Dmitri K. Efetov, and Andrea F. +Young, “Superconductivity and strong correlations in moir´e +flat bands,” Nat. Phys. 16, 725–733 (2020). +[114] Peter Cha, Aavishkar A. Patel, and Eun-Ah Kim, “Strange +metals from melting correlated insulators in twisted bilayer +graphene,” Phys. Rev. Lett. 127, 266601 (2021). +[115] Fang Xie, Jian Kang, B Andrei Bernevig, Oskar Vafek, and +Nicolas Regnault, “Phase diagram of twisted bilayer graphene +at filling factor ν = −3,” arXiv preprint arXiv:2209.14322 +(2022). +[116] Yarden Sheffer and Ady Stern, “Chiral magic-angle twisted +bilayer graphene in a magnetic field: Landau level correspon- +dence, exact wave functions, and fractional chern insulators,” +Phys. Rev. B 104, L121405 (2021). +[117] Yu Saito, Fangyuan Yang, Jingyuan Ge, Xiaoxue Liu, Takashi + +9 +Taniguchi, Kenji Watanabe, J. I. A. Li, Erez Berg, and An- +drea F. Young, “Isospin pomeranchuk effect in twisted bilayer +graphene,” Nature 592, 220–224 (2021). +[118] Asaf Rozen, Jeong Min Park, Uri Zondiner, Yuan Cao, Daniel +Rodan-Legrain, Takashi Taniguchi, Kenji Watanabe, Yuval +Oreg, Ady Stern, Erez Berg, Pablo Jarillo-Herrero, and Shahal +Ilani, “Entropic evidence for a Pomeranchuk effect in magic- +angle graphene,” Nature 592, 214–219 (2021). +[119] Oskar +Vafek +and +Jian +Kang, +“Lattice +model +for +the +coulomb interacting chiral limit of magic-angle twisted bilayer +graphene: Symmetries, obstructions, and excitations,” Phys. +Rev. B 104, 075143 (2021). +[120] Mikito Koshino, Noah F. Q. Yuan, Takashi Koretsune, +Masayuki Ochi, Kazuhiko Kuroki, +and Liang Fu, “Maxi- +mally Localized Wannier Orbitals and the Extended Hubbard +Model for Twisted Bilayer Graphene,” Phys. Rev. X 8, 031087 +(2018). +[121] Xiao Yan Xu, K. T. Law, and Patrick A. Lee, “Kekul\’e va- +lence bond order in an extended Hubbard model on the hon- +eycomb lattice with possible applications to twisted bilayer +graphene,” Phys. Rev. B 98, 121406 (2018). +[122] Zhi-Da Song and B. Andrei Bernevig, “Magic-angle twisted +bilayer graphene as a topological heavy fermion problem,” +Phys. Rev. Lett. 129, 047601 (2022). +[123] Supplementary Materials. +[124] Dumitru C˘alug˘aru, Maksim Borovkov, Zhida Song, Liam Lau, +Piers Coleman, and B. Andrei Bernevig, “TBG as Topological +Heavy Fermion: II. Analytical approximations of the model +parameters,” In preparation. +[125] Nick Bultinck, Eslam Khalaf, Shang Liu, Shubhayu Chatter- +jee, Ashvin Vishwanath, +and Michael P. Zaletel, “Ground +State and Hidden Symmetry of Magic-Angle Graphene at +Even Integer Filling,” Physical Review X 10, 031034 (2020), +publisher: American Physical Society. +[126] Oskar Vafek and Jian Kang, “Renormalization Group Study of +Hidden Symmetry in Twisted Bilayer Graphene with Coulomb +Interactions,” Phys. Rev. Lett. 125, 257602 (2020). +[127] J. R. Schrieffer and P. A. Wolff, “Relation between the an- +derson and kondo hamiltonians,” Phys. Rev. 149, 491–492 +(1966). +[128] M. A. Ruderman and C. Kittel, “Indirect exchange coupling +of nuclear magnetic moments by conduction electrons,” Phys. +Rev. 96, 99–102 (1954). +[129] Tadao +Kasuya, +“A +Theory +of +Metallic +Ferro- +and +Antiferromagnetism +on +Zener’s +Model,” +Progress +of +Theoretical +Physics +16, +45–57 +(1956), +https://academic.oup.com/ptp/article- +pdf/16/1/45/5266722/16-1-45.pdf. +[130] Kei Yosida, “Magnetic properties of cu-mn alloys,” Phys. Rev. +106, 893–898 (1957). +[131] Eslam Khalaf, Nick Bultinck, Ashvin Vishwanath, +and +Michael P. Zaletel, “Soft modes in magic angle twisted +bilayer graphene,” arXiv:2009.14827 [cond-mat] +(2020), +arXiv:2009.14827 [cond-mat]. +[132] Yang-Zhi Chou and Sankar Das Sarma, “Kondo lattice model +in magic-angle twisted bilayer graphene,” arXiv preprint +arXiv:2211.15682 (2022). +[133] Liam L.H. Lau and Piers Coleman, to be published. +[134] Geng-Dong Zhou and Zhi-Da Song, to be published. +[135] Elliott H. Lieb, “Flux phase of the half-filled band,” Phys. Rev. +Lett. 73, 2158–2161 (1994). +[136] Alexei Kitaev, “Anyons in an exactly solved model and be- +yond,” Annals of Physics 321, 2–111 (2006). +[137] Alexander Weisse, Gerhard Wellein, Andreas Alvermann, and +Holger Fehske, “The kernel polynomial method,” Rev. Mod. +Phys. 78, 275–306 (2006). +[138] Piers Coleman, Introduction to many-body physics (Cam- +bridge University Press, 2015). +[139] Milton Abramowitz, Irene A Stegun, and Robert H Romer, +“Handbook of mathematical functions with formulas, graphs, +and mathematical tables,” (1988). +[140] Alexander +Altland +and +Ben +D +Simons, +Condensed matter field theory (Cambridge university press, +2010). + +1 +Supplementary Materials +CONTENTS +References +5 +S1. Model and notation +3 +A. U(4) moments +4 +B. ψ basis and U(8) moments +5 +S2. Zero-hybridziation limit +7 +A. Symmetries of the zero-hybridization Limit +7 +B. Zero hybridization limit without ˆHJ +9 +C. Solution of the one f-electron problem above (or below) uniform integer filling +11 +D. Charge density waves? +14 +S3. Zero-hybridization Model with ˆHJ +14 +A. Coherent states of the f-moments. +15 +B. Trial wavefunction +15 +C. Perturbation theory +17 +1. Z0 +18 +2. Z1/Z0 +18 +3. Z2/Z0 +19 +4. Free energy: F +24 +D. Path integral formula +26 +E. Ground state of the zero hybridization model at M = 0, νf = 0, −1, −2 +31 +F. Ground state of the zero hybridization model at M ̸= 0, νf = 0, −1, −2 +33 +S4. Schrieffer-Wolff transformation +35 +A. Hamiltonian +35 +1. ˆH0 +36 +2. ˆH1 +38 +B. Procedure of Schrieffer–Wolff transformation +38 +C. Expression of S +39 +D. Effective Hamiltonian from SW transformation +41 +E. Effect of ˆHJ and PH ˆHfcPH +46 +F. Effective Kondo model +46 +S5. RKKY interactions at M = 0 +49 +A. −⟨(SK + SJ)Scc⟩0,con +51 +B. −⟨SKSJ⟩0,con +53 +C. − 1 +2⟨S2 +K⟩0,con +54 +D. − 1 +2⟨S2 +J⟩0,con +55 +E. RKKY interactions +56 +S6. Ground state of f-moments +58 +A. Ground state at v′ +⋆ = 0, M = 0 +58 +B. Ground state at v′ +⋆ ̸= 0, M = 0 +61 +C. Comparisons of the ground states at different limits +64 +S7. Fluctuation spectrum of f-moments based on RKKY interactions +64 +A. RKKY Hamiltonian +65 +B. Excitation spectrum from RKKY interaction at νf = 0, −2 +66 +C. Excitation spectrum from RKKY interaction at νf = −1 +70 +1. Full-empty sector +73 +2. Full-half sector +73 +3. Half-empty sector +74 + +2 +4. Half-half sector +75 +5. Number of Goldstone modes +75 +D. Discussion +75 +S8. Effective theory of f-moments +75 +A. Single-particle Green’s function of conduction electrons +81 +B. ⟨Sint⟩0 +82 +C. ⟨Sint,2⟩0 +83 +D. − 1 +2⟨S′ +KS′ +K⟩0,con +84 +E. − 1 +2⟨S′ +JS′ +J⟩0,con +85 +F. −⟨S′ +JS′ +K⟩0,con +85 +G. Effective action +86 +H. Flat U(4) symmetry and Noether’s theorem +89 +I. Symmetry +93 +J. νf = 0 +94 +K. νf = −2 +97 +L. Lagrangian at νf = 0, −2 in the long-wavelength limit +98 +M. νf = −1 +99 +1. Full-empty sector +100 +2. Full-half sector +101 +3. Half-empty sector +103 +4. Half-half sector +105 +5. Effective theory in the long-wavelength limit +106 +6. Number of Goldstone modes +106 +S9. Single-particle Green’s function +107 +A. Single-particle Green’s function at M = 0 +107 +1. g0(R, τ) +109 +2. g2,η(τ, R) +110 +B. Single-particle Green’s function at M ̸= 0 +111 +S10. Four-fermioin correlation function I +114 +S11. Four-fermioin correlation function II +114 +S12. Fourier transformations +117 +S13. Single particle Green’s function in the ordered state +119 + +3 +S1. +MODEL AND NOTATION +The topological heavy-fermion model introduced in Ref. [122] takes the following Hamiltonian +ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV +(S1) +The single-particle Hamiltonian of conduction c-electrons has the form of +ˆHc = +� +η,s,a,a′,|k|<Λc +H(c,η) +a,a′ (k)c† +kaηscka′ηs +, +H(c,η)(k) = +� +02×2 +v⋆(ηkxσ0 + ikyσz) +v⋆(ηkxσ0 − ikyσz) +Mσx. +� +(S2) +where σ0,x,y,z are identity and Pauli matrices. ckaηs represents the annihilation operator of the a(= 1, 2, 3, 4)-th conduction +band basis of the valley η(= ±) and spin s(=↑, ↓) at the moir´e momentum k. At ΓM point (k = 0) of the moir´e Brillouin zone, +ck1ηs, ck2ηs form a Γ3 irreducible representation (of P6′2′2 group), ck3ηs, ck4ηs form a Γ1 ⊕ Γ2 reducible (into Γ1 and Γ2 - +as they are written, the ck3ηs, ck4ηs are just the σx linear combinations of Γ1 ± Γ2 ) representation (of P6′2′2 group). Λc is +the momentum cutoff for the c-electrons. N is the total number of moir´e unit cells. Parameter values are v⋆ = −4.303eV · ˚A, +M = 3.697meV. +The hybridization between f and c electrons has the form of +ˆHfc = +1 +√NM +� +|k|<Λc +R +� +αaηs +� +eik·R− |k|2λ2 +2 +H(fc,η) +αa +(k)f † +Rαηsckaηs + h.c. +� +, +(S3) +where fRαηs represents the annihilation operators of the f electrons with orbital index α(= 1, 2), valley index η(= ±) and spin +s(=↑, ↓) at the moir´e unit cell R. NM is the number of moir´e unit cells and λ = 0.3376aM is the damping factor, where aM is +the moir´e lattice constant. The hybridization matrix H(fc,η) has the form of +H(fc,η)(k) = +�γσ0 + v′ +⋆(ηkxσx + kyσy), 02×2 +� +(S4) +which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2). Parameter values are γ = −24.75meV, +v′ +⋆ = 1.622eV · ˚A. +ˆHU (U = 57.89meV) describes the on-site interactions of f-electrons. +ˆHU = U +2 +� +R +: nf +R :: nf +R :, +(S5) +where nf +R = � +αηs f † +RαηsfRαηs is the f-electrons density and the colon symbols represent the normal ordered operator with +respect to the normal state: : f † +Rα1η1s1fRα2η2s2 := f † +Rα1η1s1fRα2η2s2 − 1 +2δα1η1s1;α2η2s2. +The ferromagnetic exchange interaction between f and c electrons ˆHJ is defined as +HJ = − +J +2NM +� +Rs1s2 +� +αα′ηη′ +� +|k1|,|k2|<Λc +ei(k1−k2)·R(ηη′ + (−1)α+α′) : f † +Rαηs1fRα′η′s2 :: c† +k2,α′+2,η′s2ck1,α+2,ηs1 : +(S6) +where J = 16.38meV and : c† +k2,α′+2,η′s2ck1,α+2,ηs1 := c† +k2,α′+2,η′s2ck1,α+2,ηs1 − 1 +2δk1,k2δα,α′δη,η′δs1,s2 +The repulsion between f and c electrons ˆHW has the form of +ˆHW = +� +η,s,η′,s′,a,α +� +|k|<Λc,|k+q|<Λc +Wae−iq·R : f † +R,aηsfR,aηs :: c† +k+q,aη′s′ck,aη′s′ : +(S7) +where W1 = W2, W3 = W4 [122]. We further require W1 = W2 = W3 = W4 = W = 47meV (the difference is about 10-15% +). +The Coulomb interaction between c electrons has the form of +ˆHV = +1 +2Ω0NM +� +η1s1a1 +� +η2s2a2 +� +|k1|,|k2|<Λc +� +q +|k1+q|,|k2+q|<Λc +V (q) : c† +k1a1η1s1ck1+qa1η1s1 :: c† +k2+qa2η2s2ck2a2η2s2 : +(S8) + +4 +where Ω0 is the area of the moir´e unit cell and V (q = 0)/Ω0 = 48.33meV.We will always take a mean-field treatment of ˆHV , +which gives +ˆHMF +V += −V (0) +2Ω0 +NMν2 +c + V (0) +Ω0 +νc +� +|k|<Λc,aηs +(c† +k,aηsck,aηs − 1/2) +(S9) +where νc is the filling of c-electrons and |ψ0⟩ +νc = ⟨ψ0| ˆνc|ψ0⟩ . +(S10) +|ψ0⟩ is the ground state and the density operator ˆνc of c-electrons is defined +ˆνc = +1 +NM +� +|k|<Λc,aηs +c† +k,aηsck,aηs − 1/2 +(S11) +and |ψ0⟩ is the ground state. Clearly, at the mean-field level ˆHMF +V +is equivalent to a chemical potential shifting. In addition, the +energy loss from ˆHMF +V +is +⟨ψ0| ˆHMF +V +|ψ0⟩ = V (0) +2Ω0 +NMν2 +c +(S12) +The model has a U(4) × U(4) symmetry at chiral-flat limit M = 0, v′ +⋆ = 0, a flat U(4) symmetry at flat limit M = 0 and +a chiral U(4) symmetry at v′ +⋆ = 0. In addition, the particle-hole conjugation transformation will map the ground state of the +model at ν (total filling of f and c electrons) to the ground state at −ν. Thus we only consider ν ≤ 0 in this work. +A. +U(4) moments +To observe the symmetry of the system, we introduce the following flat U(4) moments (flat U(4) symmetry at M = 0) +ˆΣ(f,ξ) +µν +(R) =δξ,(−1)α−1η +2 +Aµν +αηs,α′η′s′f † +RαηsfRα′η′s′ +ˆΣ(c′,ξ) +µν +(q) =δξ,(−1)a−1η +2NM +� +|k|<Λc +Aµν +aηs,a′η′s′c† +k+qaηscka′η′s′, (a, a′ = 1, 2) +ˆΣ(c′′,ξ) +µν +(q) =δξ,(−1)a−1η +2NM +� +|k|<Λc +Bµν +aηs,a′η′s′c† +k+qaηscka′η′s′, (a, a′ = 3, 4) +(S13) +where repeated indices should be summed over and Aµν, Bµν (µ, ν = 0, x, y, z) are eight-by-eight matrices +Aµν ={σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} +Bµν ={σ0τ0ςν, −σyτxςν, −σyτyςν, σ0τzςν} , +(S14) +with σ0,x,y,z, τ0,x,y,z, ς0,x,y,z being the Pauli or identity matrices for the orbital, valley, and spin degrees of freedom, respectively. +The ±1 valued index ξ, equal to (−1)α−1η or (−1)a−1η in the generators the f and c electrons respectively, labels different +fundamental representations of the flat-U(4) group. The global flat-U(4) rotations are generated by +ˆΣµν = +� +ξ=±1 +� +ˆΣ(f,ξ) +µν ++ ˆΣ(c′,ξ) +µν ++ ˆΣ(c′′,ξ) +µν +� +. +(S15) +The chiral U(4) moments (chiral limit with v′ +⋆ = 0 and different other parameters [122]) can be defined in a similar manner +ˆΘ(f,ξ) +µν +(R) =δξ,(−1)α−1η +2 +Θµν,f +αηs,α′η′s′f † +RαηsfRα′η′s′ +ˆΘ(c′,ξ) +µν +(q) =δξ,(−1)a−1η +2NM +� +|k|<Λc +Θµν,c′ +aηs,a′η′s′c† +k+qaηscka′η′s′, (a, a′ = 1, 2) +ˆΘ(c′′,ξ) +µν +(q) =δξ,(−1)a−1η +2NM +� +|k|<Λc +Θµν,c′′ +aηs,a′η′s′c† +k+qaηscka′η′s′, (a, a′ = 3, 4) +(S16) + +5 +where µ, ν = 0, x, y, z, and the repeated indices should be summed over. In addition, +Θ(0ν,f) = σ0τ0ςν, +Θ(0ν,c′) = σ0τ0ςν +Θ(0ν,c′′) = σ0τ0ςν , +Θ(xν,f) = σxτxςν, +Θ(xν,c′) = σxτxςν +Θ(xν,c′′) = −σxτxςν , +Θ(yν,f) = σxτyςν, +Θ(yν,c′) = σxτyςν +Θ(yν,c′′) = −σxτyςν , +Θ(zν,f) = σ0τzςν, +Θ(zν,c′) = σ0τzςν +Θ(zν,c′′) = σ0τzςν . +The global chiral-U(4) rotations are generated by ˆΘµν = � +ξ=±1 +� +ˆΘ(f,ξ) +µν ++ ˆΘ(c′,ξ) +µν ++ ˆΘ(c′′,ξ) +µν +� +. +The relations between chiral and flat U(4) moments are +ˆΣf,ξ +0ν (R) = ˆΘf,ξ +0ν (R) +ˆΣf,ξ +zν (R) = ˆΘf,ξ +zν (R) +ˆΣf,ξ +xν (R) = ξ ˆΘ(f,ξ) +yν +(R) +ˆΣf,ξ +yν (R) = −ξ ˆΘ(f,ξ) +xν +(R) +ˆΣc′,ξ +0ν (R) = ˆΘc′,ξ +0ν (q) +ˆΣc′,ξ +zν (q) = ˆΘc′,ξ +zν (q) +ˆΣc′,ξ +xν (q) = ξ ˆΘ(c′,ξ) +yν +(q) +ˆΣc′,ξ +yν (q) = −ξ ˆΘ(c′,ξ) +xν +(q) +ˆΣc′′,ξ +0ν (q) = ˆΘc′′,ξ +0ν (q) +ˆΣc′′,ξ +zν (q) = ˆΘc′′,ξ +zν (q) +ˆΣc′′,ξ +xν (q) = ξ ˆΘ(c′′,ξ) +yν +(q) +ˆΣc′′,ξ +yν (q) = −ξ ˆΘ(c′′,ξ) +xν +(q) +(S17) +We note that even though two U(4) moments are related via Eq. S17, they actually characterize two different U(4) symmetries. +B. +ψ basis and U(8) moments +In this section, we introduce more general U(8) moments, where 8=2(orbital)× 2(valley) ×2(spin). +We first consider the +following convenient basis of electron operators ψf,ξ +R,n, ψc′,ξ +k,n, ψc′′,ξ +k,n +(n = 1, 2, 3, 4, ξ = ±): +(ψf,+ +R,1, ψf,+ +R,2, ψf,+ +R,3, ψf,+ +R,4) = (fR,1+↑, fR,2−↑, fR,1+↓, fR,2−↓) +(ψf,− +R,1, ψf,− +R,2, ψf,− +R,3, ψf,− +R,4) = (fR,2+↑, −fR,1−↑fR,2+↓, −fR,1−↓) +(ψc′,+ +k,1 , ψc′,+ +k,2 , ψc′,+ +k,3 , ψc′,+ +k,4 ) = (ck,1+↑, ck,2−↑, ck,1+↓, ck,2−↓) +(ψc′,− +k,1 , ψc′,− +k,2 , ψc′,− +k,3 , ψc′,− +k,4 ) = (ck,2+↑, −ck,1−↑, ck,2+↓, −ck,1−↓) +(ψc′′,+ +k,1 , ψc′′,+ +k,2 , ψc′′,+ +k,3 , ψc′′,+ +k,4 ) = (ck,3+↑, −ck,4−↑, ck,3+↓, −ck,4−↓) +(ψc′′,− +k,1 , ψc′′,− +k,2 , ψc′′,− +k,3 , ψc′′,− +k,4 ) = (ck,4+↑, ck,3−↑, ck,4+↓, ck,3−↓) +(S18) +Here, the index ξ = η(−1)α+1 can be understood as the index of Chern basis [4]. +The single-particle Hamiltonian ˆHc (Eq. S2) and ˆHfc (Eq. S2) in the ψ basis takes the form of +ˆHc = +� +|k|<Λc,n,ξ +(Ψc,ξ +k )†v⋆ +� +04×4 +(kx + iξky)I4×4 +(kx − iξky)I4×4 +04×4 +� +Ψc,ξ +k + +� +|k|<Λc,n,ξ +(−1)n+1Mψc′′,ξ,† +k,n +ψc′′,−ξ +k,n +ˆHfc = +1 +√NM +� +|k|<Λc,R,n +� +eik·R− |k|2λ2 +2 +˜H(fc) +ξξ′ (k)ψf,ξ,† +R,n ψc′,ξ′ +k,n + h.c. +� +, +˜H(fc)(k) = +� +γ +v′ +⋆k− +v′ +⋆k+ +γ +� +(S19) +where Ψc,ξ +k += +� +ψc′,ξ +k,1,...,4 ψc′′,ξ +k,1,...,4 +�T +and k± = kx ± iky. +We now define the U(8) moments with ψ basis (µ, ν = 0, x, y, z): +ˆΣ(f,ξξ′) +µν +(R) = 1 +2 +� +mn +ψf,ξ,† +R,n [T µν]nmψf,ξ′ +R,m +ˆΣ(c′,ξξ′) +µν +(k, q) = 1 +2 +� +mn +ψc′,ξ,† +k+q,n[T µν]nmψc′,ξ′ +k,m +, +ˆΣ(c′′,ξξ′) +µν +(k, q) = 1 +2 +� +mn +ψc′′,ξ,† +k+q,n[T µν]nmψc′′,ξ′ +k,m +{T µν} = {ς′ +νρ0, ς′ +νρy, −ς′ +νρx, ς′ +νρz} +(S20) + +6 +and we let ρx,y,z,0 be the Pauli matrices and the identity matrix defined in the subspace of (1+, 2−) for ξ = +1 and (2+, 1−) +for ξ = −1. ς′ +x,y,z,0 are the Pauli matrices and identity matrix acting in the spin subspace. +The previous flat U(4) moments in Eq. S16 can be written with new electron basis as +ˆΣ(f,ξ) +µν +(R) = 1 +2 +� +mn +ψf,ξ,† +R,n [T µν]nmψf,ξ +R,m = ˆΣ(f,ξξ) +µν +(R) +ˆΣ(c′,ξ) +µν +(q) = +1 +2NM +� +mn,k +ψc′,ξ,† +k+q,n[T µν]nmψc′,ξ +k,m = +1 +NM +� +k +ˆΣ(c′,ξξ) +µν +(k, q) +ˆΣ(c′′,ξ) +µν +(q) = +1 +2NM +� +mn +ψc′′,ξ,† +k+q,n[T µν]nmψc′′,ξ +k,m = +1 +NM +� +k +ˆΣ(c′′,ξξ) +µν +(k, q) . +(S21) +The advantage of using the ψ basis is that the flat U(4) moments of f, c′, c′′ electrons have the same matrix structure T µν. +It is also useful to consider the following Fourier transformation of electron operators and f-moments. +ψc′,ξ +r,n = +1 +√NM +� +k +eik·rψc′,ξ +k,n +ψc′′,ξ +r,n = +1 +√NM +� +k +eik·rψc′′,ξ +k,n +ˆΣ(c′,ξξ′) +µν +(r, r′) = +1 +NM +� +k,q +ˆΣ(c′,ξξ′) +µν +(k, q)eik·r′−i(k+q)r = 1 +2 +� +mn +ψc′,ξ,† +r,n +[T µν]nmψc′,ξ′ +r′,m +ˆΣ(c′′,ξξ′) +µν +(r, r′) = +1 +NM +� +k,q +ˆΣ(c′′,ξξ′) +µν +(k, q)eik·r′−i(k+q)r = 1 +2 +� +mn +ψc′′,ξ,† +r,n +[T µν]nmψc′′,ξ′ +r′,m . +(S22) +Since the momentum of c-electron has a finite cutoff, this definition is for convenience. However, if we only consider the +c-electrons in the first moir´e Brillouin zone, the completeness of c-electron Fourier transformation is guaranteed and a well- +defined inverse Fourier transformation also exists for c electrons. For what follows, unless specifically mentioned, we only +consider c-electrons in the first moir´e Brillouin zone. +Now, we utilize the following relation +� +µν +[T µν]ab[T µν]cd = 4δa,dδb,c +(S23) +and find +� +µν +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(f,ξ′ +2ξ2) +µν +(R2) = +� +a,b +ψf,ξ,† +R,a ψf,ξ′ +R,bψf,ξ′ +2,† +R2,b ψf,ξ2 +R2,a +� +µν +ˆΣ(c′,ξξ′) +µν +(r1, r′ +1)ˆΣ(c′,ξ′ +2ξ2) +µν +(r′ +2, r2) = +� +a,b +ψc′,ξ,† +r1,a ψc′,ξ′ +r′ +1,b ψc′,ξ′ +2,† +r′ +2,b +ψc′,ξ2 +r2,a +� +µν +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(c′′,ξ′ +2ξ2) +µν +(r, r′) = +� +a,b,R +ψf,ξ,† +R,a ψf,ξ′ +R,bψc′′,ξ′ +2,† +r,b +ψc′′,ξ2 +r′,a +(S24) + +7 +We change from ψ basis to f and c basis, then Eq. S24 becomes(which will be used in Sec. S4) +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(f,ξ′ξ) +µν +(R2) = +� +αα′ηη′ss′ +f † +R,αηsfR,α′η′s′.f † +R2,α′η′s′fR2,αηs +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(c′,ξ′ξ) +µν +(r, r′) = +� +αα′ηη′ss′ +f † +R,αηsfR,α′η′s′.c† +r,α′η′s′cr′,αηs +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(f,−ξ′ξ) +µν +(R2) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′f † +R2,α2η′s′fR2,αηsη′[σx]α′,α2 +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(c′,−ξ′ξ) +µν +(r, r′) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′c† +r,α2η′s′cr′,αηsη′[σx]α′,α2 +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ˆΣ(c′,−ξ′−ξ) +µν +(r, r′) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′c† +r,α2η′s′cr′,α′ +2ηsη′η[σx]α′,α2[σx]α′ +2,α +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ξ′ ˆΣ(c′,−ξ′ξ) +µν +(r, r′) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′c† +r,α2η′s′cr′,αηs[iσy]α′,α2 +� +µν +� +ξ,ξ′ +(−ξ)ˆΣ(f,ξξ′) +µν +(R)ξ′ ˆΣ(c′,−ξ′ξ) +µν +(r, r′) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′c† +r′,α2η′s′cr,α′ +2ηs[iσy]α′,α2[iσy]α′ +2,α +� +µν +� +ξ,ξ′ +ˆΣ(f,ξξ′) +µν +(R)ξ′ ˆΣ(c′,−ξ′ξ) +µν +(r, r′) = +� +αηs,α′η′s′ +f † +R,αηsfR,α′η′s′c† +r,α2η′s′cr′,α′ +2ηs[iσy]α′,α2[ησx]α′ +2,α +(S25) +S2. +ZERO-HYBRIDZIATION LIMIT +The f-c hybridization parameter γ vanishes at w0/w1 = 0.9 which is close to the actual value w0/w1 = 0.8. Vanishing γ +corresponds to the gap closing between the remote and the flat bands in the continuum single particle model. We solve the model +exactly at γ = 0 and then treat it as a perturbation. In this limit, the f and c electrons are coupled only through the interacting +terms. We now neglect ˆHV except for a mean-field treatment- for the same reason as in Ref. [122], that it will cause only a +velocity renormalization of the linear conduction fermion, and keep only the remaining terms ˆHU, ˆHW , ˆHJ, ˆHMF +V +where ˆHMF +V +denotes the ˆHV with mean-field approximation. +This now becomes a polynomially solvable Hamiltonian. Some remarks: 1. This is more complicated than the zero hybridiza- +tion limit of the Anderson lattice model in which the Hilbert space factorizes. This is because in both ˆHW (we have work in +the limit W1 = W2 = W3 = W4), and ˆHJ terms, the f occupation number will influence the electron Hamiltonian, but in a +one-body fashion, since the c operators are quadratic. The strategy is then to solve the Hamiltonian in this limit and then add the +f-c hybridization perturbatively via Schrieffer–Wolff transformation (SW) transformation [127]. +To solve the Hamiltonian we first need to find the on-site configurations of the f fermions (which are good quantum numbers) +which then, after adding ˆHU + ˆHW + ˆHJ and the c-electron kinetic term +Hc = � +ηs +� +aa′ +� +|k|<Λc(H(c,η) +a,a′ (k) − µδaa′)c† +kaηscka′ηs − µNf +(S26) +give the lowest energy. This is a minimization problem similar in spirit to the Lieb theorems [135] with flux π (used in the Kitaev +model [136]) where it is shown that the flux π configuration of a noninteracting fermion model in a background plaquette flux +with each plaquette having possibly different flux is minimal at flux π per all plaquettes. Our problem, at least in the first try, +might be amenable to Monte Carlo sampling. +A. +Symmetries of the zero-hybridization Limit +We can rewrite the ˆHU + ˆHW + ˆHJ +ˆHU = U +2 +� +R : nfR :: nfR, +ˆHW = 1 +N +� +R : nfR : � +|k|,|k′|<Λc +� +η2s2a Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : , +HJ = − J +2N +� +Rs1s2 +� +αα′ηη′ +� +|k1|,|k2|<Λc ei(k1−k2)·R(ηη′ + (−1)α+α′) : f † +Rαηs1fRα′η′s2 :: c† +k2,α′+2,η′s2ck1,α+2,ηs1 : +(S27) +Where we have defined the occupation number of the electron on-site R +nfR = +� +αηs +f † +RαηsfRαηs ∈ 0, 1 . . . 8 +(S28) + +8 +which is a symmetry of the system even when W1 ̸= W3: +[ ˆHU, nfR] = 0, [ ˆHW , nfR] = 0, [ ˆHJ, nfR] = 0, [Hc, nfR] = 0 +(S29) +The system has a large U(1)NM symmetry where NM is the number of moir´e unit cells. An eigenstate of the system is first +indexed by nfR. Then this couples through the ˆHW and (with a smaller coefficient) through ˆHJ. +There are now several questions +• If first we neglect ˆHJ we have a large symmetry, U(8)NM , whose generators are ˆΣ(ξξ′) +µν +(R) = ˆΣ(f,ξξ′) +µν +(R)+ˆΣ(c′,ξξ′) +µν +(R)+ +ˆΣ(c′′,ξξ′) +µν +(R) (Eq. S20). +[ ˆHU, ˆΣ(ξξ′) +µν +(R)] = 0 +, +[ ˆHW , ˆΣ(ξξ′) +µν +(R)] = 0 +, +[Hc, ˆΣ(f,ξξ′) +µν +(R)] = 0 +(S30) +which gives huge number of degenerate states, all with the same occupation number nfR on-site distributed around the +different α, η, s. +• If we take W1 = W3 = W then we have that +ˆHW = W 1 +N +� +R : nfR : � +|k|,|k′|<Λc +� +η2s2a e−i(k−k′)·R � +ηsa : c† +kaηsck′aηs : += W 1 +N +� +R : nfR : � +|k|,|k′|<Λc +� +η2s2 e−i(k−k′)·R � +n1,n2(� +a U η,∗ +k,an1U η,∗ +k′,an2)γ† +k,n1ηsγk′,n2ηs +(S31) +where we have introduced the eigenvectors U η +k,an and eigenvalues ϵη +k,n of ˆH(c,η)(k) +� +a′ +Hc +aa′(k)U η +k,a′n = ϵη +k,nUk,an , +and the operator in the band basis +γk,nηs = +� +a +U ∗ +k,anck,aηs . +ˆHW is now a one-body term for c or γ, which depends on the distribution of nf +R. Through Monte Carlo sampling, we can +find the ground-state exactly and check whether the ground state involve uniform nf +R or not. +• If furthermore, we assume uniform nf +R = nf then the summation over R gives us a δk,k′ and we obtain an nf-dependent +chemical potential of the electrons: +ˆHW = Wnf +� +|k|,|k′|<Λc +� +nηs γ† +knηsγknηs +(S32) +We can now find easily the admixture of f, c fermions at any filling N = Nf + Nc with Nf = NMnf. This allows us to +see, as a function of the total filling, analytically, the distribution between the f, c electrons in the ground state. We can +then add the ˆHJ under the stronger assumption that +f † +Rαηs1fRα′η′s2 = Aαηs1,α′η′s2 . +(S33) +We can then find the Aαηs1,α′η′s2 which will minimize the state energy and break the U(8) on-site symmetry to U(1) +symmetry. +• We can assume nf +R not constant, and adopt charge density wave (CDW) or other types of translational symmetry breaking. +We leave it for future study +• We can then add the f †c term through Schrieffer-Wollf (SW) transformation , which will be discussed in the Sec. S4. + +9 +4 +4 +nu=0 +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A. +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�,v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +3 +5 +nu=-1 +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A. +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�,v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +2 +6 +nu=-2 +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A. +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�,v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +1 +7 +nu=-3 +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A. +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�,v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). +Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP +(k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +52 +FIG. S8. HF band structures of correlated insulator phases at ⌫ = �2. (a), (b), (c) are the one-shot HF band structures of +the VP, K-IVC, and IVC phases, respectively. (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and +IVC phases, respectively. The color represents the composition of the energy bands, where yellow corresponds to the local +orbitals and blue corresponds to the conduction bands. We have chosen w0/w1 = 0.8 in the calculation. Other parameters of +the single-particle and interaction hamiltonians are given in Tables S4 and S6. +As shown in Fig. S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in +common: There is a set of flat bands above the zero energy. Such flat bands are also observed in one of our previous +studies [6] but have not been understood. (See the particle excitation spectra in Fig. 11 of Ref. [6].) We now can +explain the origin of the flatness through our topological heavy fermion model. We consider the k·p expansion of the +one-shot mean-field Hamiltonian. For simplicity, here we mainly focus on the VP state. Following the same procedure +we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq. (S305)), we obtain +H +(MF) +VP (k) ⇡ +0 +@ +�W1�0⌧0&0 +v?(kx�0⌧z + iky�z⌧0)&0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +v?(kx�0⌧z � iky�z⌧0)&0 +�W3�0⌧0&0 + M�x⌧0&0 � J +2 �0⌧z&0 +0 +��0⌧0&0 + v0 +?(kx�x⌧z + ky�y⌧0)&0 +0 +�(U1 + 6U2)�0⌧0&0 � U1(Of � 1 +2�0⌧0&0) +1 +A . +(S332) +The density matrix Of is given in Eq. (S324). Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 +(Eq. (S305)), there are three additional terms, i.e., �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- +onal blocks. These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq. (S284)) +and HU (Eq. (S279)) and only shift the energies of the three blocks. Without the energy shift and the couplings (�, v0 +?) +between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the +f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig. 3(a) in the main text. Using +the parameters obtained at w0/w1 = 0.8 (Table S6), the average energy shift of the c-electron bands (the first two +blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV. Thus, +the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and +is approximately �U1/2 ⇡ 29meV. That means, if we turn o↵ the hybridizations, the upper branch of the f-electron +levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig. 3(b) in +the main text. Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of +f-electron bands form an isolated set of flat bands. +FIG. S1. Zero hybridization γ = 0 model at different fillings of the Heavy fermions (the state assumed as the parent state carries all the filling +in the heavy fermions, unlike the exact state at γ = 0 which contains both c and f fermions). The Fermi level is a horizontal dashed black line. +The solid lines are zero hybridization scenarios while the green dashed lines are the level splittings occurring after small increase of γ from +nonzero, but still smaller than the realistic value of γ = 24.6meV. At ν = 0 both minima of the bands are made by c electrons. At ν = −1+ϵ +the band is very flat while at ν = −1 − ϵ the minimum of the band is formed by the c electrons. This last situation changes at ν = −2, −3, at +least in the absence of hybridization: both band edges are made up of heavy fermions. At small hybridization (green dashed) the band edges +at ν = −2 ± ϵ and ν = −3 ± ϵ are still made up by heavy fermions. Once γ reaches its large 24.6meV value, we find that at ν = −2 − ϵ +the band has changed character from heavy fermion to light fermion due to large heavy-light mixing (see violet dashed line, large curvature). +At ν = −3 − ϵ however, the portion of the heavy band as a ratio of the Brillouin zone is larger and hence upon hybridization, the portion of +the mixing is smaller than at ν = −2 − ϵ, and the band has more f-character. The CDW [115] obtained at ν = −3 most likely has f-f CDW +correlations between the ν = −3 + ϵ band edge (which is around K, M points and hence made up of heavy fermions) and the ν = −3 − ϵ +band edge (around Γ but still made up of heavy fermions) +B. +Zero hybridization limit without ˆHJ +We first solve the zero hybridization model at J = 0. The total Hamiltonian of the zero-hybridization model reads +ˆHc + ˆHU + ˆHW + ˆHMF +V +(S34) +We consider the solution with uniform charge distribution of f-electrons nf +R = νf + 4, with integer νf ∈ [−4, ..., 4]. We note +that, in the zero-hybridization model, the filling of f-electron of each site is a good quantum number and takes integer values. . +The trial wavefunction of the ground state we proposed is +|νf, ν⟩ = [ +� +R +|νf⟩R]|Ψ[ν − νf, νf]⟩c. +(S35) +where νf denotes the filling of f electrons, ν denotes the total filling of f and c electrons, and νc = ν − νf is the filling of c +electrons. +• |νf⟩R describes a f state at site R with filling νf: +� +αηs +: f † +R,αηsfR,αηs : |νf⟩R = νf|νf⟩R. +and we take a uniform charge distribution, so for each site, the fillings of f-electrons are the same. + +10 +• |Ψ[νc, νf]⟩s denotes a Slater determinant state corresponding to the ground state of the following one-body Hamiltonian +of c electrons at filling νc = ν − νf: +H′ +c = ˆHc + +� +η,s,a +� +|k|<Λc +Wνf : c† +kaηsckaηs : . +Above one-body Hamiltonian can be diagonalized. (Note that Wa is taken to be orbital independent): +H′ +c = +� +|k|<Λc,η,s,n +(Eknη + Wνf)γ† +knηsγknη +where n is the band index and γ is the operator in the band basis +γknηs = +� +a +U η,∗ +kanckaηs, +ckaηs = +� +n +U η +kanγknηs, +� +a′ +H(c,η) +a,a′ (k)U η +ka′n = EknηU η +kan +(S36) +The energy of |νf, ν⟩ state is then +Eνf ,ν/NM = ⟨νf, ν| ˆHU|νf, ν⟩/NM + ⟨νf, ν| ˆHV |νf, ν⟩/NM + ⟨νf, ν| ˆHW |νf, ν⟩/NM + ⟨νf, ν| ˆHc|νf, ν⟩/NM +⟨νf, ν| ˆHU|νf, ν⟩/NM = U +2 ν2 +f +, +⟨νf, ν| ˆHMF +V +|νf, ν⟩/NM = V0 +2Ω0 +ν2 +c = V0 +2Ω0 +(ν − νf)2 +⟨νf, ν| ˆHW |νf, ν⟩/NM = Wνfνc = Wνf(ν − νf) +⟨νf, ν| ˆHc|νf, ν⟩/NM = +1 +NM +� +k,nηs +(Eknη)⟨Ψ[νc, νf|γ† +knηsγknηs|Ψ[νc, νf]⟩ +Eνf ,ν/NM = U1 +2 ν2 +f + V0 +2Ω0 +(ν − νf)2 + Wνf(ν − νf) + +1 +NM +� +k,nηs +(Eknη)⟨Ψ[νc, νf|γ† +knηsγknηs|Ψ[νc, νf]⟩ +For a given total filling ν, we compare the energies of different νf (νf can only be an integer) and take the one with the lowest +energy as our ground state. +In the M = 0 limit, the analytical expression of Eνf ,ν/NM can be easily given At M = 0, c electron dispersion becomes +±v⋆|k|. Filling νc = ν − νf conduction electrons is equivalent to fill the 8-fold (8 = 2 × 2 × 2, 2 for spin, 2 for valley, 2 for +orbital) linear-dispersive bands up to certain momentum k0. Depending on the sign of νc, we either fill electrons (νc > 0) or +holes (νc < 0) to the Dirac sea. This gives the following equations to determine k0 +8 +AMBZ +� +|k| −µ). For |µ| < M, we find +νc = − +4π +AMBZ +|µ|(|µ| + M) +|v⋆|2 +(S40) +For |µ| > M, we find +νc = − +4π +AMBZ +|µ|(|µ| + M) +|v⋆|2 +− +4π +AMBZ +|µ|(|µ| − M) +|v⋆|2 +(S41) +Combining Eq. S40 and Eq. S41, we find +νc = − +4π +AMBZ +|µ|(|µ| + M) +|v⋆|2 +− +4π +AMBZ +|µ|(|µ| − M) +|v⋆|2 +θ(|µ| − M) +(S42) +Then the energy loss from conduction c electron is +Ec = 1 +NM +� +k,nηs +(Eknη)⟨Ψ[νc, νf|γ† +knηsγknηs|Ψ[νc, νf]⟩ += +4 +AMBZ +� +|k|<Λc +θ(|µ| − −M + +� +M 2 + 4|v⋆|2|k|2 +2 +)−M + +� +M 2 + 4|v⋆|2|k|2 +2 +dkxdky ++ +4 +AMBZ +� +|k|<Λ +θ(|µ| − M + +� +M 2 + 4|v⋆|2|k|2 +2 +)M + +� +M 2 + 4|v⋆|2|k|2 +2 +dkxdky +(S43) +For |µ| < M, we find +Ec = +4π +AMBZ +|µ|(|µ| + M) +|v⋆|2 +(S44) +For |µ| > M, we find +Ec = +4π +AMBZ +µ2 + M|µ| +|v⋆|2 ++ +4π +AMBZ +|µ|(|µ| − M) +|v2⋆| +(S45) +Combing Eq. S44 and Eq. S45, we have +Ec = +4π +AMBZ +µ2 + M|µ| +|v⋆|2 ++ +4π +AMBZ +|µ|(|µ| − M) +|v2⋆| +θ(|µ| − M) +(S46) +Then the energy of the system is +Eνf ,ν/NM = U1 +2 ν2 +f + V0 +2Ω0 +(ν − νf)2 + Wνf(ν − νf) + Ec +(S47) +where Ec is given in Eq. S46 and depends on µ, with µ solved from Eq. S42. Solutions at nonzero M are shown in Fig. S3. +We notice two rules: +• At integer ν, the states with νf = ν would minimize the energy. +• ν = −3 is very close to the transition point (see Fig. S3) between νf = −3 and νf = −2. +We will consider ν = 0, −1, −2 in this work. The ν = −3 needs separate treatment due to it being near the transition point +between νf = −2, −3. We point out that the ν = −3 case is also known to exhibit a competition of orders [115]. +C. +Solution of the one f-electron problem above (or below) uniform integer filling +We now consider one more electron than integer filling. The state reads +|R, α⟩ = f † +R,α|νf⟩|c⟩ +(S48) + +12 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +t +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +E f, +t/NM(eV) +f = 0 +f = -1 +f = -2 +f = -3 +f = -4 +FIG. S2. Evolution of energy as a function of ν at different νf and M = 0. For fixed ν, the ground state is determined by νf that gives the +lowest energy. +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +f +c +FIG. S3. Evolution of the f and c filling as a function of ν. We observe that ν = −3 is near the transition point between νf = −2, −3 states. +where |νf⟩ is the state with νf + 4 fermions per site and |c⟩ is the conduction electron ground state of ˆHc (Eq. S2) with filling +νc = 0. We then have: +� +α +: f † +R1,αfR1,α : |R, α⟩ = (δR1,R + νf)|R, α⟩ +(S49) +(here α = 1 . . . 8 over all valleys and spins and flavors) +ˆHU|R, α⟩ = U +2 +� +R +: nfR :: nfR : |R, α⟩ = (NMν2 +f +U +2 + U +2 (1 + 2νf))|R, α⟩ +ˆHW |R, α⟩ = 1 +N +� +R +: nfR : +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |R, α⟩ = += 1 +N +� +R +(δR1,R + νf) +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |R, α⟩ = += νf +� +|k|<Λc +� +η2s2a +Wa : c† +kaη2s2ckaη2s2 : |R, α⟩ + 1 +N +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |R, α⟩ +(S50) +This gives the one electron Hamiltonian for the c-electrons without the ferromagnetic exchange interaction term ˆHJ (Eq. S6). +The Coulomb repulsion ˆHV has been treated at the mean-field level (Eq. S9) and is absorbed by the chemical potential of +conduction c-electrons. + +13 +H|R, α⟩ += (NMν2 +f +U +2 + U +2 (1 + 2νf))|R, α⟩ + ( +� +ηs +� +aa′ +� +|k|<Λc +(H(c,η) +a,a′ (k) − µδaa′ + νfWaδaa′)c† +kaηscka′ηs + ++ 1 +N +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : )|R, α⟩ += (NMν2 +f +U +2 + U +2 (1 + 2νf))|R, α⟩ + +� +ηs,η′s′ +� +a,a′ +� +|k|,|k′|<Λc +(Akaηs,k′a′η′s′ + Bkaηs,k′a′η′s′(R))c† +kaηsck′a′η′s′ +(S51) +We see that the Hamiltonian action on the one f particle above the integer state contains two c-electron terms. First: +Akaηs,k′a′η′s′ = (H(c,η) +a,a′ (k) − µδaa′ + νfWaδaa′)δk,k′δηη′δs,s′ +(S52) +has a continuous spectrum, given by H(c,η) +a,a′ (k) and shifted by an a-dependent chemical potential Wa. When W1 = W2, the +term is actually transformed into the density, and it is a bona-fide chemical potential. Let us call the eigenvalues of this term in +Eq. S52 ν1 ≥ ν2 ≥ . . . ν16NM as the Hamiltonian has 16NM eigenvalues, 16 for each momentum. +The second term is +Bkaηs,k′a′η′s′(R) = +1 +NM +� +|k|,|k′|<Λc +Wae−i(k−k′)·Rδa,a′δη,η′δs,s′ +(S53) +Unfortunately Eq. S52 and Eq. S53 do not commute, hence obtaining the spectrum analytically is hard. However, we can still +derive clear theorems. First we rewrite +Bkaηs,k′a′η′s′(R) = +� +η′′,s′′a′′ +Pkaηs;η′′,s′′a′′(R)P † +η′′,s′′a′′;k′a′;η;s′(R), +Pkaηs;η′′,s′′a′′(R) = +� +Wa +NM +e−ik·Rδaa′′δηη′′δss′′ +(S54) +Since B = P(R)P(R)†, where PR is a Rank 16 (a = 1 . . . 4, η = ±, a = ±) matrix, it is highly rank deficient and has a large +number of zero eigenvalues. The nonzero eigenvalues are the same as those of the matrix P †(R)P(R): +(P †(R)P(R))η′s′a′;ηsa = Waδaa′δηη′δss′ +(S55) +where we have summed over all the momentum values k in the first moir´e Brillouin zone (i.e we took Λc to be entire first moir´e +Brillouin zone). Since B is a positive semidefinite matrix, we confirm all eigenvalues are non-negative. We now know that there +are 16 nonzero eigenvalues, 8 of them being W3, larger than the other 8 of them which are W1. We now order them as such: +ρ1 = . . . = ρ8 = W3 > ρ9 = . . . = ρ16 = W1 > ρ17 = . . . ρ16NM = 0. +The existence of only a finite (Norbitals = Norb = 16) number of nonzero eigenvalues of the second matrix B with eigen- +values ρi and a continuum spectrum of eigenvalues of the first matrix A with eigenvalues νi allows for strong theorems of the +spectrum. We employ Weyl’s inequalities that say that the eigenvalues µ1 ≥ µ2 ≥ . . . ≥ µNorbNM of the sum of two matrices +A, B satisfy: +νj + ρk ≤ µi ≤ νr + ρs, +if +j + k − NorbNM ≥ i ≥ r + s − 1 +(S56) +For the A, B matric in Eq. S52 and Eq. S53, we pick k = NorbNM +=⇒ ρk = 0 and s = Norb + 1 =⇒ ρs = 0. We then +have: +νj ≥ µi ≥ νr, +if +j ≥ i ≥ r + Norb +(S57) +In Eq. S56, we pick j = i, r = i − Norb and we have +νi ≤ µi ≤ νi−Norb, +i > Norb +(S58) +Now we see that most of the µ eigenvalues form a continuum themselves, as they are bounded between the eigenvalues of +the matrix A, which form a continuum. In fact, µNorb+1 ≥ µNorb+2 ≥ . . . ≥ µNorbNM are bounded by the eigenvalues +of A, νi, νi−Norb; since these latter eigenvalues form a continuum, the most (most of the times they are degenerate as they + +14 +represent equal energies in k-space, i.e. Fermi surfaces of the continuum Hamiltonian) they can have in energy separation is +Norbv⋆2π/√NM where we have used the linear spectrum of A and the momentum quantum 2π/√NM. When NM → ∞ the +two eigenvalues are the same and hence to a good approximation µi ≈ νi, +i > Norb. (A similar proof is an observed reason +why the particle-particle continuum is formed by sums of the particle-particle energies, and the bound/anti bound states form a +finite number). The number of eigenvalues that can be away from a νi is small, µi, i = 1, 2 . . . Norb: νi ≤ µi ≤ νi + W. These +are anti-bound states (i = 1, 2 . . . Norb and hence they are at the top of the spectrum, as they should be since B (Eq. S53) is a +positive semidefinite matrix. +Since the c-spectrum of the one-f fermion state in Eq. S48 is unchanged except for the upper parts of the spectrum, adding +one electron to the integer-filled heavy fermion ground-state at integer total fillings costs ν2 +f +U +2 + U +2 (1 + 2νf) per unit cell. As +such it is beneficial to add c electrons, at least at νf = 0. +We next comment on the effect of ˆHV . At the mean-field level ( ˆHV ≈ ˆHMF +V +, Eq. S9), ˆHMF +V +corresponds to a chemical +potential term. The c-spectrum remains gapless and forms a continuum even after including ˆHMF +V +. Therefore, our statement +remains valid. +D. +Charge density waves? +Consider charge neutrality νf = νc = 0. As we showed, it costs U +2 to add one f electron, hence upon changing the filling, it +is easy to add c electrons. Suppose we add c electron up to k0 defined in Eq. S37, adding one more c electron will cost +Ec = v⋆k0 + V (0) +Ω0 +νc = v⋆k0 + V (0) +Ω0 +νc = v⋆ +� +|νc|AMBZ +8π ++ V (0) +Ω0 +νc +(S59) +where the second term comes from the ˆHMF +V +(Eq. S9). For sufficient large νc, we have Ec > U1/2. Then it becomes advanta- +geous to add f-electrons, and they can be added (if we neglect W) from : νf := 0 all the way to : νf := 1 at W = 0 limit. At +non-zero W, the process of adding electrons is not so straightforward. We use a trial state +|ψ⟩ = +� +(R,αηs)∈Sfill +f † +R,αηs| : νf := 0⟩|k0, c-filled⟩ +(S60) +where Sfill is a set with NMνf elements, that characterize the site and flavor (orbital, valley, and spin) where the f-electron is +filled. We have that: +ˆHW |ψ⟩ = 1 +N +� +R +: nf +R : +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |ψ⟩ += 1 +N +� +R,αηs +� +(R1,α1η1s1)∈Sfill +δR1,Rδα,α1δs,s1δη,η1 +� +|k|,|k′|<Λc +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |ψ⟩ = += 1 +N +� +|k|,|k′|<Λc +� +(R,αηs)∈Sfill +� +η2s2a +Wae−i(k−k′)·R : c† +kaη2s2ck′aη2s2 : |ψ⟩ +(S61) +The spectrum of this matrix is again very simple, only W1, W3 and zero eigenvalues, by the same proof as in the previous +section (Sec. S2 C). However, there is now a thermodynamic number of them, Norb : νf : NM and the Weyl theorems do +not give nontrivial results, as they only bound the new eigenvalues in between νi and νi−Norb:νf :NM which can now be a large +interval. At large fillings (say νf = −3), it could become convenient [115] to fill unbounded eigenvalues rather than just fill the +particle continuum and hence a CDW or other translational breaking states could be competitive. +S3. +ZERO-HYBRIDIZATION MODEL WITH ˆHJ +We next study the zero hybridization model with ˆHJ. The Hamiltonian is given below +ˆHc + ˆHU + ˆHW + ˆHV + ˆHJ +(S62) +where ˆHJ describes a ferromagnetic exchange coupling between flat U(4) moments (Eq. S16 of f-electrons and Γ1 ⊕ Γ2 c- +electrons (a = 3, 4) [122] +ˆHJ = −J +� +Rq +� +µν +� +ξ=± +e−iq·R : ˆΣ(f,ξ) +µν +(R) :: ˆΣ(c′′,ξ) +µν +(q) : +(S63) + +15 +A. +Coherent states of the f-moments. +Since J ∼ 16.38meV is relatively small compared to U and W, we assume that ˆHJ term will not destroy the uniform charge +distribution of the f-electrons in the plateau region of Fig. S3 which includes ν = 0, −1, −2. To find the ground state at small +but non-zero J, we do the following +• Find νf at given ν at J = 0 limit (which is given in Fig. S3). +• Turn on a small but nonzero J and assume the filling of f electron at each site is fixed to be νf. Find the optimal solution. +We now describe the second step in detail. We start by introducing the coherent state in a similar manner as spin coherent +state of SU(2) spin. We assume that, for each site R, we have a fixed integer number of f electrons (νf + 4), which allows us +to label each site via its expectation value of chiral U(4) moments. In other words, for a given a state |ψR⟩ of the f-electron at +site R, we label it as |{θ(f,ξ) +µν +(R)}µ,ν,ξ⟩, where θ(f,ξ) +µν +(R) is given by +⟨ψR| : ˆΘ(f,ξ) +µν +(R) : |ψR⟩ = θ(f,ξ) +µν +(R) +(S64) +We will call the states with such labeling coherent states. +B. +Trial wavefunction +Using the coherent state defined in the previous section, we now propose the trial wavefunction of the ground state. Again, +we fix the filling of f to be νf and the total filling to be ν, and assume uniform charge distribution of f. +We first consider the following state of f electrons +|ϑ⟩ = +� +R +|{θf,ξ +µν (R)}µνξ⟩R +(S65) +where we simply tensor product the coherent state of each site and we use ϑ (= {{θf,ξ +µν (R)}µνξ}R) to denote the set of θf,ξ +µν (R). +We now act the Hamiltonian on it and calculate: +⟨ϑ| ˆH|ϑ⟩. +(S66) +|ϑ⟩ only describes f states, so ⟨ϑ| ˆH|ϑ⟩ is an operator acting on the Hilbert space of c electrons. Each term in ⟨ϑ| ˆH|ϑ⟩ becomes +⟨ϑ| ˆHU|ϑ⟩ = NM +U1 +2 ν2 +f +⟨ϑ| ˆHV |ϑ⟩ = NM +V0 +2Ω0 +ˆνc +2 +(mean-field level) +⟨ϑ| ˆHW |ϑ⟩ = NMWνf ˆνc +⟨ϑ| ˆ +Hc|ϑ⟩ = ˆ +Hc +(Hc only contains conduction electron operators) +⟨ϑ| ˆHJ|ϑ⟩ = −J +� +Rq +� +µν +� +ξ=± +e−iq·Rθf,ξ +µν (R) : ˆΘ(c′′,ξ) +µν +(q) : +(S67) +Here, we rewrite ˆHJ in terms of chiral U(4) moment. θf,ξ +µν (R) is a complex number but : ˆΘ(c′′,ξ) +µν +(q) : is an operator. Since we +fix the total filling to be ν and the Hamiltonian without f-c hybridization preserves c-electron particle numbers , we can replace +operator ˆνc with a number νc = ν − νf. Now, ⟨ϑ| ˆH|ϑ⟩ can be separated into two parts: a constant term E0[νf, ν] (which is a +function of νf, ν) and a one-body Hamiltonian ˆH′′ +c [ϑ] of c-electron: +⟨ϑ| ˆH|ϑ⟩ = E0[νf, ν] + ˆH′′ +c [ϑ] +E0[νf, ν] = NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc +ˆH′′ +c [ϑ] = ˆHc − J +� +Rq +� +µν +� +ξ=± +e−iq·Rθf,ξ +µν (R) : ˆΘ(c′′,ξ) +µν +(q) : +(S68) +We then write ˆH′′ +c [ϑ] in a more compact form + +16 +ˆH′′ +c [ϑ] = +� +k,k′,aa′,ηη′,ss′ +c† +kaηs +� +[h0]kaηs,k′a′η′s′ + [h1]kaηs,k′a′η′s′ +� +ck′η′s′ + +J +NM +� +Rξ=±.|k|<Λc +θ(f,ξ) +00 +(R) +(S69) +h0, h1 are understood as a matrix with row index kaηs and column index k′a′η′s′. The explicit form of h0, h1 are +[h0]kaηs,k′a′η′s′ = δk,k′δηη′δss′H(c,η) +aa′ (k) +[h1]kaηs,k′a′η′s′ = −J +� +ξ=±,R +e−iq·Rθf,ξ +µν (R)[Θ(c′′,ξ) +µν +]aηs,a′η′s′ δξ,(−1)a−1ηδξ,(−1)a′−1η′ +2NM +δk,k′+q +(S70) +h0 describes the noninteracting part of conduction electrons. h1 describes the contribution from ˆHJ. The last term of Eq. S69, +comes from normal ordering. To observe the origin of the last term, we note +− J +� +Rq,µν,ξ +e−iq·Rθ(f,ξ) +µν +(R) : ˆΘ(c′′,ξ) +µν +(q) : += −J +2NM +� +Rq,µν,ξ +e−iq·Rθ(f,ξ) +µν +(R) +� +|k|<Λc +� +aηs,a′η′s′ +δξ,(−1)a−1η +Θµν,c′′ +aηs,a′η′s′ +2 +[ck+q,αηsck,α′η′s′ − 1 +2δq,0δα,α′δη,η′δs,s′] += − J +� +Rq,µν,ξ +e−iq·Rθ(f,ξ) +µν +(R)ˆΘ(c′′,ξ) +µν +(q) − +1 +4NM +� +k,R +� +aηs,a′η′s′ +θ(f,ξ) +µν +(R)Θµν,c′′ +aηs,a′η′s′ +=ˆΘ(c′′,ξ) +µν +(q) − δq,0 +NM +δµν,00 +� +k<Λc +θ(f,ξ) +µν +(R) +(S71) +h0 + h1 in Eq. S69 can be diagonalized. Let En and [vn]kaηs be n-th eigenvalue and n-th eigenvector +� +k′a′η′s′ +[h0 + h1]kaηs,k′a′η′s′[vn]k′a′η′s′ = En[vn]kaηs +(S72) +where we assume E1 ≤ E2 ≤ .... Then we have +ˆH′′ +c [ϑ] = +� +n +Enγ† +nγn + +J +NM +� +Rξ=±.|k|<Λc +θf,ξ +00 (R) +γn = +� +kaηs +ckaηs +� +[vn]kaηs +�∗ +(S73) +For fixed νc, the Slater determinant states that minimize the energy of ˆH′′ +c are +|Ψ[ϑ, νc]⟩ := +� +n≤Nc +γ† +n|0⟩ +(S74) +where γn|0⟩ = 0 for all n and Nc = (νc + 8)NM denotes the total particle numbers of c (without normal ordering). Based on +the states we constructed in Eq. S65 and Eq. S74 , we define the following trial wavefunction of the ground state: +|ϑ⟩|Ψ[ϑ, νc]⟩. +(S75) +Its energy is +E[ϑ, νf, ν] = ⟨Ψ[ϑ, νc]|⟨ϑ| ˆH|ϑ⟩|Ψ[ϑ, νc]⟩ += NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc + ⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩. += NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc + +J +NM +� +R|k|<Λc,ξ=± +θf,ξ +00 (R) + +� +n≤(νc+8)NM +Enγ† +nγn +(S76) +where νc = ν − νf. For given ϑ, νf, ν, the ground state will minimize the energy. Remember, the energy contains two terms: +⟨ϑ| ˆH|ϑ⟩ = E0[νf, ν] + ˆ +Hc +′′[ϑ]. The first term is fixed for given νf, ν, and the Slater determinant state we choose will always +minimize the energy from ˆ +Hc +′′[ϑ] at fixed filling. +In order to find the optimal states at fixed νf, ν, we need to vary ϑ and find the configuration of f states that minimize +E[ϑ, νf, ν]. In principle, this could be efficiently done numerically via simulated annealing and kernel polynomial meth- +ods [137]. Here, instead, we tackle this problem via perturbation theory. + +17 +C. +Perturbation theory +Instead of numerically solving the problem, we could expand the energy function E[ϑ, νf, ν] in powers of J. We note that +E[ϑ, νf, ν] = NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc + ⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩ . +For fixed νf, ν, the ground state minimize ⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩. Then we find the ground state by calculating the ground +state energy of ˆH′′ +c [ϑ] at fixed filling νc. To do so, it is more convenient to first calculate the free energy at the fixed filing of +c electrons νc and at a finite temperature. We then set the temperature to zero, and find the ground state energy. We take the +following partition function +Z = Tr[e−β ˆ +H′′ +c [ϑ]P] +(S77) +where P is a projection operator that fix the total filling of c electrons to be νc = ν − νf being an integer. β is the inverse +temperature. For given νc, P is defined as +P = +1 +Aνc +� +ν′c∈Sνc +( ˆνc − ν′ +c) +, +Aνc = +� +ν′c∈Sνc +(νc − ν′ +c) +(S78) +where ˆνc is the density operator of c electrons (defined in Eq. S11) and Sνc denotes the set of all possible fillings of c electrons +that not equals to νc. The free energy is +F = − 1 +β log(Z) +and the energy is +⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩ = EH′′ +c = lim +β→∞ F +We calculate the free energy by performing expansion in J. To do an expansion in powers of J, we separate ˆH′′ +c [ϑ] into two +parts, J independent and J dependent part. +ˆH′′ +c [ϑ] = ˆH0 + ˆH1 +ˆH0 = ˆHc +ˆH1 = −J +� +Rq +� +µν +� +ξ=± +e−iq·Rθf,ξ +µν (R) : ˆΘ(c′′,ξ) +µν +(q) : +(S79) +Then we expand the partition functions as +Z = +∞ +� +n=0 +Zn +Zn = (−1)n +� β +0 +dτ1 +� β +τ1 +dτ2... +� β +τn−1 +dτnTr +� +e−(β−τn) ˆ +H0 ˆH1e−(τn−τn−1) ˆ +H0...e−(τ2−τ1) ˆ +H0 ˆH1e−τ1 ˆ +H0P +� +(S80) +We now prove Eq. S80. For convenience, we introduce an artificial parameter g = 1. We first perform a Suzuki–Trotter +expansion. +Z = lim +N→∞ Tr[ +N +� +n=i +e−∆τ( ˆ +H0+g ˆ +H1)P] +(S81) +where ∆τ = β/N. We next expand partition functions in powers of g via a Taylor expansion (Z = � +n Zn). The n-th order +term is +Zn = lim +N→∞ +1 +n! +∂n +∂gn Z +���� +g=0 += (−∆τ)n +n! +n! +� +i1≤i2...≤in +Tr[ +� +N +� +n=in+1 +e−∆τ ˆ +H0 +� +ˆH1 +� +in +� +n=in−1+1 +e−∆τ ˆ +H0 +� +ˆH1... ˆH1 +� +i1 +� +n=1 +e−∆τ ˆ +H0 +� +P] += lim +N→∞(−∆τ)n +� +i1≤i2...≤in +Tr[e−∆τ(N−in+1) ˆ +H0 ˆH1e−∆τ(in−in−1) ˆ +H0 ˆH1... ˆH1 +� +e−∆τi1 ˆ +H0 +� +P] + +18 +where i1, ..., in denotes the time-slices where the derivative has acted on. Since we have assumed a specific order of i1 ≤ i2... ≤ +in, permutation of i1, ..., in leads to an additional n! factor. In the limit of N → ∞, we reach Eq. S80. +Since ˆH1 has a coefficient J, Zn ∝ Jn. We truncate the expansion to the second order: Z = Z0 + Z1 + Z2 + o(J2). The +free energy becomes +F = − 1 +β log[Z0 + Z1 + Z2 + o(J2)] = − 1 +β log[Z0] − 1 +β +Z1 +Z0 ++ 1 +2 +1 +β [Z1 +Z0 +]2 − 1 +β +Z2 +Z0 ++ o(J2) +(S82) +Now we want to compute Z0, Z1/Z0, Z2/Z0. Before moving on, we first introduce the following notation. For an operator +ˆO, its expectation value with respect to the Hamiltonian ˆH0(= ˆHc) at inverse temperature β and filling νc = ν − νf is +⟨ ˆO⟩0 := 1 +Z0 +Tr[e−βH0OP]. +where Z0 = Tr log[e−βH0P]. The ground state information can be obtained by setting β → ∞. The operator in the interaction +picture ˆO(τ) is defined as +ˆO(τ) := eτH0 ˆOe−τH0 +Now, we calculate Z0, Z1/Z0, Z2/Z0 +1. +Z0 +Z0 = Tr[e−βH0P]. This is the partition of the conduction electrons at filling νc with J = 0. +2. +Z1/Z0 +Z1 +Z0 += − 1 +Z0 +� β +0 +dτTr +� +e−β ˆ +H0[eτ ˆ +H0 ˆH1e−τ ˆ +H0]P +� += − 1 +Z0 +� β +0 +dτTr +� +e−β ˆ +H0 ˆH1(τ)P +� += − +� β +0 +⟨ ˆH1(τ)⟩0dτ +Using the explicit expression of ˆH1, we find +Z1 +Z0 += J +� +Rqk +� +µν +� +ξ=± +� +a,a′=3,4 +� +ηη′ss′ +e−iq·Rδξ,(−1)a−1η +θ(f,ξ) +µν +(R) +2NM +� β +0 +dτ⟨: c† +k+qaηs(τ)Θµν,c′′ +aηs,a′η′s′cka′η′s′(τ) :⟩0dτ +where c† +kaηs(τ) is defined as ckaηs(τ) = eτH0ckaηse−τH0. Now we want to evaluate +1 +2NM +� +k,q +� +a,a′=3,4 +� +ηη′ss′ +δξ,(−1)a−1η +� β +0 +dτ⟨: c† +k+qaηs(τ)Θµν,c′′ +aηs,a′η′s′cka′η′s′(τ) :⟩0 +Due to the space translation symmetry of H0, only q = 0 gives a non-zero contribution. In addition, for any operator ˆO, +⟨ ˆO(τ)⟩0 is τ-independent due to the imaginary-time translational symmetry. This is because ⟨ ˆO(τ)⟩ = ⟨eτ ˆ +H0 ˆOe−τ ˆ +H0⟩0 = +Tr[e−(β−τ) ˆ +H0 ˆOe−τ ˆ +H0]/Z0 = Tr[e−β ˆ +H0 ˆO]/Z0 = ⟨ ˆO⟩0. Therefore, we have +1 +2NM +� +k +� +a,a′=3,4 +� +ηη′ss′ +δξ,(−1)a−1η +� β +0 +dτ⟨: c† +k+qaηs(τ)Θµν,c′′ +aηs,a′η′s′cka′η′s′(τ) :⟩0 +=δq,0β⟨: +1 +2NM +� +k +� +a,a′=3,4 +� +ηη′ss′ +δξ,(−1)a−1ηc† +kaηsΘµν,c′′ +aηs,a′η′s′cka′η′s′ :⟩0 +=β⟨: ˆΘ(c′′,ξ) +µν +(q = 0) :⟩ + +19 +This term gives the expectation value of chiral U(4) moment of a = 3, 4 orbitals. +Since the fermion-bilinear Hamiltonian +ˆH0 has chiral U(4) symmetry, not all the operators have a non-zero ex- +pectation value. +We utilize the following statement: +Given operator +ˆO, if there exists a chiral U(4) transforma- +tion g, such that g ˆOg−1 += +− ˆO, then ⟨ ˆO⟩0 += +⟨g ˆOg−1⟩0 += +−⟨ ˆO⟩0 += +0. +We prove that, for ˆΘ(c′′,ξ) +µν +(q) += +� +k,aηs,a′η′s′ ck+q,aηsΣ(µν,c′′) +aηs,a′η′s′ck,a′η′s′δξ,(−1)a−1η/(2NM) with µν ̸= 00, we can always find such chiral U(4) transforma- +tion. +e−iπ ˆΘz0 +: +ˆΘ(c′′,ξ) +xµ +(q), ˆΘ(c′′,ξ) +yµ +(q) → −ˆΘ(c′′,ξ) +xµ +(q), −ˆΘ(c′′,ξ) +yµ +(q) +e−iπ ˆΘx0 +: +ˆΘ(c′′,ξ) +zµ +(q) → −ˆΘ(c′′,ξ) +zµ +(q) +e−iπ ˆΘ0z +: +ˆΘ(c′′,ξ) +0x +(q), ˆΘ(c′′,ξ) +0y +(q) → −ˆΘ(c′′,ξ) +0x +(q), −ˆΘ(c′′,ξ) +0y +(q) +e−iπ ˆΘ0x +: +ˆΘ(c′′,ξ) +0z +(q) → −ˆΘ(c′′,ξ) +0z +(q) +(S83) +Therefore, +⟨: ˆΘ(c′′,ξ) +µν +(q) :⟩0 = 0 +, +µν ̸= 00 +(S84) +Here, we comment that, in order to prove ⟨O⟩0 = 0, we require the chiral U(4) transformation to flip the sign of the operator. +Suppose we consider an chiral U(4) transformation that changes ˆO(one components of U(4) momentum) to ˆO′ (another U(4) +momentum). Symmetry only indicates ⟨ ˆO⟩0 = ⟨ ˆO′⟩0, but does not indicate the expectation value goes to zero. +We now conclude the only non-zero component is +δq,0β⟨: ˆΘ(c′′,ξ) +00 +(q = 0) :⟩ = δq,0β +� +a=3,4 +1 +2NM +� +ξ,s +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 +(S85) +By only including the non-vanishing contributions. +Z1 +Z0 += Jβ +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 +(S86) +3. +Z2/Z0 +Z2/Z0 = 1 +Z0 +� β +0 +dτ2 +� τ2 +0 +dτ1Tr +� +e−β ˆ +H0(eτ2 ˆ +H0 ˆH1e−τ2 ˆ +H0)(eτ1 ˆ +H0 ˆH1e−τ1 ˆ +H0)P +� += 1 +Z0 +� β +0 +dτ2 +� τ2 +0 +dτ1Tr +� +e−β ˆ +H0 ˆH1(τ2) ˆH1(τ1)P +� += +� β +0 +dτ2 +� τ2 +0 +dτ1⟨ ˆH1(τ2) ˆH1(τ1)⟩0 +(S87) +Using the explicit formula of ˆH1(Eq. S79), we have +Z2/Z0 =J2 +� +R,R2,q,q2,µν,µ2ν2 +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µ2ν2 (R2)e−iq·R−iq2·R2 +� β +0 +dτ2 +� τ2 +0 +dτ1⟨: ˆΘc′′,ξ +µν (q, τ2) :: ˆΘc′′,ξ2 +µ2ν2 (q2, τ1) :⟩0 +(S88) +We note that +⟨: ˆΘc′′,ξ +µν (q) :: ˆΘc′′,ξ2 +µ2ν2 (q2) :⟩0 ∝ δµ,µ2δν,ν2 +(S89) +To prove Eq. S89, we first define +ˆNµν,µ2ν2 =: ˆΘc′′,ξ +µν (q) :: ˆΘc′′,ξ2 +µ2ν2 (q2) : . +(S90) +We now show, for µν ̸= µ2ν2, there exists an chiral-U(4) transformation g, such that g ˆNµν,µ2ν2g−1 = − ˆNµν,µ2ν2, Then +⟨ ˆNµν,µ2ν2⟩0 = −⟨ ˆNµν,µ2ν2⟩0 = 0. We consider different cases of µν, µ2ν2, and show that in the following cases ⟨ ˆNµν,µ2ν2⟩0 = +0 if µν ̸= µ2ν2. + +20 +• µν = 00, µ2ν2 ̸= 00 or µν ̸= 00, µ2ν2 = 00. We take the case of µν = 00, µ2ν2 ̸= 00 as an example. : ˆOc′′,ξ +µν=(00)(q2) : +is invariant under all the chiral U(4) transformation. However, : ˆΘc′′,ξ2 +µ2ν2 (q2, τ1) : would change sign under certain chiral +U(4) transformation (as shown in Eq.S83). Therefore, the expectation value of ˆNµν,µ2ν2 =: ˆΘc′′,ξ +µν (q) :: ˆΘc′′,ξ2 +µ2ν2 (q2) : goes +to zero. +• µ ̸= µ2 and µ ̸= 0, µ2 ̸= 0. For g = e−iπ ˆΘµ0, we have ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = −⟨ ˆNµν,µ2ν2⟩0 = 0. +• µ ̸= µ2 and µ = 0, µ2 ̸= 0. For g = e−iπ ˆΘµ30 with µ3 ̸= 0, µ3 ̸= µ2, we find ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = +−⟨ ˆNµν,µ2ν2⟩0 = 0. +• µ ̸= µ2 and µ ̸= 0, µ2 = 0. For g = e−iπ ˆΘµ30 with µ3 ̸= 0, µ3 ̸= µ, we find ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = +−⟨ ˆNµν,µ2ν2⟩0 = 0. +• ν ̸= ν2 and ν ̸= 0, ν2 ̸= 0. For g = e−iπ ˆΘ0ν, we ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = −⟨ ˆNµν,µ2ν2⟩0 = 0. +• ν ̸= ν2 and ν = 0, ν2 ̸= 0. For g = e−iπ ˆΘ0ν3 with ν3 ̸= 0, ν3 ̸= ν2, we find ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = +−⟨ ˆNµν,µ2ν2⟩0 = 0. +• ν ̸= ν2 and ν ̸= 0, ν2 = 0. For g = e−iπ ˆΘ0ν3 with ν3 ̸= 0, ν3 ̸= ν, we find ⟨ ˆNµν,µ2ν2⟩0 = ⟨g ˆNµν,µ2ν2g−1⟩0 = +−⟨ ˆNµν,µ2ν2⟩0 = 0. +Therefore, the only nonvanishing term comes from the case with µν = µ2ν2. Then we can rewrite Eq. S88 as +Z2/Z0 =J2 +� +R,R2,µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R−iq2·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +µν (q, τ1) :: ˆΘc′′,ξ2 +µν +(q2, τ2) :⟩ +One can further show the expectation value for all the µν with µν ̸= 00 are the same. This is because, for a chiral U(4) +transformation g and operator ˆO, we have +⟨g−1 ˆOg⟩0 = ⟨ ˆO⟩0 +(note that ˆH0 has chiral U(4) symmetry). +We let ˆOµν +ξ,ξ2,q,q2,τ1,τ2 :=: ˆΘc′′,ξ +µν (q, τ1) :: ˆΘc′′,ξ2 +µν +(q2, τ2) :. +We then show +ˆOµν +ξ,ξ2,q,q2,τ1,τ2 with different µν index are connected by chiral U(4) symmetry (except for µν = 00). +This is because +{ˆΘc′′,ξ +µν (q, τ)}µν ̸= 00 form a irreducible representation of chiral SU(4) group. We can find a chiral SU(4) transformation +g, such that g : ˆΘc′′,ξ +µν (q, τ) → ˆΘc′′,ξ +0z (q, τ) and then +g : ˆOµν +ξ,ξ2,q,q2 → ˆO0z +ξ,ξ2,q,q2 +(S91) +, where we have pick µν = 0z as a representative. To show it we first introduce the following chiral transformations: +eiπ/2 ˆΘy0 : ˆOxµ +ξ,ξ2,q,q2 → ˆOzµ +ξ,ξ2,q,q2 +, +e−iπ/2 ˆΘx0 : ˆOyµ +ξ,ξ2,q,q2 → ˆOzµ +ξ,ξ2,q,q2 +(S92) +eiπ/2 ˆΘ0y : ˆOµx +ξ,ξ2,q,q2 → ˆOµz +ξ,ξ2,q,q2 +, +e−iπ/2 ˆΘ0x : ˆOµy +ξ,ξ2,q,q2 → ˆOµz +ξ,ξ2,q,q2 +(S93) +e−iπ ˆΘxx/2+iπ ˆΘyy/2 : ˆO0z +ξ,ξ2,q,q2 → ˆOz0 +ξ,ξ2,q,q2 +, +e−iπ/2 ˆΘx0+iπ/2 ˆΘxz : ˆOzz +ξ,ξ2,q,q2 → ˆOz0 +ξ,ξ2,q,q2 +(S94) +Then +ˆOxµ +ξ,ξ2,q,q2, ˆOyµ +ξ,ξ2,q,q2 +Eq. S92 +−−−−→ ˆOzµ +ξ,ξ2,q,q2 +Eq. S93 +−−−−→ ˆOzz +ξ,ξ2,q,q2 or ˆOz0 +ξ,ξ2,q,q2 +Eq. S94 +−−−−→ ˆOz0 +ξ,ξ2,q,q2 +ˆOµx +ξ,ξ2,q,q2, ˆOµy +ξ,ξ2,q,q2 +Eq. S93 +−−−−→ ˆOµz +ξ,ξ2,q,q2 +Eq. S92 +−−−−→ ˆOzz +ξ,ξ2,q,q2 or ˆO0z +ξ,ξ2,q,q2 +Eq. S94 +−−−−→ ˆOz0 +ξ,ξ2,q,q2 +ˆO0z +ξ,ξ2,q,q2, ˆOzz +ξ,ξ2,q,q2 +Eq. S94 +−−−−→ ˆOz0 +ξ,ξ2,q,q2 +(S95) +We thus prove Eq. S91. Therefore +⟨: ˆΘc′′,ξ +µν (q, τ1) :: ˆΘc′′,ξ2 +µν +(q2, τ2) :⟩0 = ⟨ : ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(q2, τ2) :⟩0, +whenever µν ̸= 00 +(S96) +(Here we pick µν = 0z as a representative.) + +21 +Now the expression becomes +Z2/Z0 =J2 � +R,R2 +� +q,q2 +� +µν̸=00 +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R−iq2·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(q2, τ2) :⟩0 ++ J2 � +R,R2 +� +q,q2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R−iq2·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(q2, τ2) :⟩0 +(S97) +Since ˆH0 is also translational invariant, the moment conservation requires q = −q2. This leads to +Z2/Z0 =J2 � +R,R2 +� +q +� +µν̸=00 +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 ++ J2 � +R,R2 +� +q +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +(S98) +Then we only need to evaluate the following four-fermion correlation functions: +⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 +where +: ˆΘc′′,ξ +00 (q, τ1) := +� +k, +� +a1=3,4 +� +ηs +1 +2NM +δξ,(−1)a−1η : c† +aηs,k+q(τ1)caηs,k(τ1) : +: ˆΘc′′,ξ +0z (q, τ1) := +� +k, +� +a1=3,4 +� +ηs +s +1 +2NM +δξ,(−1)a−1η : c† +aηs,k+q(τ1)caηs,k(τ1) : +one is the U(1) charge, the other one is the spin moment along z direction of a = 3, 4 orbital with index ξ, momentum q. +We will evaluate the four-fermion correlation functions using Wick’s theorem. To do so we first introduce single-particle +Green’s function +Gaa′,ηη′,ss′(k, τ1 − τ2) = −⟨Tτckaηs(τ1)c† +ka′η′s′(τ2)⟩0 +where Tτ denotes time-ordering. We now explore the structure of the single-particle Green’s function. The single-particle +Green’s functions are calculated with respect to the ground state of Hamiltonian ˆH0 = ˆHc (We also calculate the single-particle +Green’s function with respect to the symmetry broken state in Sec. S8 and Sec. S13). The ground state of non-interacting +conduction-electron Hamiltonian ˆHc (Eq. S2) always has chiral-U(4) symmetry, and thus has SU(2) spin and U(1) valley +symmetries. The corresponding single-particle Green’s function will also have SU(2) spin and U(1) valley symmetries. Due to +the SU(2) spin symmetry, Gk,aa′,ηη′,ss′(τ1 − τ2) ∝ δss′ and Gk,aa′,ηη′,↑↑(τ1 − τ2) = Gk,aa′,ηη′,↓↓(τ1 − τ2) Due to the U(1) +valley symmetry, Gk,aa′,ηη′,ss′ ∝ δηη′. +This leads to a simplified notation of the Green’s function: +Gaa′,ηη′,ss′(k, τ1 − τ2) = δηη′δss′Gaa′,η(k, τ1 − τ2) +Gaa′,η(k, τ1 − τ2) = −⟨Tτckaηs(τ1)c† +ka′ηs(τ2)⟩0. +The analytical formula of the Green’s function at zero temperature and infinity momentum cutoff is given in Sec. S9. + +22 +Now we are in the position to calculate two correlation functions. The first correlation function +⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 += +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨: c† +aηs,k+q(τ1)caηs,k(τ1) :: c† +a2η2s2k′−q(τ2)ca2η2s2k′(τ2) :⟩0 +[Use Wick’s theorem] += +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨: c† +aηs,k+q(τ1)caηs,k(τ1) :⟩0⟨: c† +a2η2s2k′−q(τ2)ca2η2s2k′(τ2) :⟩0 +− +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨ca2η2s2k′(τ2)c† +aηs,k+q(τ1)⟩0⟨caηsk(τ1)c† +a2η2s2k′−q(τ2)⟩0 +[Rewrite 1st term] +=⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +− +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨ca2η2s2k′(τ2)c† +aηs,k+q(τ1)⟩0⟨caηsk(τ1)c† +a2η2s2k′−q(τ2)⟩0 +[Rewrite second term via single-particle Green’s function] +=⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +− +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2)δη2,ηδs,s2δk′,k+q +[Further simplification] +=⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +− +� +k +� +a1,a2=3,4 +� +ηs +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2) +(S99) +The second correlation functions +⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 += +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1η2ss2⟨: c† +aηs,k+q(τ1)caηs,k(τ1) :: c† +a2η2s2k′−q(τ2)ca2η2s2k′(τ2) :⟩0 +[Use Wick’s theorem] += +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +ss2δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨: c† +aηs,k+q(τ1)caηs,k(τ1) :⟩0⟨: c† +a2η2s2k′−q(τ2)ca2η2s2k′(τ2) :⟩0 +− +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +ss2δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨ca2η2s2k′(τ2)c† +aηs,k+q(τ1)⟩0⟨caηsk(τ1)c† +a2η2s2k′−q(τ2)⟩0 +[Rewrite 1st term] +=⟨: ˆΘc′′,ξ +0z (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 +− +� +k,k′ +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +ss2δξ,(−1)a−1ηδξ2,(−1)a2−1η2⟨ca2η2s2k′(τ2)c† +aηs,k+q(τ1)⟩0⟨caηsk(τ1)c† +a2η2s2k′−q(τ2)⟩0 +[1st term goes to zero (as shown in Eq. S84). Rewrite the second term via single-particle Green’s function] += − +� +k,k′ +� +a1,a2=3,4 +� +ηs +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηss2Ga2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2)δη2,ηδs,s2δk′,k+q +[Further simplification] += − +� +k +� +a1,a2=3,4 +� +ηs +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2) +(S100) + +23 +Comparing two correlation functions (Eq.S99 and Eq.S100), we find +⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 − ⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 = ⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 += − +� +k +� +a1,a2=3,4 +� +ηη2ss2 +1 +2NM +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2) +(S101) +We then define χc(q, τ1 − τ2) +χc(q, τ1 − τ2, ξ, ξ2) +=⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 − ⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 = ⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 += − +� +k +� +a1,a2=3,4 +� +ηs +1 +2 +1 +2NM +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, τ2 − τ1)Gaa2,η(k, τ1 − τ2) +(S102) +such that +⟨: ˆΘc′′,ξ +00 (q, τ1) :: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 = χc(q, τ1 − τ2, ξ, ξ2) + ⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +⟨: ˆΘc′′,ξ +0z (q, τ1) :: ˆΘc′′,ξ2 +0z +(−q, τ2) :⟩0 = χc(q, τ1 − τ2, ξ, ξ2) +(S103) +The Fourier transformation of χc(q, τ1 − τ2) is defined as +χc(R, τ1 − τ2, ξ, ξ2) = +1 +NM +� +k +χc(q, τ1 − τ2, ξ, ξ2)eiq·R +(S104) +The analytical formulas of χc(R, τ, ξ, ξ2) at zero temperature are derived in Sec. S10, Eqs. S564, S565 and are given below +χc(R, τ, ξ, ξ) ≈ +π2 +A2 +MBZ +|v⋆τ|2 +(|v⋆τ|2 + r2)3 +χc(R, τ, ξ, −ξ) ≈ − +π2M 2 +4A2 +MBZ|v⋆|2 +r4 +� +|v⋆τ|2 + r2 +�3 +(S105) +where we perform an expansion in powers of M and truncate to second orders. +Combining Eq. S98 and Eq. S103, +Z2/Z0 =J2 � +R,R2 +� +µν̸=00 +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) +� β +0 +dτ1 +� τ1 +0 +dτ2χc(R2 − R, τ1 − τ2, ξ, ξ2) ++ J2 � +R,R2 +� +µν=00 +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) +� β +0 +dτ1 +� τ1 +0 +dτ2χc(R2 − R, τ1 − τ2, ξ, ξ2) ++ J2 +� +R,R2,Q +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +=J2 � +R,R2 +� +µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) +� β +0 +dτ1 +� τ1 +0 +dτ2χc(R2 − R, τ1 − τ2, ξ, ξ2) ++ J2 � +R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 + +24 +The last term can be written as +J2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +=J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 ++ J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +=J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 ++ J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +τ2 +dτ1 +� β +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +=J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 ++ J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +τ1 +dτ2 +� β +0 +dτ1⟨: ˆΘc′′,ξ +00 (q, τ2) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ1) :⟩0 +=J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 ++ J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e+iq·R−iq·R2 +� β +τ1 +dτ2 +� β +0 +dτ1⟨: ˆΘc′′,ξ +00 (−q, τ2) :⟩0⟨: ˆΘc′′,ξ2 +00 +(q, τ1) :⟩0 +=J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 ++ J2 1 +2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 +� β +τ1 +dτ2 +� β +0 +dτ1⟨: ˆΘc′′,ξ +00 (−q, τ2) :⟩0⟨: ˆΘc′′,ξ2 +00 +(q, τ1) :⟩0 +=J2 +� +q,R,R2 +� +ξ=±,ξ2=± +θ(f,ξ) +00 +(R)θ(f,ξ2) +00 +(R2)e−iq·R+iq·R2 1 +2 +� β +0 +dτ1 +� β +0 +dτ2⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0⟨: ˆΘc′′,ξ2 +00 +(−q, τ2) :⟩0 +=1 +2 +� � β +0 +dτ +� +q,R +� +ξ=± +θ(f,ξ) +00 +(R)e−iq·R⟨: ˆΘc′′,ξ +00 (q, τ1) :⟩0 +�2 += 1 +2 +�Z1 +Z0 +�2 +Therefore +Z2/Z0 = J2 � +R,R2 +� +µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) +� β +0 +dτ1 +� τ1 +0 +dτ2χc(R2 − R, τ1 − τ2, ξ, ξ2) + 1 +2(Z1 +Z0 +)2 +(S106) +4. +Free energy: F +In summary +Z0 = Tr[e−βH0P] +Z1/Z0 = Jβ +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 +Z2/Z0 = J2 +� +q,R,R2 +� +µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2χc(q, τ1 − τ2, ξ, ξ2) + 1 +2(Z1 +Z0 +)2 + +25 +Combining Eqs. S82, S86 and S106, we have +F = − 1 +β log[Z0] − 1 +β +Z1 +Z0 ++ 1 +2 +1 +β [Z1 +Z0 +]2 − 1 +β +Z2 +Z0 ++ o(J2) +F =F0 − J +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 + 1 +2β (Z1 +Z0 +)2 +− 1 +β J2 +� +q,R,R2 +� +µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2χc(q, τ1 − τ2, ξ, ξ2) − 1 +2β (Z1 +Z0 +)2 +=F0 − J +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 +− 1 +β J2 +� +q,R,R2 +� +µν +� +ξ=±,ξ2=± +θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2)e−iq·R+iq·R2 +� β +0 +dτ1 +� τ1 +0 +dτ2χc(q, τ1 − τ2, ξ, ξ2) +where F0 = − 1 +β log[Z0] denotes the free energy of the conduction electron system at J = 0 (with Hamiltonian ˆH0 = ˆHc). +Finally we set β = ∞. The energy is +⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩ += lim +β→∞ F +=E0 − J +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 + +� +RR′,ξξ2 +JRKKY (R − R2, ξ, ξ2)θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) ++ o(J2) +where E0 = limβ→∞ F0 is the energy of conduction electron system with Hamiltonian ˆH0 = ˆHc and filling νc. The RKKY +type of interactions are defined as +JRKKY (R − R2, ξ, ξ2) = lim +β→∞ +� +q +− 1 +β J2 +� β +0 +dτ1 +� τ1 +0 +dτ2χc(q, τ1 − τ2, ξ, ξ2)e−iq·R+iq·R2 += lim +β→∞ +� +q +− 1 +β J2 1 +2 +� β +0 +dτ1 +� β +0 +dτ2χc(q, τ1 − τ2, ξ, ξ2)e−iq·R+iq·R2 +[Using∆τ = τ2 − τ2 and χ(q, τ, +β, ξ, ξ2) = χ(q, τ, ξ, ξ2)] += lim +β→∞ +� +q +− 1 +β J2 1 +2 +� β +0 +dτ1 +� β/2 +−β/2 +d∆τχc(q, ∆τ, ξ, ξ2)e−iq·R+iq·R2 +[Replace ∆τ with τ ] += lim +β→∞ +� +q +−J2 +2 +� β/2 +−β/2 +dτχc(q, τ, ξ, ξ2)e−iq·(R−R2) +(S107) +where χc(q, τ, ξ, ξ2) is defined in Eq. S102. +In summary, for a given ϑ, νf, ν, the energy of the proposed trial wavefunction can be written as +E[ϑ, νf, ν] = ⟨Ψ[ϑ, νc]|⟨ϑ| ˆH|ϑ⟩|Ψ[ϑ, νc]⟩ += NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc + ⟨Ψ[ϑ, νc]| ˆH′′ +c [ϑ]|Ψ[ϑ, νc]⟩ +≈ NM +U1 +2 ν2 +f + NMWνfνc + NM +V0 +2Ω0 +νc + E0 +− J +� +R +� +ξ=± +θf,ξ +00 (R) +2NM +� +k,ηs +� +a=3,4 +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 ++ +� +RR′,ξξ2,µν +JRKKY (R − R2, ξ, ξ2)θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) + +26 +where E0 is the ground state energy of one-particle Hamiltonian ˆH0 = ˆHc with filling νc. The expectation value ⟨⟩0 is taken for +the ground state of ˆH0 at filling νc. +We note that, for both M = 0 or M ̸= 0, the electrons equally fill two types of fermion(ξ = ±). When νc = 0 the filling of +each type of fermion is zero. Therefore. +1 +NM +� +kaηs +δξ,(−1)a−1η⟨: c† +kaηsckaηs :⟩0 = 0, +when νc = 0 +In other words the linear J term in E[ϑ, νf, ν] vanishes. Therefore, for fixed integer fillings νf = ν = 0, −1, −2, the ground +state energy is determined by RKKY interactions. We define the energy coming from RKKY interactions as +ERKKY [ϑ] := +� +RR′,ξξ2 +JRKKY (R − R2, ξ, ξ2)θ(f,ξ) +µν +(R)θ(f,ξ2) +µν +(R2) +(S108) +Using the analytical expressions of χc(R, τ, ξ, ξ2) derived in section S10, Eqs. S564, S565 and the definition of RKKY inter- +actions (Eq. S107, we find the analytical expression of JRKKY (R, ξ, ξ2). JRKKY (R − R2, ξ, ξ2) at νc = 0, zero temperature +β = ∞ and infinity momentum cutoff Λc = ∞: +JRKKY (R, ξ, ξ) = − +π3 +16A2 +MBZ|v⋆| +1 +|R|3 + o(M 2) +JRKKY (R, ξ, −ξ) = +3π3M 2 +32A2 +MBZ|v⋆|3|R| + o(M 2) +(S109) +where AMBZ is the area of the first moir´e Brillouin zone and we expand the results in powers of M and keep the leading-order +contributions. +Here, we comment that a non-zero M hybridizes two conduction electrons with opposite ξ indices. Therefore, it induces an +RKKY interaction between two chiral U(4) moments with opposite ξ indices. However, such RKKY interaction still preserves +the chiral U(4) symmetry but breaks flat U(4) symmetry, as M ̸= 0, v′ +⋆ = 0 preserves the chiral U(4) but not flat U(4) +symmetry. +At νf = ν = 0, −1, −2, we observe +• JRKKY (R, ξ, ξ) decays as 1/|R|3 and is always ≤ 0, in other words, is ferromagnetic. +• JRKKY (R, ξ, −ξ) decays as 1/|R| and is always ≥ 0 (antiferromagnetic). In addition, it goes to zero at M = 0. +• In the derivation of the above analytical formula, we take the momentum cutoff to be infinity (therefore, it diverges at +r = |R| = 0. This divergence will be regularized to a finite value once we take a finite cutoff in momentum space Λc). +In addition, the energy function in Eq. S108 can be described by the following effective Hamiltonian +ˆHRKKY = +� +RR′,ξξ2 +JRKKY (R − R2, ξ, ξ2) : ˆΘf,ξ +µν (R) :: ˆΘf,ξ2 +µν (R2) : . +(S110) +where we simply replace θ(f,ξ) +µν +(R) with the corresponding operator ˆΘf,ξ +µν (R). +D. +Path integral formula +The effective Hamiltonian in Eq. S110 can also be derived by directly integrating out conduction c-electrons. The action of +the system is +S = Sf + Sc + SJ +Sf = +� +R,αηs +� β +0 +dτf † +R,αηs(τ)(∂τ − µ)fR,αηs(τ) + +� β +0 +HU(τ)dτ +Sc = +� +k,aηs +� β +0 +dτc† +k,aηs(τ)(∂τ − µ)ck,aηs(τ) + +� β +0 +(Hc(τ) + HV (τ) + HW (τ))dτ +SJ = +� β +0 +ˆHJ(τ)dτ . +(S111) + +27 +Here, fR,αηs(τ), ck,aηs(τ) are τ dependent Grassmann numbers. HU(τ), Hc(τ), HV (τ), HW (τ) are defined by replacing +fR,αηs, ck,aηs with fR,αηs(τ), ck,aηs(τ) in the expressions of ˆHU, ˆHc, ˆHV , ˆHW , ˆHJ shown in Eq. S34 and Eq. S63. µ is +the chemical potential. The partition function then reads +Z = +� +D[f † +R,αηs, fR,αηs, c† +k,a,η,s, ck,a,η,s]e−Sf +Sc+SJδ( +� +αηs +f † +R,αηs(τ)fR,αηs(τ) − 4 − νf)) += +� +D[f † +R,αηs, fR,αηs, c† +k,a,η,s, ck,a,η,s]e−Sf +Sc+SJ +� +D[λR(τ)]e−i +� β +0 λR(τ)[� +αηs f † +R,αηs(τ)fR,αηs(τ)−4−νf )dτdτ += +� +D[f † +R,αηs, fR,αηs, c† +k,a,η,s, ck,a,η,s, λR(τ)]e−Sf +Sc+SJ−i +� β +0 λR(τ)[� +αηs f † +R,αηs(τ)fR,αηs(τ)−4−νf ) +(S112) +where we use the Dirac Delta function δ(x) to fix the filling of f-electron and then replace the Dirac-Delta function with a +Lagrangian multiplier in the second line. +We combine the Lagrangian multiplier term and Sf, and then define the following +new Sf +Sf = +� +R,αηs +� β +0 +dτf † +R,αηs(τ)∂τfR,αηs(τ)dτ + i +� +R +� β +0 +λR(τ)[ +� +αηs +f † +R,αηs(τ)fR,αηs(τ) − 4 − νf)]dτ +(S113) +where we drop the constant contribution from ˆHU and µ (Note that the filling of f is fixed). We also comment that the Lagrangian +multiplier λR(τ) is only introduced to fix the filling of f electrons in the path integral formula, and will not be explicitly used in +the calculations of this section. +We can also rewrite HW as +HW = +� β +0 +Wνf +� +k,a,ηs +: c† +kaηsckaηs : dτ +where we have take Wa=1,2,3,4 = W. As for HV , we treat this term at the mean-field level: +HV → V0 +Ω0 +νc +� +η,s,a,k +c† +kaηscka′ηs − V0NM +2Ω0 +ν2 +c − V0 +Ω0 +νc +� +k +8 +At the mean-field level, the action of conduction electron only contains fermion bilinear term +Sc = +� +k,aηs +� β +0 +dτc† +k,aηs(τ)∂τck,aηs(τ)dτ + +� β +0 +� +η,s,a,a′,k +� +H(c,η) +a,a′ + (−µ + Wνf + V0 +Ω0 +νc)δa,a′ +� +c† +k,aηs(τ)ck,a′ηs(τ)dτ +− V0NM +2Ω0 +ν2 +c − (Wνf + V0 +Ω0 +νc) +� +k +8 +(S114) +where the constant term comes from the normal ordering. +Using Sf and Sc defined in Eq. S113 and Eq. S114, the partition function is written as +Z = +� +D[c† +k,aηs(τ), ck,aηs(τ)]D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Sf −Sc−SJ +We next integrate out conduction c-electrons: +Z = +� +D[c† +k,aηs(τ), ck,aηs(τ)]D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Sf −Sc−SJ +=Z0 +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Sf +� 1 +Z0 +� +D[c† +k,aηs(τ), ck,aηs(τ)]e−SJe−Sc +� +=Z0 +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Sf ⟨e−SJ⟩0 +=Z0 +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Seff + +28 +where the expectation value of a given operator ˆO is defined as +⟨ ˆO⟩0 = 1 +Z0 +� +D[c† +k,aηs(τ), ck,aηs(τ)]Oe−Sc , +and in the path integral formula, ˆO has been replaced by the corresponding c-numers or Grassmann numbers O. Z0 = +� +D[c† +k,aηs, ck,aηs]e−Sc and the effective action is defined as +Seff = Sf − log⟨e−SJ⟩0. +Since SJ ∝ J, we expand Seff in powers of J and truncate to the second order of J +Seff = Sf − log⟨e−SJ⟩0 = Sf − log +� +1 − ⟨SJ⟩0 + 1 +2⟨S2 +J⟩0 + o(J2) +� += Sf + ⟨SJ⟩0 − 1 +2 +� +⟨S2 +J⟩0 − (⟨SJ⟩0)2 +� ++ o(J2) +The first-order term is +⟨SJ⟩0 = −J +� β/2 +−β/2 +dτ +� +Rq +� +µν +� +ξ=± +e−iqR : ˆΘ(f,ξ) +µν +(R, τ) : ⟨: ˆΘ(c′′,ξ) +µν +(q, τ) :⟩0 +where +: ˆΘ(f,ξ) +µν +(R, τ) := +� +α,α′,η,η′,s,s′ +δξ,(−1)α−1η +2 +Θ(µν,f) +αηs,α′η′s′ +� +f † +R,αηs(τ)fR,α′η′s′(τ) − 1 +2δα,α′δη,η′δs,s′ +� +: ˆΘ(c′′,ξ) +µν +(q, τ) := +� +a,a′∈{3,4} +� +η,η′,s,s′ +δξ,(−1)a−1η +2NM +Θ(µν,c′′) +aηs,a′η′s′ +� +c† +k+q,aηs(τ)ck,a′η′s′(τ) − 1 +2δq,0δa,a′δη,η′δs,s′ +� +. +Due to the time translational symmetry, ⟨: ˆΘ(c′′,ξ) +µν +(τ, q) :⟩0 is time independent. Due to momentum conservation only q = 0 +components are finite. Then we have +⟨SJ⟩0 = −J +� β/2 +−β/2 +dτ +� +R +� +µν +� +ξ=± +: ˆΘ(f,ξ) +µν +(R, τ) : ⟨: ˆΘ(c′′,ξ) +µν +(0, 0) :⟩0 +As proved in Eq. S84, only µν = 00 component are finite, we have (at νc = 0) +⟨SJ⟩0 = −J +� β/2 +−β/2 +dτ +� +R +� +ξ=± +: ˆΘ(f,ξ) +00 +(R, τ) : ⟨: ˆΘ(c′′,ξ) +00 +(0, 0) :⟩0 +For the next order +⟨S2 +J⟩0 = J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +Rq,R2q2 +� +µν,µ2ν2 +� +ξ=±,ξ2=± +e−iqR−iq2R2⟨: ˆΘ(c′′,ξ) +µν +(q, τ) :: ˆΘ(c′′,ξ2) +µ2ν2 +(q2, τ2) :⟩0 +: ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µ2ν2 (R2, τ2) : +As described around Eq. S89, only terms with µν = µ2ν2 remain finite. Due to momentum conservation, only q = −q2 +component produces a non-zero contribution. Then we have +⟨S2 +J⟩0 = J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +Rq,R2 +� +µν,µ2ν2 +� +ξ=±,ξ2=± +e−iq(R−R2)⟨: ˆΘ(c′′,ξ) +µν +(q, τ) :: ˆΘ(c′′,ξ2) +µν +(−q, τ2) :⟩0 +: ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : + +29 +We calculate ⟨S2 +J⟩0 and ⟨S2 +J⟩0 at integer νf and νc = 0. The first-order term becomes +⟨SJ⟩0 = − J +� β/2 +−β/2 +dτ +� +R +� +ξ=± +: ˆΘ(f,ξ) +00 +(R, τ) : ⟨: ˆΘ(c′′,ξ) +00 +(0, 0) :⟩0 += − J +� β/2 +−β/2 +adτ +� +R +� +ξ=± +: ˆΘ(f,ξ) +00 +(R, τ) : +� +a=3,4,η,s +1 +2NM +⟨c† +aηsck,aηs − 1 +2⟩0 += − J +� β/2 +−β/2 +dτ +� +R +� +ξ=± +: ˆΘ(f,ξ) +00 +(R, τ) : 0 = 0 +(S115) +where we use the fact that the filling of c-electrons in orbital 3, 4 are 0 at νc = 0. +The second order term is +⟨S2 +J⟩0 = J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +Rq,R2 +� +µν +� +ξ=±,ξ2=± +e−iq(R−R2)⟨: ˆΘ(c′′,ξ) +µν +(τ, q) :: ˆΘ(c′′,ξ2) +µν +(τ2, −q) :⟩0 +: ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +For the same reason as we give in Eq. S96, all the correlators with µν ̸= 00 are the same as the one with µν = 0z +⟨: ˆΘ(c′′,ξ) +µν +(τ, q) :: ˆΘ(c′′,ξ2) +µν +(τ2, −q) :⟩0 = ⟨: ˆΘ(c′′,ξ) +0z +(τ, q) :: ˆΘ(c′′,ξ2) +0z +(τ2, −q) :⟩0, +µν ̸= 00 +(S116) +Then we have +⟨S2 +J⟩0 =J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +Rq,R2 +� +µν̸=00 +� +ξ=±,ξ2=± +e−iq(R−R2)⟨: ˆΘ(c′′,ξ) +0z +(τ, q) :: ˆΘ(c′′,ξ2) +0z +(τ2, −q) :⟩0 +: ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : ++ J2 +� β +0 +dτ +� β +0 +dτ2 +� +Rq,R2 +� +ξ=±,ξ2=± +e−iq(R−R2)⟨: ˆΘ(c′′,ξ) +00 +(τ, q) :: ˆΘ(c′′,ξ2) +00 +(τ2, −q) :⟩0 +: ˆΘ(f,ξ) +00 +(R, τ) :: ˆΘ(f,ξ2) +00 +(R2, τ2) : +Using the expression of χc(q, τ, ξ, ξ2) defined in Eq. S104, we find +⟨S2 +J⟩0 = J2 +NM +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +Rq,R2 +� +µν +� +ξ=±,ξ2=± +e−iq(R−R2)χc(q, τ − τ2, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +=J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +R,R2 +� +µν +� +ξ=±,ξ2=± +χc(R2 − R, τ − τ2, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +(S117) +Combining Eq. S115 and Eq. S117, we find the following effective action +Seff =⟨SJ⟩0 − 1 +2 +� +⟨S2 +J⟩0 − (⟨SJ⟩0)2 +� +=Sf − 1 +2J2 +� β/2 +−β/2 +dτ +� β/2 +−β/2 +dτ2 +� +R,R2 +� +µν +� +ξ=±,ξ2=± +χc(R2 − R, τ − τ2, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +(S118) +where χc is the same correlation function given before in Eq. S105. +We next take the low-frequency limit of χc. We first perform Fourier transformation +χc(R2 − R, iωn, ξ, ξ2) = +� β/2 +−β/2 +χc(R2 − R, τ, ξ, ξ2)eiωnτdτ +(S119) + +30 +with ωn = 2nπ/β. The inverse Fourier transformation gives +χc(R2 − R, τ, ξ, ξ2) = 1 +β +� +n +χc(R2 − R, iωn, ξ, ξ2)e−iωnτ +(S120) +We only keep the iωn = 0 contributions and then have +χc(R2 − R, τ, ξ, ξ2) ≈ 1 +β χc(R2 − R, iωn = 0, ξ, ξ2) = 1 +β +� β/2 +−β/2 +χc(R2 − R, τ ′, ξ, ξ2)dτ ′ +(S121) +Using Eq. S121, the second term of the effective action (Eq. S118) now becomes +− 1 +2J2 +� β +0 +dτ +� β +0 +dτ2 +� +Rq,R2 +� +µν,µ2ν2 +� +ξ=±,ξ2=± +e−iq(R−R2)χc(q, τ1 − τ2, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +≈ − 1 +2J2 1 +β +� β/2 +−β/2 +dτ ′ +� β +0 +dτ +� β +0 +dτ2 +� +Rq,R2 +� +µν,µ2ν2 +� +ξ=±,ξ2=± +e−iq(R−R2)χc(q, τ ′, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R, τ) :: ˆΘ(f,ξ2) +µν +(R2, τ2) : +≈ − J2β +2 +� +ξ,ξ2,µν.R,R2 +� 1 +β +� β +0 +: ˆΘ(f,ξ) +µν +(R, τ) : dτ +�� 1 +β +� 0 +β +: ˆΘ(f,ξ2) +µν +(R2, τ2) : dτ2 +� +� +q +� β/2 +−β/2 +χc(q, τ ′, ξ, ξ2)e−iq(R−R2)dτ ′ +=β +� +ξ,ξ2,µν,R,R2 +� 1 +β +� β/2 +−β/2 +: ˆΘ(f,ξ) +µν +(R, τ) : dτ +�� 1 +β +� β/2 +−β/2 +: ˆΘ(f,ξ2) +µν +(R2, τ2) : dτ2 +� +JRKKY (R − R2, ξ, ξ2) +where we have defined JRKKY (R − R2, ξ, ξ2) as +JRKKY (R − R2, ξ, ξ2) = − +� β/2 +−β/2 +J2 +2 +1 +β +� +q +χc(q, τ ′, ξ, ξ2)e−iq(R−R2)dτ ′ +Transforming the action to the Hamiltonian, we find the following RKKY interaction terms which is the same Hamiltonian as +we derived in Eq. S110 for both M = 0, M ̸= 0, at integer filling ν = νf = 0, −1, −2 +ˆHRKKY = +� +R,R2,µν,ξ,ξ2 +JRKKY (R − R2, ξ, ξ2) : ˆΘ(f,ξ) +µν +(R) :: ˆΘ(f,ξ2) +µν +(R2) : +(S122) +We can also rewrite the RKKY interaction with flat U(4) moment, using the relation we derived in Eq. S17 +ˆHRKKY = +� +R,R2,ξ +� +µν +JRKKY (R − R2, ξ, ξ2) : ˆΣ(f,ξ) +µν +(R) :: ˆΣ(f,ξ) +µν +(R2) : ++ +� +R,R2,ξ +� +µν∈{00,0x,0y,0z,z0,zx,zy,zz} +JRKKY (R − R2, ξ, −ξ) : ˆΣ(f,ξ) +µν +(R) :: ˆΣ(f,−ξ) +µν +(R2) : ++ +� +R,R2,ξ +� +µν∈{x0,xx,xy,xz,y0,yx,yy,yz} +−JRKKY (R − R2, ξ, −ξ) : ˆΣ(f,ξ) +µν +(R) :: ˆΣ(f,−ξ) +µν +(R2) : +(S123) +We observe that +• The interactions between two chiral U(4) moment with different ξ indices are antiferromagnetic. +• The interactions between two chiral (or two flat) U(4) momentum with the same ξ indices are ferromagnetic. +• The interactions between two chiral U(4) moments with the opposite ξ indices are antiferromagnetic. However, the +interactions between two flat U(4) moments are antiferromagnetic for the 00, 0x, 0y, 0z, z0, zx, zy, zz components and +are ferromagnetic for the x0, xx, xy, xz, y0, yx, yy, yz components. + +31 +• The interactions between two chiral (or flat) U(4) moments with opposite ξ indices are induced by M term and vanish in +the flat limit M = 0. +• We also notice that , in both the zero-hybridization model (γ = v′ +⋆ = 0) and the effective spin model in Eq. S122, the +density operator of f electrons with index ξ at each site R, ˆνξ +f(R) = � +αηs δξ,(−1)α−1η : f † +RαηsfRαηs :, commutes +with all the terms in the Hamiltonian and is good quantum number. The average filling of each ξ is also a good quantum +number: ˆνξ +f = � +R ˆνξ +f(R)/NM. Therefore, we can replace ˆνf +ξ (R), ˆνξ +f with real numbers νf +ξ (R), νξ +f respectively. +E. +Ground state of the zero hybridization model at M = 0, νf = 0, −1, −2 +We now discuss the ground state of RKKY Hamiltonian in Eq. S110 at M = 0 and νc = 0, νf = 0, −1, −2. At M = 0, +JRKKY (R − R2, ξ, −ξ) = 0. In this case we only have a ferromagnetic interaction between U(4) moments with same ξ. The +Hamiltonian becomes +ˆHRKKY +���� +M=0 += +� +R1,R2,µν +JRKKY (R1 − R2, ξ, ξ) : ˆΘf,ξ +µν (R) :: ˆΘf,ξ +µν (R2) : +(S124) +with JRKKY (R − R2, ξ, ξ) ≤ 0. We next introduce the bond operators +Bξ,ξ +R,R2 = +� +α,η,s +f † +R,αηsfR2,αηsδξ,(−1)α+1η +(S125) +where (Bξ,ξ +R,R)†Bξ,ξ +R,R = (ˆνξ +f(R) + 2)2. We note that the ˆνξ +f(R) is measured with respect to the charge neutrality point, and at +the charge neutrality point, each ξ sector has 2 f-electrons, so we have a +2 coming with ˆνξ +f(R). Via bond operators, we find +� +µν +: ˆΘf,ξ +µν (R) :: ˆΘf,ξ +µν (R2) : += +� +µν,αηs,α′η′s′,α2η2s2,α′ +2η′ +2s′ +2 +1 +4δξ,(−1)α−1ηδξ,(−1)α2−1η2 +(f † +R,αηsfR,α′η′s′ − 1 +2δα,α′δη,η′δs,s′)(f † +R2,α2η2s2fR2,α′ +2η′ +2s′ +2 − 1 +2δα2,α′ +2δη2,η′ +2δs2,s′ +2)Θµν,f +αηs,α′η′s′Θµν,f +α2η2s2,α′ +2η′ +2s′ +2 +=1 − 2 +4 × (ˆνξ +f(R) + ˆνξ +f(R2) + 4) + 1 +4 +� +αηs,α′η′s′ +δξ,(−1)α−1ηδξ,(−1)α′−1η′4f † +R,αηsfR,α′η′s′f † +R2,α′η′s′fR2,αηs += − 1 − 1 +2(ˆνξ +f(R) + ˆνξ +f(R2)) + +� +αηs,α′η′s′ +δξ,(−1)α−1ηδξ,(−1)α′−1η′f † +R,αηsfR,α′η′s′(δα′η′s′,αηs − fR2,αηsf † +R2,α′η′s′) += − 1 − 1 +2(ˆνξ +f(R) + ˆνξ +f(R2)) + (ˆνξ +f(R) + 2)+ +� +αηs,α′η′s′ +δξ,(−1)α−1ηδξ,(−1)α′−1η′f † +R,αηsfR2,αηs(δR,R2 − f † +R2,α′η′s′fR,α′η′s′) += − 1 − 1 +2(ˆνξ +f(R) + ˆνξ +f(R2)) + (ˆνξ +f(R) + 2) + 4δR,R2(ˆνξ +f(R) + 2) +− +� +αηs,α′η′s′ +δξ,(−1)α−1ηδξ,(−1)α′−1η′f † +R,αηsfR2,αηsf † +R2,α′η′s′fR,α′η′s′ += − 1 − 1 +2(ˆνξ +f(R) + ˆνξ +f(R2)) + (ˆνξ +f(R) + 2) + 4δR,R2(ˆνξ +f(R) + 2) − (Bξ,ξ +R2,R)†Bξ,ξ +R2,R +(S126) + +32 +Therefore, we can rewrite the Hamiltonian as +ˆHRKKY +���� +M=0 += +� +R1,R2 +JRKKY (R − R2, +1, +1) +� +2 + δR,R2(νf + 4) − +� +ξ +(Bξ,ξ +R2,R)†Bξ,ξ +R2,R +� += +� +R +JRKKY (0, +1, +1) +� +2 + νf + 4 − +� +ξ +(νξ +f + 2)2 +� ++ +� +R1̸=R2 +JRKKY (R − R2, +1, +1) +� +2 − +� +ξ +(Bξ,ξ +R2,R)†Bξ,ξ +R2,R +� +=const − +� +R +JRKKY (0, +1, +1) +� +ξ +(νξ +f + 2)2 − +� +R̸=R2 +JRKKY (R − R2, +1, +1) +� +ξ +(Bξ,ξ +R2,R)†Bξ,ξ +R2,R +(S127) +where we use the fact that JRKKY (R − R2, +1, +1) = JRKKY (R − R2, −1, −1) and replace ˆνξ +f(R) with νξ +f. const denotes +the constant term that only depends on νf. We next prove that the following state is the ground state of ˆHRKKY at M = 0 (with +chiral U(4) and flat U(4) symmetry): +|ψ0⟩ = +� +R +� ν+1 +f ++2 +� +i=1 +f † +R,αiηisi +νf +4 +� +i=ν+1 +f ++3 +f † +R,αiηisi +� +|0⟩ +(S128) +with the filling being +νf = −2 : (ν+1 +f , ν−1 +f ) = (−1, −1) +νf = −1 : (ν+1 +f , ν−1 +f ) = (−1, 0) or (0, −1) +νf = 0 : (ν+1 +f , ν−1 +f ) = (0, 0) +(S129) +and the orbital, valley indices satisfying +1 = ηi(−1)αi+1 +for +i = 1, ..., ν+ +f +− 1 = ηi(−1)αi+1 +for +i = ν+ +f + 1, ..., νf + 4 +(S130) +We note that the filling requirements in Eq. S129 minimize the second term in Eq. S127: +Eν+ +f ,ν− +f = − +� +R +JRKKY (0, +1, +1) +� +ξ +(νξ +f + 2)2 +(S131) +We show the values of Eν+ +f ,ν− +f at different fillings +νf = 0 :E−2,2 = E2,−2 = − +� +R +16JRKKY (0, +1, +1), +E−1,1 = E1,−1 = − +� +R +10JRKKY (0, +1, +1), +E0,0 = − +� +R +8JRKKY (0, +1, +1) +νf = −1 :E−2,1 = E1,−2 = − +� +R +9JRKKY (0, +1, +1), +E0,−1 = E−1,0 = − +� +R +5JRKKY (0, +1, +1), +νf = −2 :E−2,0 = E0,−2 = − +� +R +4JRKKY (0, +1, +1), +E−1,−1 = − +� +R +2JRKKY (0, +1, +1) +nuf = −3 :E−2,−1 = E−1,−2 = − +� +R +JRKKY (0, +1, +1) +(S132) +From Eq. S132, we proved that the filling requirements in Eq. S129 minimize Eν+ +f ,ν− +f . +We next prove that, for R ̸= R2, Bξ,ξ +R2,R|ψ0⟩ = 0. We first consider +f † +R,αηsfR2,αηs|ψ0⟩ +(S133) + +33 +with R ̸= R2. Clearly, f † +R,αηsfR2,αηs will move one f electron at R2 in orbital α valley η spin s to site R and same orbital, +valley, spin. There are two possibilities, |ψ0⟩ has zero electron at R2, αηs, then f † +R,αηsfR2,αηs|ψ0⟩ = 0. If |ψ0⟩ has one electron +at R2, αηs, then there must also be one electron at R, αηs (from Eq. S128). Consequently, f † +R,αηsfR2,αηs|ψ0⟩ = 0. Thus we +conclude f † +R,αηsfR2,αηs|ψ0⟩ = 0 for R ̸= R2. Then we have +Bξ,ξ +R2,R|ψ0⟩ = +� +αηs +f † +R,αηsfR2,αηsδξ,(−1)α+1η|ψ0⟩ = 0 +(S134) +and also ⟨ψ0|Bξ,ξ +R2,R|ψ0⟩. Since +⟨ψ0|(Bξ,ξ +R2,R)†Bξ,ξ +R2,R|ψ0⟩ = 0 +and JRKKY (R − R2, +1, +1) ≤ 0. +|ψ0⟩ minimize energy of the third term in Eq. S278: − � +R̸=R2 JRKKY (R − +R2, +1, +1) � +ξ(Bξ,ξ +R2,R)†Bξ,ξ +R2,R. In summary, |ψ0⟩ (Eq. S128) is the ground state that minimizes the energy. We mention +that the ground states we derived here form a subset (that of zero Chern number) of the ground states of the projected Coulomb +model in the chiral-flat limit [85]. We mention that in the topological heavy-fermion model, the effect of remote bands is +included, whose effect is absent in the projected Coulomb model. +F. +Ground state of the zero hybridization model at M ̸= 0, νf = 0, −1, −2 +We now solve the ground state at M ̸= 0. The Hamiltonian is +ˆHRKKY = +� +RR2,ξξ2 +JRKKY (R − R2, ξ, ξ2)ˆΘf,ξ +µν (R)ˆΘf,ξ2 +µν (R2) . +(S135) +where JRKKY (R − R2, ξ, ξ) ≤ 0 and JRKKY (R − R2, ξ, ξ) ≥ 0. However, since JRKKY (R − R2, +1, −1) are induced by +the non-zero M, and JRKKY (R − R2, ξ, −ξ) is relatively weak comparing to JRKKY (R − R2, ξ, ξ). Therefore, we can treat +JRKKY (R − R2, ξ, ξ) perturbatively. We separate the Hamiltonian into two parts +ˆHRKKY = ˆHM=0 + ˆH+− +ˆHM=0 = +� +RR2,ξ +JRKKY (R − R2, ξ, ξ)ˆΘf,ξ +µν (R)ˆΘf,ξ +µν (R2) +ˆH+− = +� +RR2,ξ +JRKKY (R − R2, ξ, −ξ)ˆΘf,ξ +µν (R)ˆΘf,−ξ +µν +(R2) +(S136) +Becaue the Hamiltonian ˆHRKKY breaks flat U(4) symmetry but keeps the chiral U(4) symmetry, so we work with chiral U(4) +moment ˆΘf,ξ +µν . Flat U(4) breaking can be observed from Eq. S123, where we find JRKKY (R − R2, ξ, −ξ) leads to anisotropic +interactions (different µν components have different interaction strength) between flat U(4) moments. +ˆHM=0 has degenerate ground states as defined in Eq. S128. We let |ψ0,i⟩ be the ground states of ˆHM=0. Based on the +perturbation theory of degenerate levels, we define the following matrix +[H+−]ij = ⟨ψ0,i| ˆH+−|ψ0,j⟩ . +(S137) +and determine the ground state by finding the lowest eigenstate of ˆH+−. +We first note that |ψ0,i⟩ (given in Eq. S128) can be written as a product state +|ψ0,i(R)⟩ = +� +R +|ψ0,i(R)⟩ +(S138) +where |ψ0,i(R)⟩ is the corresponding state at R and +⟨ψ0,j|ψ0,i(R)⟩ = 0, +when i ̸= j . +(S139) + +34 +Then, for (R1 ̸= R2) i ̸= j, we find +⟨ψ0,j|ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R2)|ψ0,i⟩ += +� +R̸=R1,R̸=R2 +⟨ψ0,j(R)|ψ0,i(R)⟩⟨ψ0,j(R1)|ˆΘf,ξ +µν (R1)|ψ0,i(R1)⟩⟨ψ0,j(R2)|ˆΘf,−ξ +µν +(R2)|ψ0,i(R2)⟩ = 0 +⟨ψ0,j|ˆΘf,ξ +µν (R1)|ψ0,i⟩ += +� +R̸=R1,R̸=R2 +⟨ψ0,j(R)|ψ0,i(R)⟩⟨ψ0,j(R1)|ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R1)|ψ0,i(R1)⟩ = 0 +(S140) +Combining Eq. S140 and Eq. S136, we have ⟨ψ0,i| ˆH+−|ψ0,j⟩ = 0 for i ̸= j. +For i = j, we find +⟨ψ0,i|ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R2)|ψ0,i⟩ += +� +R̸=R1,R̸=R2 +⟨ψ0,i(R)|ψ0,i(R)⟩⟨ψ0,i(R1)|ˆΘf,ξ +µν (R1)|ψ0,i(R1)⟩⟨psi0,j(R2)|ˆΘf,−ξ +µν +(R2)|ψ0,i(R2)⟩ = 0 +⟨ψ0,i|ˆΘf,ξ +µν (R1)|ψ0,i⟩ += +� +R̸=R1,R̸=R2 +⟨ψ0,i(R)|ψ0,i(R)⟩⟨ψ0,i(R1)|ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R1)|ψ0,i(R1)⟩ +(S141) +According to Eq. S16, ˆΘf,−ξ +µν +(R1) with µν ̸= 00, 0z, zz will move f electron from one αηs flavor with (−1)α+1η = −ξ to +another α′η′s′ flavor (−1)α′+1η′ = −ξ at the same site. Therefore ⟨ψ0,i(R1)|ˆΘf,−ξ +µν +(R1)|ψ0,i(R1)⟩ = 0 because ˆΘf,−ξ +µν +(R1) +will change the wavefunction of |ψ0,i(R1)⟩ or annihilate it. Similarly, ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R1) with µν ̸= 00, 0z, zz will either +change the configuration of |ψ0,i⟩ (and leads to a state orthogonal to |ψ0,i⟩) or annihilate |ψ0,i⟩. Thus +⟨ψ0,i|ˆΘf,ξ +µν (R1)ˆΘf,−ξ +µν +(R2)|ψ0,i⟩ = 0, +µν /∈ {00, 0z, zz} +⟨ψ0,i|ˆΘf,ξ +µν (R1)|ψ0,i⟩ = 0, +µν /∈ {00, 0z, zz} +(S142) +Combining Eq. S140 and Eq. S142, the only non-vanishing components of ˆH+− in Eq. S137 are +[ ˆH+−]ij = ⟨ψ0,i| ˆH+−|ψ0,i⟩ +=δi,j +� +RR2,ξ +� +µν∈{00,0z,z0,zz} +JRKKY (R − R2, ξ, −ξ)⟨ψ0,i|ˆΘf,ξ +µν (R)ˆΘf,−ξ +µν +(R2)|ψ0,i⟩ +(S143) +Therefore [ ˆH+−]ij is a diagonal matrix. +We next take the state in Eq. S128 and calculate +E+− = [ ˆH+−]ii . +(S144) +The ground states are the states that minimize E+−. +We provide the values of E+− of the states shown in Eq. S128. +At νf = 0, we find the following states have E+− = � +RR2 2JRKKY (R − R2, +, −) +{1+ ↑, 2− ↑, 2+ ↑, 1− ↑}, {1+ ↑, 1+ ↓, 2+ ↑, 2+ ↓}, {1+ ↑, 2− ↓, 2+ ↑, 1− ↓}, +{2− ↑, 2− ↓, 1− ↑, 1− ↓}, {1+ ↓, 2− ↓, 1− ↓, 2+ ↓} +(S145) +where we have characterized the states with {αiηisi} (Eq. S128). +At ν += +0, the following states have E+− += +− � +RR2 2JRKKY (R − R2, +, −) +{1+ ↓, 2− ↓, 2+ ↑, 1− ↑}, {2− ↑, 2− ↓, 2+ ↑, 2+ ↓}, {1+ ↓, 2− ↑, 2+ ↑, 1− ↓}, {1+ ↑, 2− ↓, 1− ↑, 2+ ↓}, +{1+ ↑, 1+ ↓, 1− ↑, 1− ↓}, {1+ ↑, 2− ↑, 1− ↓, 2+ ↓} +(S146) +All other states in Eq. S128 at ν = 0 have E+− = 0. Since JRKKY (R − R2, +, −) ≥ 0, Eq. S146 gives the ground states at +νf = 0. + +35 +At νf = −1, we find the following states in Eq. S128 have energy E+− = � +RR2 JRKKY (R − R2, +, −) +{1+ ↑, 1+ ↓, 1− ↑}, {1+ ↑, 1+ ↓, 1− ↓} +{1+ ↑, 2− ↑, 2+ ↓}, {1+ ↑, 2− ↑, 1− ↓} +{1+ ↑, 2− ↓, 1− ↑}, {1+ ↑, 2− ↓, 2+ ↓} +{1+ ↓, 2− ↑, 2+ ↑}, {1+ ↓, 2− ↑, 1− ↓} +{1+ ↓, 2− ↓, 2+ ↑}, {1+ ↓, 2− ↓, 1− ↑} +{2− ↑, 2− ↓, 2+ ↑}, {2− ↑, 2− ↓, 2+ ↓} +(S147) +The remaining states in Eq. S128 at νf = −1 have E+− = − � +RR2 JRKKY (R−R2, +, −). Thus the ground states at ν = −1 +are the states in Eq. S147. +At νf = −2, we find the following states have E+− = 0 +{1+ ↑, 2+ ↓}, {1+ ↑, 1− ↑}, {1+ ↑, 1− ↓}, {1+ ↓, 2+ ↑}, {1+ ↓, 2+ ↓}, {1+ ↓, 1− ↓}, +{2− ↑, 2+ ↓}, {2− ↑, 2+ ↑}, {2− ↑, 1− ↓}, {2− ↓, 2+ ↑}, {2− ↓↓, 2+ ↓}, {2− ↓, 1− ↑}. +(S148) +The remaining states in Eq. S128 at νf = −2 have E+− = � +RR2 2JRKKY (R−R2, +, −). Thus the ground states at νf = −2 +are the states in Eq. S148. +At νf = −3, all states Eq. S128 in have the same E+− +In summary, we find f electrons at the same site tend to fill different spin-valley flavors, in order to minimize the energy from +JRKKY (R − R2, ξ, −ξ). In a more compact form, the following states (from Eq. S146,Eq. S147,Eq. S148) are the ground state +at M ̸= 0 +� +R +� ν+ +f +2 +� +i=1 +f † +R,1ηisi +νf +4 +� +i=ν+ +f +3 +f † +R,2ηisi +� +|0⟩ +(S149) +where ν+ +f and {αiηisi}i=1,...,νf +4 need to satisfy the Eq. S130, Eq. S129 and +(ηi, si) ̸= (ηj, sj) +for +i ̸= j. +(S150) +We point out that the ground states of f-electrons here give rise to the same ground states of the projected Coulomb model in +the chiral-nonflat limit [85]. +S4. +SCHRIEFFER-WOLFF TRANSFORMATION +In this section, we treat the model of non-zero hybridization γ ̸= 0, v′ +⋆ ̸= 0 with the Schrieffer-Wolff transformation [127]. +We focus on the νf = 0, −1, −2. At νf = −3 and in the zero-hybridization limit, νf = −3 is close to the transition point as +shown in Fig. S3. Therefore, a uniform charge distribution of f electrons may not be energetically favorable at νf = −3 and the +SW transformation could fail at this filling. +A. +Hamiltonian +We first separate the Hamiltonian with non-zero hybridization into two parts +ˆH = ˆHcomplete +0 ++ ˆH1 +(S151) +In the heavy-fermion model with non-zero hybridization, we let +ˆHcomplete +0 += ˆHc + ˆHU + ˆHV + ˆHW − µ +� +R,α,η,s +f † +R,αηsfR,αηs − µ +� +R,a,η,s +c† +k,aηsck,aηs+PH ˆHfcPH + ˆHJ +ˆH1 = PL ˆHfc + ˆHfcPL +ˆHfc = +1 +√NM +� +k,R,αηs +eikRH(fc,η) +αa +(k)f † +Rαηsckaηs + h.c +, +H(fc,η)(k) = +�γσ0 + v′ +⋆(ηkxσx + kyσy) 02×2 +� +. +(S152) + +36 +where µ is the chemical potential. The projection operator for a given filling of f-electrons νf is defined as +PL = +� +R +� +ανf +� +n̸=νf +4 +� � +αηs +f † +R,αηsfR,αηs − n +�� +, +PH = I − PL . +(S153) +The projection operator PL will only keep the states with filling of f electron being νf for each site and α−1 +νf = � +n̸=νf +4(νf + +4 − n)] = (νf + 4)!(4 − νf)!(−1)νf is the normalization factor ensuring P 2 +L = PL. Here we also introduce the notation of +low-energy space and high-energy scape. Low-energy space is defined as +HL = +� +|ψ⟩ +����PL|ψ⟩ = |ψ⟩ +� +For all states in HL, the filling of f-electrons is νf at each site. We define the high-energy space as +HH = +� +|ψ⟩ +����PL|ψ⟩ = 0 +� +(S154) +which is formed by all states that are not in HL. We use low-energy (high-energy) states to denote the state in low-energy +(high-energy) space. Then the Hamiltonian ˆHcomplete +0 +maps a low-energy state to a low-energy state, or maps a high-energy +state to a high-energy state; ˆH1 maps a low-energy state to a high-energy state or a high-energy state to a low-energy state [138]. +Then after SW transformation, we are able to derive an effective Hamiltonian that maps a low-energy state to a low-energy +state or a high-energy state to a high-energy state. In other words, we eliminate the off-diagonal term, ˆH1, during the SW +transformation [138]. +However, we comment that the original hybridization term ˆHfc can map a high-energy state to either a high-energy state or a +low-energy state. To observe this, we act ˆHfc on a state with filling νf + 1. The resulting state can has filling νf − 1 or νf + 2 +(if νf + 2 ≤ 4). Therefore, it can map a high-energy state with filling νf + 1 to another high-energy state with filling νf + 2. +The same argument also applies to the state with νf − 1. It violates the requirement of ˆH1 that we explained earlier. However +, we could separate ˆHfc into two parts: ˆHfc = +� +PL ˆHcPH + PL ˆHcPH +� ++ PH ˆHfcPH (Note that ˆHfc will change the filling +of f electrons by 1, so PL ˆHfcPL = 0). We treat the first part PL ˆHcPH + PL ˆHcPH as ˆH1, and put the second part PH ˆHfcPH +into ˆHcomplelte +0 +. Then we can perform SW transformation by treating ˆH1 as a perturbation. +One may wonder what happens if we take ˆHfc as ˆH1 and perform SW transformation. In principle, we can find the operator S +that satisfies [ ˆH0, S] = ˆH1. However, [ ˆH1, S], which appears in the effective Hamiltonian, would contain terms with the form of +f †f †cc or ffc†c†. f †f †cc or ffc†c† which maps a low-energy state to a high-energy stat. And the effective Hamiltonian after +transformation can still map a low-energy state to a high-energy state or a high-energy state to a low-energy state. Therefore, we +can not treat f as a local moment in the effective Hamiltonian. The such issue will not emerge if we let ˆH1 = PL ˆHfc + ˆHfcPL. +We remark that, in the standard one-orbital Anderson model, we always have ˆHfc = PL ˆHfc + ˆHfcPL and it is not necessary to +explicitly introduce the projection operator. +In addition, instead of working with ˆHcomplete +0 +, we first drop both PH ˆHfcPH and ˆHJ terms in the procedure of SW transfor- +mation, and define +ˆH0 = ˆHc + ˆHU + ˆHV + ˆHW − µ +� +R,α,η,s +f † +R,αηsfR,αηs − µ +� +R,a,η,s +c† +k,aηsck,aηs . +(S155) +If we keep both terms, it would be complicated to derive the effective Hamiltonian, since it contains f-c hybridization and f-c +interactions. +1. +ˆH0 +We provide the explicit form of ˆH0 in this section. We simplify the notation by using single index m to label orbital α, valley +η and spin σ and let fR,m := fR,αηs, ck,m := ck,aηs. The corresponding density operator is defined as +nf +R,m = f † +R,mfR,m +, +nf +R = +� +m +nf +Rm +, +Nf = +� +R,m +f † +R,mfR,m +Nc = +� +k,a +γ† +k,aγk,a = +� +k,a +c† +k,ack,a . +(S156) + +37 +The kinetic term of conduction electrons now becomes +ˆHc = +� +k,m,m′ +Hc +mm′(k)c† +k,mck,m′. +(S157) +where Hc +mm′(k) is the hopping matrix of conduction electrons with new label m, m′. We next introduce eigenvalues and +eigenvectors of Hc +mm′(k), +� +α +Hc +mn(k)Uk,na = ϵk,aUk,ma , +(S158) +and the operator in the band basis +γk,a = +� +n +U ∗ +k,nack,n . +(S159) +Then we have ˆHc = � +k,a ϵk,aγ† +k,aγk,a. +The Hubbard interactions between f electrons now become +ˆHU =U +2 +� +R +� +: +� +m +f † +R,mfR,m : +�2 += U +2 +� +R +( +� +m +(f † +R,mfR,m − 1/2))2 +=U +2 +� +R +� +m′,m +nf +R,mnf +R,m′ + U +2 +� +R +42 − U +2 +� +R,m +2f † +R,mfR,m4 +=U +2 +� +R +� +m′,m,m′̸=m +nf +R,mnf +R,m′ + U +2 +� +R,m +nf +R,m + 8UNM − U +2 +� +R,m +8nf +R,m +=U +2 +� +R,m′,m,m̸=m′ +nf +R,mnf +R,m′ − 7U +2 Nf + 8UNM . +(S160) +where NM is the total number of moir´e unit cells. +For the Coulomb interactions between c electrons, we only include the q = 0 contribution: V (q) = V0δq,0. This leads to +ˆˆHV = V0 +2Ω0 +1 +NM +( +� +k,m +: c† +k,mck,m :)2 = V0 +2Ω0 +1 +NM +� +Nc − +� +k +8 +�2 += +V0 +2Ω0NM +N 2 +c − +� 8V0 +Ω0NM +� +k +1 +� +Nc + +V0 +2Ω0NM +( +� +k +8)2 +(S161) +where Ω0 is the area of the first moir´e Brillouin zone. +The density-density interactions between f, c are +ˆHW = 1 +NM +W +� +a,R,q,k +: nf +R :: c† +k+q,ack,a : e−iqR = W +1 +NM +� +a,R,q,k,a +(nf +R − 4)(c† +k+q,ack,a − δq,0 +2 )e−iqR += +1 +NM +W +� +a,R,q,k +nf +Rc† +k+q,ack,ae−iqR + W +NM +� +a,R,k +nf +R(−1 +2) − 4W +NM +� +a,R,q,k +c† +k+q,ack,ae−iqR ++ W +NM +� +a,R,q,k +(−4)(−δq,0 +2 )e−iqR += +1 +NM +W +� +a,R,q,k +nf +Rc† +k+q,ack,ae−iqR − 8W +1 +NM +� +R,k +nf +R − 4WNc + 32W +� +k +1 +Here is the derivation, we only consider the c-electrons in the first moir´e Brillouin by taking a specific momentum cutoff Λc. Due +to the first term in ˆHW (which contains c† +k+q,ack,a), it is complicated to find an analytical expression of the SW transformation. +However, if we stick to the case, where the fillings of f fermion are the same at each site, we can replace nf +R by its average value +1 +NM +� +R′ nf +R′. This leads to +W +NM +� +a,R,q,k +nf +Rc† +k+q,ack,ae−iqR → W +NM +� +a,R,q,k +�� +R′ nf +R′ +NM +� +c† +k+q,ack,ae−iqR = W +NM +� +R′ +nf +R′ +� +a,q,k +c† +k+q,ack,aδq,0 += W +NM +� +R +nf +R +� +k,a +c† +k,ack,a = W +NM +� +R +nf +R +� +k,a +γ† +k,aγk,a + +38 +Now, we define our new ˆHW as +ˆHW = W +NM +� +R,k,a +nf +Rγ† +k,aγk,a − 8W +1 +NM +� +R,k +nf +R − 4WNc + 32W +� +k +1 +Combining all terms and rearranging them, we have +ˆH0 = ˆhU + ˆhEf + ˆhVc + ˆhc + ˆhW + ˆhconst +ˆhU = U +2 +� +R +� +m,m′,m̸=m′ +nf +Rmnf +Rm′ +, +ˆhEf = EfNf +ˆhVc = EcNc + VcN 2 +c +, +ˆhc = +� +k,a +ϵk,aγ† +k,aγk,a +ˆhW = +1 +NM +W +� +a,R,q,k +nf +Rc† +k+q,ack,ae−iqR +, +ˆhconst = 8UNM + +V0 +2Ω0NM +( +� +k +8)2 + 32W +� +k +1 +(S162) +where +Ef = −7U +2 − 8W +NM +� +k +1 − µ +, +Ec = −4W − +8V0 +Ω0NM +� +k +1 − µ +, +Vc = +V0 +2Ω0NM +(S163) +2. +ˆH1 +ˆH1 can be written as +ˆH1 = +1 +√NM +� +k,R +� +m,m′ +eikRHfc +mm′(k)(PLf † +R,mck,m′ + f † +R,mck,m′PL) + h.c. += +1 +√NM +� +k,R +� +m,n +eikR( +� +m′ +Hfc +mm′(k)Uk,m′n)(PLf † +R,mγk,n + f † +R,mγk,nPL) + h.c. += +� +k,R +� +m,n +VRk,mn(PLf † +R,mγk,n + f † +R,mγk,nPL) + h.c. +(S164) +where Hfc(k) is the hybridization matrix (with damping factor included, Eq. S3, Eq. S4) with new label m, m′. Vk,mn is defined +as VRk,mn = +1 +√NM +� +m′ eikRHfc +mm′(k)Uk,m′n. +B. +Procedure of Schrieffer–Wolff transformation +We now perform SW transformation. We aim to find operator S such that +S = −S† +, +[ ˆH0, S] = ˆH1 . +(S165) +This allows us to introduce the following unitary transformations +eS ˆHe−S ≈ ˆH0 + ( ˆH1 − [ ˆH0, S]) + [S, ˆH1] + 1 +2[[ ˆH0, S], S] = ˆH0 + 1 +2[S, ˆH1] , +(S166) +and obtain the effective Hamiltonian +ˆHeff = ˆH0 + 1 +2[S, ˆH1] +(S167) + +39 +C. +Expression of S +In this section, we provide the analytical expression of S. We first decompose S based on its anti-hermitian property +S = S1 − S† +1 . +(S168) +We then assume +S† +1 = PLs† +1 + s† +1PL +s† +1 = +� +R,k,m,n +VRk,mnRRk,mnf † +R,mγk,n +(S169) +where VRk,ma is the hybridization matrix between f-fermion and γ-fermion. RRk,mn is an operator with assumed commuta- +tion relations [RRk,mn, γ† +k+q,nγk′,n′] = 0, [RRk,mn, f † +R′,n′fR′,n′] = 0 and [RRk,mn, PL] = 0. RRk,mn are determined by +requiring [ ˆH0, S] = ˆH1. +To find the expression of RRk,mn, we directly calculate [ ˆH0, s1]. Before moving to the calculations, we first list several useful +commutation relations: +[nf +R,m, f † +R′,m′] = δR,R′δm,m′f † +R′,m′ +[γ† +k+q,aγk,a′, γk′,b] = −δk+q,k′δa,bγk′−q,a′ +Given operators A, B, C, D with [A, D] = [B, C] = 0, we have [AB, CD] = [A, C]DB + CA[B, D] + [A, C][B, D] +(S170) +For each term in ˆH0 (Eq. S162), we use Eq. S170 and find +[ˆhU, s† +1] = +� +R′,m,m′,m̸=m′ +� +R,k,a +UVRk,maRRk,manf +R,m′[nf +R,m, f † +R,m]γk,a += +� +R,k,m,a +UVRk,maRRk,ma +� +m′,m′̸=m +nf +R,m′f † +R,mγk,a += +� +R,k,m,a +UVRk,maRRk,ma( +� +m′ +nf +R,m − 1)f † +R,mγk,a = +� +R,k,m,a +UVRk,maRRk,ma(nf +R − 1)f † +R,mγk,a +[ˆhEf , s† +1] = +� +R,k,m,a +EfVRk,maRRk,ma[nf +R,m, f † +R,m]γk,a = +� +R,k,m,a +EfVRk,maRRk,maf † +R,mγk,a +[ˆhc, s† +1] = +� +R,k,m,a +VRk,maRRk,ma +� +k′,a′ +ϵk′,a′(−f † +R,mγk,aγ† +k′,a′γk′,a′ + γ† +k′,a′γk′,a′f † +R,mγk,a) += +� +R,k,m,a +VRk,maRRk,ma +� +k′,a′ +ϵk′,a′f † +R,m(−γk,aγ† +k′,a′ − γ† +k′,a′γk,a)γk′,a′ += +� +R,k,m,a +VRk,maRRk,ma +� +k′,a′ +ϵk,a(−f † +R,mδk,k′δa,a′γk′,a′) = − +� +R,k,m,a +VRk,maRRk,maϵk,af † +R,mγk,a + +40 +[ˆhVc, s† +1] = +� +R,k,m,a +VRk,maRRk,maf † +R,m[VcN 2 +c + EcNc, γk,a] += +� +R,k,m,a +VRk,maRRk,maf † +R,m[Vc(N 2 +c − (Nc + 1)2)γk,a − Ecγk,a] += +� +R,k,m,a +VRk,maRRk,maf † +R,m[Vc(−2Nc − 1)γk,a − Ecγk,a] +[ˆhW , s† +1] = +1 +NM +W +� +a,R,q,k +� +R′,k′,m,n +VR′k′,mnRR′k′,mne−iqR[nf +Rγ† +k+q,aγk,a, f † +R′,mγk′,n] += +1 +NM +W +� +a,R,q,k +� +R′,k′,m,n +VR′k′,mnRR′k′,mne−iqR +� +nf +Rf † +R′,mγ† +k+q,aγk,a, γk′,n − f † +R′,mnf +Rγk′,nγ† +k+q,aγk,a +� += +1 +NM +W +� +a,R,q,k +� +R′,k′,m,n +VR′k′,mnRR′k′,mne−iqRδR,R′f † +R′,m +� +γ† +k+q,aγk,a, γk′,n − γk′,nγ† +k+q,aγk,a +� += +1 +NM +W +� +a,R,q,k +� +R′,k′,m,n +VR′k′,mnRR′k′,mne−iqRδR,R′f † +R′,m(−δk+q,k′δn,a)γk,a += +1 +NM +W +� +R,q,k +� +m,n +VR(k+q),mnRRk+q,mne−iqR(−1)f † +R,mγk,n +[ˆhconst, s† +1] = 0 +We notice that PL only contains the density operator of f electron, and it commutes with ˆH0. Then we have [ ˆH0, S† +1] = +[ ˆH0, PLs† +1 + s† +1PL = PL[ ˆH0, s† +1] + [ ˆH0, s† +1]PL. +Combining all terms, we have +[ ˆH0, S† +1] = +� +R,k,m,a +VRk,maRRk,ma +� +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec +�� +PLf † +R,mγk,a + f † +R,mγk,aPL +� ++ +1 +NM +W +� +R,q,k +� +m,n +VR(k+q),mnRRk+q,mne−iqR(−1) +� +PLf † +R,mγk,n + f † +R,mγk,nPL +� +(S171) +From the definition of PL (Eq. S153), we note that +nf +RPL = (νf + 4)PL +with νf the integer number that characterizes the filling of the low-energy state. We let RRk,ma +RRk,ma = − +1 +U(νf + 3) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec − W +(S172) +Combining the definition of RRk,ma (Eq. S172) and Eq. S171, we have +[ ˆH0, S† +1] = − +� +R,k,m,a +VRk,ma +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec +U(νf + 3) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec − W +� +PLf † +R,mγk,a + f † +R,mγk,aPL +� +− +1 +NM +W +� +q,k +� +m,n +1 +U(νf + 3) + Ef − ϵk+q,a + Vc(−2Nc − 1) − Ec − W +which gives the correct commutation relations +[ ˆH0, −S† +1] = +� +R,k,m,a +VRk,maf † +R,mγk,a +[ ˆH0, S1] = +� +[ ˆH0, −S† +1] +�† += +� +R,k,m,a +V ∗ +Rk,maγ† +k,afR,m , ‘ +and +[ ˆH0, S] = [ ˆH0, S1 − S† +1] = ˆH1 + +41 +D. +Effective Hamiltonian from SW transformation +We now derive the effective Hamiltonian ˆHeff = ˆH0 + 1 +2[S, ˆH1]. Combining Eq. S168 and Eq. S169, we have +1 +2[S, ˆH1] = 1 +2(PLs1 + s1PL − PLs† +1 − s† +1PL)(PL ˆHfc + ˆHfcPL) − 1 +2(PL ˆHfc + ˆHfcPL)(PLs1 + s1PL − PLs† +1 − s† +1PL) +Since ˆHfc and s1 change the filling of f by ±1, both ˆHfc and s1 map a low-energy to high-energy state, which indicates +PLs1PL = PL ˆHfcPL = 0 +PL ˆHfc = PL ˆHfcPH +, +ˆHfcPL = PH ˆHfcPL +, +PLs1 = PLs1PH +, +s1PL = PHs1PL +Then we find +1 +2[S, ˆH1] = −1 +2PL +� +(s† +1 − s1) ˆHfc + ˆHfc(−s† +1 + s2) +� +PL − 1 +2PH +� +(s† +1 − s1)PL ˆHfc + ˆHfcPL(s1 − s† +1) +� +PH +(S173) +We note that the effective Hamiltonian ˆHeff = ˆH0 + 1 +2[S, ˆH1] will only map a low-energy state to a low-energy state or a +high-energy state to a high-energy state. We focus on the effective Hamiltonian that controls the low-energy physics, which is +defined in the low-energy Hilbert space: ˆHeff = PL ˆH0PL + 1 +2PL[S, ˆH1]PL. Combining Eq. S173, Eq. S172 and Eq. S164, we +find +1 +2PL[S, ˆH1]PL = − 1 +2PL +� +� +R,k,m,a,R′,k′,m′,a′ +� +� +VRk,maRRk,maf † +R,mγk,a − V ∗ +Rk,maγ† +k,afR,mR† +Rk,ma +�� +VR′k′,m′a′f † +R′,m′γk′,a′ + V ∗ +R′k′,m′a′γ† +k′,a′fR′,m′ +� ++ +� +VR′k′,m′a′f † +R′,m′γk′,a′ + V ∗ +R′k′,m′a′γ† +k′,a′fR′,m′ +�� +− VRk,maRRk,maf † +R,mγk,a + γ† +k,afR,mV ∗ +Rk,maR† +Rk,ma +�� +� +PL +(S174) +Annihilating or creating two electrons would obviously violate the uniform charge distribution of f, and will map low-energy +space to high energy space. Therefore, +PLfR,m′fR′,mPL = 0 +, +PLf † +R,m′f † +R′,mPL = 0 +PLf † +R,m′fR′,m′PL ∝ δR,R′ +, +PLfR,m′f † +R′,m′PL ∝ δR,R′ , +and Eq. S174 can be written as +1 +2PL[S, ˆH1]PL += − 1 +2 +� +R,k,k′,ma,m′a′ +� +PL(V ∗ +Rk′,m′a′VRk,maRRk,maf † +R,mfR,m′γk,aγ† +k′,a′)PL +− PL(VRk′,m′a′V ∗ +Rk,maγ† +k,afR,mR† +Rk,maf † +R,m′γk′,a′)PL ++ PL(VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,mγk′,a′γ† +k,aR† +Rk,ma)PL − PL(V ∗ +R′k′,m′a′VRk,maγ† +k′,a′fR,m′RRk,maf † +R,mγk,a)PL +� +(S175) + +42 +We next evaluate each term in the above equation (Eq. S175). The first term of Eq. S175 reads +− V ∗ +Rk′,m′a′VRk,maRRk,maf † +R,mfR,m′γk,aγ† +k′,a′ +=V ∗ +Rk′,m′a′VRk,ma +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +f † +R,mfR,m′γk,aγ† +k′,a′ +=V ∗ +Rk′,m′a′VRk,maf † +R,mfR,m′γk,aγ† +k′,a′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +=V ∗ +Rk′,m′a′VRk,maf † +R,mfR,m′(δk,k′δa,a′ − γ† +k′,a′γk,a) +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) += − V ∗ +Rk′,m′a′VRk,maf † +R,mfR,m′γ† +k′,a′γk,a +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) ++ δk,k′δa,a′V ∗ +Rk,m′aVRk,maf † +R,mfR,m′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +(S176) +The second term of Eq. S175 reads +VRk′,m′a′V ∗ +Rk,maγ† +k,afR,mR† +Rk,maf † +R,m′γk′,a′ += − VRk′,m′a′V ∗ +Rk,maγ† +k,afR,m +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +f † +R,m′γk′,a′ += − VRk′,m′a′V ∗ +Rk,maγ† +k,afR,mf † +R,m′γk′,a′ +1 +U(nf +R + 1 − 1) + Ef − ϵk,a + Vc(−2(Nc − 1) − 1) − Ec + W/NM((Nc − 1) − (Nf + 1) + 1) += − VRk′,m′a′V ∗ +Rk,maγ† +k,afR,mf † +R,m′γk′,a′ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) += − VRk′,m′a′V ∗ +Rk,mafR,mf † +R,m′γ† +k,aγk′,a′ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) += − VRk′,m′a′V ∗ +Rk,ma(δm,m′ − f † +R,m′fR,m)γ† +k,aγk′,a′ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +=VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,mγ† +k,aγk′,a′ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +− δm,m′VRk′,ma′V ∗ +Rk,maγ† +k,aγk′,a′ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +(S177) +The third term of Eq. S175 reads +− VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,mγk′,a′γ† +k,aR† +Rk,ma +=VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,mγk′,a′γ† +k,a +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +=VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,m(δk,k′δa,a′ − γ† +k,aγk′,a′) +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) += − VRk′,m′a′V ∗ +Rk,maf † +R,m′fR,mγ† +k,aγk′,a′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) ++ δk,k′δa,a′VRk,m′aV ∗ +Rk,maf † +R,mfR,m′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +(S178) + +43 +The fourth term of Eq. S175 reads +V ∗ +R′k′,m′a′VRk,maγ† +k′,a′fR,m′RRk,maf † +R,mγk,a += − V ∗ +R′k′,m′a′VRk,maγ† +k′,a′fR,m′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +f † +R,mγk,a += − V ∗ +R′k′,m′a′VRk,maγ† +k′,a′fR,m′f † +R,mγk,a +(S179) +1 +U(nf +R + 1 − 1) + Ef − ϵk,a + Vc(−2(Nc − 1) − 1) − Ec + W/NM(Nc − 1 − (Nf + 1) + 1) += − V ∗ +R′k′,m′a′VRk,maγ† +k′,a′fR,m′f † +R,mγk,a +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) += − V ∗ +R′k′,m′a′VRk,mafR,m′f † +R,mγ† +k′,a′γk,a +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) += − V ∗ +R′k′,m′a′VRk,ma(δm,m′ − f † +R,mfR,m′)γ† +k′,a′γk,a +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +=V ∗ +R′k′,m′a′VRk,maf † +R,mfR,m′γ† +k′,a′γk,a +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +− δm,m′V ∗ +R′k′,ma′VRk,maγ† +k′,a′γk,a +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +(S180) +Summing over all terms (Eqs. S176, S177, S178, S180), we have +1 +2PL(S ˆH1 + ˆH1S†)PL += +� +R,kk′,m,a,m′,a′ +V ∗ +Rk′,m′a′VRk,maPLf † +R,mfR,m′γ† +k′,a′γk,a +� +− +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) ++ +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +� +PL ++ +� +� +R,k,,m,m′,a +V ∗ +Rk,m′aVRk,maPLf † +R,mfR,m′ +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) +PL +− +� +R,k,k′,m,a,a′ +VRk′,ma′V ∗ +Rk,maPLγ† +k,aγk′,a′ +� +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) +PL +(S181) +Since we fix the filling of f at each site, we can replace nf +R with νf + 4. We also replace Nc by Nc = � +k,a γ† +k,aγk,a = +� +k,a c† +k,ack,a = NM ˆνc + � +k 8 . Based on this, the pre-factors that appear in the effective Hamiltonian can be written as +1 +D1,k,a[ˆνc, νf] = +1 +U(nf +R − 1) + Ef − ϵk,a + Vc(−2Nc − 1) − Ec + W/NM(Nc − Nf + 1) += +1 +U(νf + 3) + Ef − ϵk,a + Vc(−2NM ˆνc − 16 � +k 1 − 1) − Ec + W(−νf − 4 + ˆνc + 8 � +k 1/NM + 1/NM) +(S182) +and +1 +D2,k,a[ˆνc, νf] = +1 +U(nf +R) + Ef − ϵk,a + Vc(−2Nc + 1) − Ec + W/NM(Nc − Nf − 1) += +1 +U(νf + 4) + Ef − ϵk,a + Vc(−2NM ˆνc − 16 � +k 1 + 1) − Ec + W(−νf − 4 + ˆνc + 8 � +k 1/NM − 1/NM)PL +(S183) + +44 +Using explicit formula of Ef, Vc, Ec in Eq. S163, the denominators becomes +D1,k,a[ˆνc, νf] =U(νf + 3) − 7U +2 − 8W +NM +� +k +1 − ϵk,a + +V0 +2Ω0NM +(−2NM ˆνc − 16 +� +k +1 − 1) − (−4W − +8V0 +Ω0NM +� +k +1) +− W(νf − ˆνc − 4 + 8 +� +k +1 + 1/NM) +=(U − W)νf + (−V0 +Ω0 ++ W)ˆνc − ϵk,a − U +2 + +1 +2NM +(− V0 +2Ω0 +− W) +and +D2,k,a[ ˆνc, νf] = = U(νf + 4) − 7U +2 − 8W +NM +� +k +1 − ϵk,a + Vc(−2NM ˆνc − 16 +� +k +1 + 1) − (−4W − +8V0 +Ω0NM +� +k +1) +− W(νf − ˆνc + 4 − +� +k +1/NM − 1/NM) +=(U − W)νf + (−V0 +Ω0 ++ W)ˆνc − ϵk,a + U +2 + +1 +2NM +( V0 +2Ω0 +− W) +We comment on the value of denominators: +• We drop the term at the order of 1/NM which is negligibly small. +• In the original model, there is a damping term, which suppresses the hybridization between f fermions and c-fermions +with large momentum. Then the dominant hybridization comes from the c electron with momentum near ΓM. These +conduction electrons have small |ϵk,a|, so we drop this term in the denominators. (A similar approximation has also been +taken in the standard Kondo model.). +• We also replace the operator ˆνc by a real number νc which represents the average filling of conduction electrons. +Taking above approximations, we can replace D1/2,k,a[ˆνc, νf] with real numbers D1/2,νc,νf +D1,k,a[νc, νf] ≈ D1,νc,νf = (U − W)νf − U +2 + (−V0 +Ω0 ++ W)νc +D2,k,a[νc, νf] ≈ D2,νc,νf = (U − W)νf + U +2 + (−V0 +Ω0 ++ W)νc +(S184) +and we define Dνc,νf as +Dνc,νf = +� +− +1 +D1,νc,νf ++ +1 +D2,νc,νf +�−1 +(S185) +We give the value at νc = 0, U = 58meV, W = 48meV +νf +0 +-1 +-2 +1 +Dνc=0,νf 0.0690 (meV)−1 0.0805 (meV)−1 0.161 (meV)−1 +The final formula of the effective Hamiltonian is +ˆHeff = ˆH0 + +� +R,k,k′,m,a,m′,a′ +ei(k−k′)R +NM +V ∗ +Rk′,m′a′VRk,ma +Dνc,νf +f † +R,mfR,m′γ† +k′,a′γk,a ++ +� +� +R,k,m′,a +V ∗ +Rk,m′a′VRk,ma +NM +f † +R,mfR,m′ +1 +D1,νc,νf +− +� +k,m,a,a′ +V ∗ +Rk′,m′a′VRk,maγ† +k,aγk,a′ +1 +D2,νc,νf +� +. +(S186) +Since, we only consider the state with fixed fillings of f at each site, we drop the projection operator PL. However, we need to +remember that our Hilbert space is spanned by the states with the fixed filling of f-electrons νf. + +45 +We now transform everything from band basis γk,m to conduction electron basis ck,m. Using γk,a = � +a U ∗ +k,back,b and +VRk,mn = +1 +√NM +� +m′ eikRHfc +mm′(k)Uk,m′n +ˆHeff = ˆH0 + +� +R,k,k′,m,a,m′,a′ +ei(k−k′)R +NM +Hfc,∗ +m′b′(k′)Hfc +mb(k) +Dνc,νf +f † +R,mfR,m′c† +k′,b′ck,b ++ +� +� +R,k,,m,m′,b +Hfc,∗ +m′b (k)Hfc +mb(k) +NM +f † +R,mfR,m′ +1 +D1,νc,νf +− +� +k,m,b,b′ +Hfc +ma′(k)Hfc,∗ +ma (k)c† +k,bck,b′ +1 +D2,νc,νf +� +. +(S187) +We next go to the original labeling. +ˆHeff = ˆH0 ++ +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)F(|k|)2 +Dνc,νf +� +f † +R,αηsfR,α′η′s′c† +k′,a′η′s′ck,aηs +� ++ +� +� +k,α,α′,η,s +� +a∈{1,2} +F(|k|)2H(fc,η),∗ +α′a +(k)H(fc,η) +αa +(k) +NMD1,νc,νf +� +R +f † +R,αηsfR,α′ηs +− +� +k,α,η,s +� +a,a′∈{1,2} +F(|k|)2H(fc,η) +αa′ +(k)H(fc,η),∗ +αa +D2,νc,νf +(k)c† +k,aηsck,a′ηs +� +. +(S188) +where +F(|k|) = e−λ2|k|2/2 +(S189) +is the damping factor of f-c hybridization (Eq. S3). With normal ordering, we have +ˆHeff = ˆH0 ++ +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)F(|k|)2 +Dνc,νf +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : +� ++ +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)|F(|k|)|2 +Dνc,νf +�|F(|k|)|2 +2 +δα,α′δη,η′δs,s′ : c† +k′,a′η′s′ck,aηs : +� ++ +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)|F(|k|)|2 +Dνc,νf +� +: f † +R,αηsfR,α′η′s′ : δk,k′δa,a′δη,η′δs,s′ +� ++ +� +� +R,k,α,α′,η,s +� +a∈{1,2} +|F(|k|)|2H(fc,η),∗ +α′a +(k)H(fc,η) +αa +(k) +NMD1,νc,νf +: f † +R,αηsfR,α′ηs : +− +� +k,α,η,s +� +a,a′∈{1,2} +|F(|k|)|2H(fc,η) +αa′ +(k)H(fc,η),∗ +αa +(k) +D2,νc,νf +: c† +k,aηsck,a′ηs : +� ++ const += ˆH0 + +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)|F(|k|)|2 +Dνc,νf +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : +� ++ +� +� +k,α,α′,η,s +� +a∈{1,2} +|F(|k|)|2H(fc,η),∗ +α′a +(k)H(fc,η) +αa +(k) +2NM +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� � +R +: f † +R,αηsfR,α′ηs : +− +� +k,α,η,s +� +a,a′∈{1,2} +|F(|k|)|2H(fc,η) +αa′ +(k)H(fc,η),∗ +αa +(k) +2 +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: c† +k,aηsck,a′ηs : +� ++ const +(S190) + +46 +E. +Effect of ˆHJ and PH ˆHfcPH +We have ignored the effect of ferromagnetic exchange coupling term ˆHJ and also part of the hybridization term PH ˆHfcPH. +We now add these terms to ˆH0 +ˆHcomplete +0 += ˆH0 + λ ˆHλ +ˆHλ = ˆHJ + PH ˆHfcPH +(S191) +with λ = 1 a dimensionless parameter introduced to keep track of the ˆHλ. We can perform SW transformation based on +ˆHcomplete +0 +and ˆH1. The corresponding S′ operator needs to satisfy +[S′, ˆHcomplete +0 +] = − ˆH1 . +(S192) +The analytical expression of S′ is complicated to find, but we can expand S′ in powers of λ +S′ = +∞ +� +n=0 +λnSn . +Then we have +[S′, ˆHcomplete +0 +] = − ˆH1 +⇔[ +∞ +� +n=0 +λnSn, ˆH0 + λ ˆHλ] = − ˆH1 +⇔[S0, ˆH0] + +∞ +� +n=1 +λn +� +[Sn, ˆH0] + [Sn−1, ˆHλ] +� += − ˆH1. +We can find Sn iteratively by requiring +[S0, ˆH0] = − ˆH1 +[Sn, ˆH0] = −[Sn−1, ˆHλ] +, +n ≥ 1. +S0 = S is the operator we derived previously (Eq. S168). +The effective Hamiltonian after SW transformation now becomes +( ˆH0 + ˆHλ) + 1 +2([S′, ˆH1]) = ˆH0 + λ ˆHJ + 1 +2[S0, ˆH1] + 1 +2 +∞ +� +n=1 +λn[Sn, ˆH1] . +We only keep the leading-order (λ0) contribution and the effective Hamiltonian is +ˆH′ +eff = +� +ˆH0 + ˆHJ + 1 +2[S, ˆH1] +� += ˆHeff + ˆHJ +(S193) +where ˆHeff is defined in Eq. S190. We are only interested in the states in the low-energy space, where the filling of f at each +site is νf, so we drop PH ˆHfcPH term. +F. +Effective Kondo model +We are now in the position to write down the effective Kondo model derived from SW transformation. Combining Eq. S193 +and Eq. S190, we have the following Kondo Hamiltonian +ˆHKondo = ˆH′ +eff = ˆH0 + ˆHJ ++ +� +R,k,k′,η,η′,s,s′α,α +� +a,a′∈{1,2} +ei(k−k′)R +NM +H(fc,η),∗ +α′a′ +(k′)H(fc,η′) +αa +(k)F(|k|)2 +Dνc,νf +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : +� ++ +� +� +R,k,α,α′,η,s +� +a∈{1,2} +H(fc,η),∗ +α′a +(k)H(fc,η) +αa +(k)F(|k|)2 +2NM +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: f † +R,αηsfR,α′ηs : +− +� +k,α,η,s +� +a,a′∈{1,2} +H(fc,η) +αa′ +(k)H(fc,η),∗ +αa +(k)F(|k|)2 +2 +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: c† +k,aηsck,a′ηs : +� +(S194) + +47 +We expand the hybridization matrix in powers of k and only keep the zeroth-order and linear order terms: +H(fc,η)(k) = +� +γ + O(|k|2) +v′ +⋆(ηkx − iky) + O(|k|2) 0 0 +v′ +⋆(ηkx + iky) + O(|k|2) +γ + O(|k|2) +0 0 +� +(S195) +The hybridization between f-fermions and Γ1 ⊕ Γ2 c-fermions is relatively weak and has been omitted. +Besides the four-fermion interactions term, the SW transformation introduces two additional fermion-bilinear term as shown +in Eq. S194. The first one is +� +R,k,α,α′,η,s +� +a +F(|k|)2H(fc,η),∗ +α′a +(k)H(fc,η) +αa +(k) +2NM +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: f † +R,αηsfR,α′ηs : += +� +R,k,α,η,s +F(|k|)2 γ2 + O(|k|2) +2NM +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: f † +R,αηsfR,αηs :≈ +� +k +F(|k|)2 γ2 +2 +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +νf +which is a constant. The second one corresponds to a conduction electron scattering term +ˆHcc = − +� +k,η,s +� +a,a′∈{1,2} +F(|k|)2H(fc,η) +αa′ +(k)H(fc,η),∗ +αa +(k) +2 +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +: c† +k,aηsck,a′ηs : += − +� +k,η,s +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +F(|k|)2 +� +a,a′∈{1,2} +� � +γ2/2 +γv′ +⋆(ηkx − iky) +γv′ +⋆(ηkx + iky) +γ2/2 +� +a,a′ ++ O(|k|2) +� +: c† +k,aηsck,a′ηs : += +� +k,η,s +� +a,a′∈{1,2} +[Hη +cc(k)]aa′ : c† +k,aηsck,a′ηs : +(S196) +where Hη +cc(k) = − +� +1 +D1,νc,νf + +1 +D2,νc,νf +� +F(|k|)2 +� +γ2 +2 σ0 + γv′ +⋆(ηkxσx + kyσy) +� +We define the four-fermion interaction induced by SW transformation in Eq. S194 as Kondo interactions ˆHK and rewrite it in +a more compact form: +ˆHK = +� +R,k,k′,α,α′,a,a′,η,η′,s,s′ +ei(k−k′)R H(fc,η) +αa +(k)(H(fc,η′) +α′a′ +(k′))∗ +NMDνc,νf +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : += +� +R,k,k′,α,α′,a,a′,η,η′,s,s′ +ei(k−k′)RF(|k|)F(|k′|) +�γ2δα′,a′δα,a +NMDνc,νf ++ γv′ +⋆δα,a[η′k′ +xσx − k′ +yσy]α′a′ +NMDνc,νf ++ γv′ +⋆δα′,a′[ηkxσx + kyσy]αa +NMDνc,νf ++ O(|k|2, |k′|2, |k||k′|) +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : +≈ +� +R,k,k′,α,α′,a,a′,η,η′,s,s′ +ei(k−k′)RF(|k|)F(|k′|) +�γ2δα′,a′δα,a +NMDνc,νf ++ γv′ +⋆δα,a[η′k′ +xσx − k′ +yσy]α′a′ +NMDνc,νf ++ γv′ +⋆δα′,a′[ηkxσx + kyσy]αa +NMDνc,νf +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : +(S197) +Using the U(4) momentum defined in Sec. S1 B and Eq. S25, we rewrite ˆHK as +ˆHK = +� +R,k,k′,α,α′,a,a′,η,η′,s,s′ +ei(k−k′)RF(|k|)F(|k′|) +�γ2δα′,a′δα,a +NMDνc,νf ++ γv′ +⋆δα,a[η′k′ +xσx − k′ +yσy]α′a′ +NMDνc,νf ++ γv′ +⋆δα′,a′[ηkxσx + kyσy]αa +NMDνc,νf +� +: f † +R,αηsfR,α′η′s′ :: c† +k′,a′η′s′ck,aηs : += +� +R,k,k′ +e−i(k′−k)·R F(|k|)F(|k′|) +Dνc,νf NM +� +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′ξ) +µν +(k, k′ − k) : ++ γv′ +⋆(k′ +x + iξ′k′ +y) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,−ξ′ξ) +µν +(k, k′ − k) : +γv′ +⋆(kx − iξky) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′,−ξ) +µν +(k, k′ − k) : +� +(S198) + +48 +where we have neglected the (v′ +⋆γ)2 which is the second order in k, and outsiede the scope of the Hamiltonian approximation in +Eq. S195. +We next use the real-space representation of U(8) moments as defined in Eq. S22, which indicates +ˆΣ(c′,ξξ′) +µν +(k, k′) = +1 +NM +� +k,k′ +e−ik·r′+ik′·r ˆΣ(c′,ξξ′) +µν +(r, r′) +Then Eq. S198 becomes +ˆHK = +� +R,r,r′,k,k′ +F(|k|)F(|k′|) +Dνc,νf N 2 +M +e−ik′·(R−r)+ik·(R−r′) � +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′ξ) +µν +(r, r′) : ++ γv′ +⋆(−i∂rx + ξ′∂ry) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,−ξ′ξ) +µν +(r, r′) : +γv′ +⋆(i∂r′x + ξ∂r′y) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′,−ξ) +µν +(r, r′) : +� +(S199) +where the derivative is defined as +∂rα : Σ(c′,ξ′ξ) +µν +(r, r′) := ∂rα +1 +NM +� +k +eik·r′−ik′·r : Σ(c′,ξ′ξ) +µν +(k, k′ − k) = +1 +NM +� +k +(−ik′ +α) : Σ(c′,ξ′ξ) +µν +(k, k′ − k)eik·r′−ik′·r +∂r′α : Σ(c′,ξ′ξ) +µν +(r, r′) := ∂r′α +1 +NM +� +k +eik·r′−ik′·r : Σ(c′,ξ′ξ) +µν +(k, k′ − k) = +1 +NM +� +k +(ikα) : Σ(c′,ξ′ξ) +µν +(k, k′ − k)eik·r′−ik′·r +The k summation appearing in Eq. S199 takes form of +1 +NM +� +k +F(|k|)eik·(R−r′) ≈ +1 +AMBZ +� +exp(−1 +2|k|2λ2)eik·(R−r′)d2k = +2π +AMBZ +e−|R−r′|2/λ2/2 +λ2 +. +For an arbitrary function of r, F(r), we notice +� +r +F(r) +� 1 +NM +� +k +F(|k|)eik·(R−r′) +� += +� +r +F(r) +2π +AMBZ +e−|R−r′|2/λ2/2 +λ2 +≈ 1 +Ω0 +� +r +F(r) +2π +AMBZ +e−|R−r′|2/λ2/2 +λ2 +For small λ = 0.3375aM, we can approximate F(r) with its value at R. Inserting above equations into Eq. S199, we have +� +r +F(r) +� 1 +NM +� +k +F(|k|)eik·(R−r′) +� +≈ 1 +Ω0 +F(R) +� +r +2π +AMBZ +e−|R−r′|2/λ2/2 +λ2 += F(R) = +� +r +δr,RF(r) +Finally, we have the following approximated real-space expression of ˆHK +ˆHK ≈ +� +R,k,k′,,ξ,ξ′ +e−i(k′−k)·R +1 +Dνc,νf NM +� +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) : +γv′ +⋆(k′ +x − iξ′k′ +y) : ˆΣ(f,ξ−ξ′) +µν +(R) : +γv′ +⋆(kx + iξky) : ˆΣ(f,−ξξ′) +µν +(R) : +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k) : += +� +R,ξ,ξ′ +γ2 +Dνc,νf +: ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′,ξ) +µν +(R, R) : ++ +� +R,r,ξ,ξ′ +γv′ +⋆ +Dνc,νf +δr,R +� +: ˆΣ(f,ξ−ξ′) +µν +(R) : (i∂rx + ξ′∂ry) : ˆΣ(c′,ξ′ξ) +µν +(R, r) : + : ˆΣ(f,−ξξ′) +µν +(R) : (−i∂rx + ξ∂ry) : ˆΣ(c′,ξ′ξ) +µν +(r, R) : +� +(S200) +In summary, we have the following two four-fermion interactions: Kondo coupling ˆHK and ferromagnetic exchange ˆHJ: +ˆHK = +� +R,r,r′,ξ,ξ′ +δr,Rδr′,R +Dνc,νf +� +µν +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) : ++ γv′ +⋆ : ˆΣ(f,ξ−ξ′) +µν +(R) : (i∂r′ +x + ξ′∂r′ +y) + γv′ +⋆ : ˆΣ(f,−ξξ′) +µν +(R) : (−i∂rx + ξ∂ry) +� +: ˆΣ(c′,ξ′,ξ) +µν +(r, r′) : +ˆHJ =(−J) +� +R,µν,ξ +: ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(c′′,ξξ) +µν +(R, R) +(S201) + +49 +Here, we also provide the interactions in the momentum space (k-space of c-electrons) +ˆHK = +� +R,k,k′ +e−i(k′−k)·R F(|k|)F(|k′|) +Dνc,νf NM +� +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′ξ) +µν +(k, k′ − k) : ++ γv′ +⋆(k′ +x + iξ′k′ +y) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,−ξ′ξ) +µν +(k, k′ − k) : +γv′ +⋆(kx − iξky) : ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(c′,ξ′,−ξ) +µν +(k, k′ − k) : +� +ˆHJ = − J +� +R,k,k′,ξ +e−i(k′−k)·R +NM +� +µν,ξ +: ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(c′′,ξξ) +µν +(k, k′ − k) : +(S202) +The one-body scattering term from SW transformation is given in Eq. S196, and is listed below +ˆHcc = +� +k,η,s +� +a,a′∈{1,2} +[Hη +cc(k)]aa′ : c† +k,aηsck,a′ηs : +Hη +cc(k) = − +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +F(|k|)2 +�γ2 +2 σ0 + γv′ +⋆(ηkxσx + kyσy) +� +(S203) +The effective Kondo model now reads +ˆHKondo = ˆH0 + ˆHK + ˆHJ + ˆHcc . +(S204) +Here, we also want to mention that the model is defined with respect to the Hilbert space with the filling of f electrons being +νf at each site. We estimate the coupling strength here +νf +0 +-1 +-2 +J +16meV +16meV 16meV +γ2 +Dνc=0,νf +42meV +49meV 99meV +γ|v′ +⋆|/aM +Dνc=0,νf 20.6meV 24meV 48meV +where γ = −24.75meV , v′ +⋆/aM = 12.08meV , aM is the moir´e lattice constant and 1/D = 1/Dνc=0,νf . +S5. +RKKY INTERACTIONS AT M = 0 +Based on the Kondo model defined in Eq. S204, we derive the corresponding RKKY interactions by integrating out conduction +electrons. +The action of the effective Hamiltonian ˆHeff (Eq. S204) can be separated into +S = S0 + S1 +S0 = Sf + Sc +, +S1 = SK + SJ + Scc . +Sf and Sc denote the effective action of f- and c- electrons that have been introduced in Eq. S113 and Eq. S114 and are also +given below +Sf = +� +R,αηs +� β +0 +dτf † +R,αηs(τ)∂τfR,αηs(τ)dτ + i +� +R +� β +0 +λR(τ)[ +� +αηs +f † +R,αηs(τ)fR,αηs(τ) − 4 − νf)]dτ +Sc = +� +k,aηs +� β +0 +dτc† +k,aηs(τ)∂τck,aηs(τ)dτ + +� β +0 +� +η,s,a,a′,k +� +H(c,η) +a,a′ + (−µ + Wνf + V0 +Ω0 +νc)δa,a′ +� +c† +k,aηs(τ)ck,a′ηs(τ)dτ +− V0NM +2Ω0 +ν2 +c − (Wνf + V0 +Ω0 +νc) +� +k +8 +(S205) +where we have introduced Lagrangian multiplier λR(τ) to fix the filling of f-electrons. +ˆHU and chemical potential term of +f-electrons are a constants after fixing the filling of f-electrons and have been omitted. We also treat ˆHV via a mean-field +approximation. SK, SJ, Scc are defined as +SK = +� β +0 +ˆHK(τ)dτ, +SJ = +� β +0 +ˆHJ(τ)dτ, +Scc = +� β +0 +ˆHcc(τ)dτ . +(S206) + +50 +ˆHK(τ), ˆHJ(τ), ˆHcc(τ) take the same for as ˆHK (Eq. S202), ˆHJ (Eq. S202), ˆHcc (Eq. S203), but with fR,αηs replaced by +fR,αηs(τ), ck,aηs replaced by ck,aηs(τ), where τ denotes imaginary time. +We now integrate out the c-electrons in the partition functions and perform the cumulant expansion: ⟨eO⟩ = exp +� +⟨O⟩ + +1 +2⟨O2⟩ − 1 +2⟨O⟩2 + ... +� +. +Z = +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)] +� +D[c† +k,aηs(τ), ck,aηs(τ)]e−S0−S1 += +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)]e−Sf ⟨e−S1⟩0 +≈ +� +D[f † +R,αηs(τ), fR,αηs(τ), λR(τ)] exp +� +− Sf − ⟨S1⟩0 + 1 +2⟨S2 +1⟩0 − 1 +2(⟨S1⟩0)2 + ... +� +where +⟨O⟩0 := 1 +Z0 +� +D[c† +k,aηs(τ), ck,aηs(τ)]Oe−Sc +, +Z0 := +� +D[c† +k,aηs(τ), ck,aηs(τ)]e−Sc +The effective actions of f fermions then becomes +Seff = Sf + ⟨SK + SJ + Scc⟩0 − +�⟨S2 +J⟩0,con +2 ++ ⟨S2 +K⟩0,con +2 ++ ⟨SJSK⟩0,con + ⟨S2 +cc⟩0,con +2 ++ ⟨(SK + SJ)Scc⟩0,con +� +(S207) +with ⟨AB⟩0,con = ⟨AB⟩0 − ⟨A⟩0⟨B⟩0. We treat S1 as perturbation, since all terms in S1 are either generated from SW +transformation that is proportional to γ2, (v′ +⋆)2, γv′ +⋆, or coming from the ferromagnetic exchange coupling which is proportional +to J. We derive the effective action at M = 0 and integer filling ν = 0, −1, −2 with νf = ν, νc = 0. +We first note that, Scc (see Eq. S206 and Eq. S203) does not contain any f electron operators, so ⟨Scc⟩0 and ⟨S2 +cc⟩0,con only +give constant contributions. As for the linear term: ⟨SK + SJ⟩0, ⟨SK⟩0 gives +⟨SK⟩0 = +� β +0 +� +R,k,k′,,ξ,ξ′ +e−i(k′−k)·R +1 +Dνc,νf NM +⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :⟩0 +� +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R, τ) : +γv′ +⋆(k′ +x − iξ′k′ +y) : ˆΣ(f,ξ−ξ′) +µν +(R, τ) : +γv′ +⋆(kx + iξky) : ˆΣ(f,−ξξ′) +µν +(R, τ) : +� +dτ +(S208) +We need to calculate ⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :⟩0. Due to the flat U(4) symmetry, only µν = 00 component can be non-zero +for the same reason proved around Eq. S84. The momentum conservation requires k′ = k. Due to imaginary-time translational +symmetry, we only need to consider τ = 0 +⟨: ˆΣ(c′,ξ′ξ) +µν +(k, 0)⟩ = δµ,0δν,0⟨ +� +m +ψc′,ξ,† +k,m ψc′,ξ′ +k,m ⟩0 = δµ,0δν,0δξ,ξ′⟨ +� +m +ψc′,ξ,† +k,m ψc′,ξ +k,m⟩0 +(S209) +The above quantity is just the filling of conduction electrons in orbitals 1, 2 with index ξ. At νc = 0, this becomes zero. Then +⟨SK⟩0 = 0. Another linear term is ⟨SJ⟩0. We have +⟨SJ⟩0 = +� β +0 +(−J) +� +R,µν,ξ +: ˆΣ(f,ξξ) +µν +(R, τ) : ⟨: ˆΣ(c′′,ξξ) +µν +(R, R, τ)⟩0dτ +(S210) +which requires the calculation of ⟨: ˆΣ(c′′,ξξ) +µν +(k, k′ − k)⟩0. Similarly, the only non-zero component corresponds to µν = 00 and +k′ − k. We have +⟨: ˆΣ(c′′,ξξ) +µν +(k, 0)⟩ = ⟨ +� +m +ψc′′,ξ,† +k,m ψc′′,ξ +k,m⟩0 +(S211) +which is the filling of conduction electrons in orbitals 3, 4 with index ξ. At νc = 0, this becomes zero. Then ⟨SJ⟩0 = 0. + +51 +The remaining terms give the following spin-spin interaction +SRKKY = − +�⟨S2 +J⟩0,con +2 ++ ⟨S2 +K⟩0,con +2 ++ ⟨SJSK⟩0,con +� +(S212) +The effective action becomes +Seff = Sf + SRKKY . +(S213) +Before calculating SRKKY , here, we first introduce the following single-particle Green’s function at M = 0, as also derived +in Sec. S9, +Gaa′,η(k, τ) = −⟨Tτck,aηs(τ)c† +k,a′ηs(0)⟩ +(S214) +where Tτ denotes time ordering. We further introduce g0(k, τ) and g2,η(k, τ) (Sec. S9, Eq. S544) as +g0(k, τ) = G11,η(k, τ) = G22,η(k, τ) = G33,η(k, τ) = G44,η(k, τ) +g2,η(k, k) = G13,η(k, τ) = G∗ +24,η(k, −τ) = G∗ +31,η(k, −τ) = G42,η(k, τ) +(S215) +A. +−⟨(SK + SJ)Scc⟩0,con +We now prove −⟨(SK + SJ)Scc⟩0,con only produces a constant contribution. For ⟨SKScc⟩0,con, we have +⟨SKScc⟩0,con = +� β +0 +� β +0 +� +R,k,k′,k2,ξ,ξ′ +e−i(k′−k)·R � +η2s2 +� +a2,a′ +2∈1,2 +� +µν +1 +Dνc,νf NM +[Hη2 +cc (k2)]a2a′ +2 +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) : +γv′ +⋆(k′ +x − iξ′k′ +y) : ˆΣ(f,ξ−ξ′) +µν +(R, τ1) : +γv′ +⋆(kx + iξky) : ˆΣ(f,−ξξ′) +µν +(R, τ1) : +� +⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,condτ1dτ2 +(S216) +We need to evaluate ⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con. For ξ′ = ξ, we have +⟨: ˆΣ(c′,ξ,ξ) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +η,η′,s,s′ +� +a,a′∈{1,2} +1 +2Aµν +aηs,a′η′s′δξ,(−1)a+1ηδξ,(−1)a′+1η′⟨: c† +k′,aηs(τ1)ck,a′η′s′(τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +η,η′,s,s′ +� +a,a′∈{1,2} +1 +2Aµν +aηs,a′η′s′δξ,(−1)a+1ηδξ,(−1)a′+1η′δa,a′ +2δs,s2δs′,s2δη,η2δη′,η2δk,k2δk′,k2 +(−1)g0(τ1 − τ2, k2)g0(τ2 − τ1, k2) += − 2δµ,0δν,0g0(τ1 − τ2, k2)g0(τ2 − τ1, k2)δk,k2δk′,k2δa2,a′ +2 +(S217) +where we use Wick’s theorem and the definition of Green’s function in Eq.S215. As for ξ′ = −ξ = −1, we have +⟨: ˆΣ(c′,−1,+1) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +η,η′,s,s′ +� +a,a′∈{1,2} +1 +2Aµν +aηs,a′η′s′δ−1,(−1)a+1ηδ+1,(−1)a′+1η′⟨: η′c† +k′,aηs(τ1)ck,a′η′s′(τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +a,a′∈{1,2} +1 +2Aµν +a′ +2η2s2,a2η2s2δ−1,(−1)a′ +2+1η2δ+1,(−1)a2+1η2η2(−1)g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) +=g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) +� +a,a′∈{1,2} +1 +2Aµν +a′ +2η2s2,a2η2s2δ−1,(−1)a′ +2+1η2δ+1,(−1)a2+1η2η2(−1) +=g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) × 0 = 0 +(S218) + +52 +where we use Wick’s theorem and the definition of Green’s function in Eq.S215. Similarly, for ξ′ = −ξ = +1 +⟨: ˆΣ(c′,+1,+−1) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +η,η′,s,s′ +� +a,a′∈{1,2} +1 +2Aµν +aηs,a′η′s′δ1,(−1)a+1ηδ−1,(−1)a′+1η′⟨: ηc† +k′,aηs(τ1)ck,a′η′s′(τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += +� +a,a′∈{1,2} +1 +2Aµν +a′ +2η2s2,a2η2s2δ1,(−1)a′ +2+1η2δ−1,(−1)a2+1η2η(−1)g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) +=g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) +� +a,a′∈{1,2} +1 +2Aµν +a′ +2η2s2,a2η2s2δ1,(−1)a′ +2+1η2δ−1,(−1)a2+1η2η(−1) +=g0(τ1 − τ2, k2)g2(τ2 − τ1, k2) × 0 = 0 +(S219) +where we use Wick’s theorem and the definition of Green’s function in Eq.S215 +Combining Eq. S216, Eq. S217, Eq. S218 and Eq. S219, we have +⟨SKScc⟩0,con += +� β +0 +� β +0 +� +R,k,k′,k2ξ +e−i(k′−k)·R � +η2s2 +� +a2,a′ +2∈1,2 +� +µν +1 +Dνc,νf NM +[Hη2 +cc (k2)]a2a′ +2 +� +γ2 : ˆΣ(f,ξξ) +µν +(R) : +γv′ +⋆(k′ +x − iξk′ +y) : ˆΣ(f,ξ−ξ) +µν +(R, τ1) : +γv′ +⋆(kx + iξky) : ˆΣ(f,−ξξ) +µν +(R, τ1) : +� +⟨: ˆΣ(c′,ξ,ξ) +µν +(k, k′ − k, τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,condτ1dτ2 += +� β +0 +� β +0 +� +R,k,ξ +� +η2s2 +� +a2∈1,2 +� +µν +1 +Dνc,νf NM +[Hη2 +cc (k2)]a2a2 +� +γ2 : ˆΣ(f,ξξ) +00 +(R) : +γv′ +⋆(kx − iξky) : ˆΣ(f,ξ−ξ) +00 +(R, τ1) : +γv′ +⋆(kx + iξky) : ˆΣ(f,−ξξ) +00 +(R, τ1) : +� +(−2)g0(τ1 − τ2, k)g0(τ2 − τ1, k)dτ1dτ2 += +� � β +0 +� β +0 +� +R,k +� +η2s2 +� +a2∈1,2 +� +µν +1 +Dνc,νf NM +[Hη2 +cc (k2)]a2a2γ2(−1)g0(τ1 − τ2, k)g0(τ2 − τ1, k) . +(S220) +We note that g0(k, τ) only depends on |k| and τ (Sec. S9, Eq. S553). Then the terms proportional to γv′ +⋆ vanish after k +summation due to the linear-k factor. We find ⟨SKScc⟩0,con is a constant. +For ⟨SJScc⟩0,con, we have +⟨SJScc⟩0,con = +� β +0 +� β +0 +� +R,k,k′,k2,ξ +e−i(k′−k)·R � +η2s2 +� +a2,a′ +2∈1,2 +� +µν +−J +NM +[Hη2 +cc (k2)]a2a′ +2 : ˆΣ(f,ξξ) +µν +(R, τ) : +⟨: ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,condτ1dτ2 +(S221) +We need to evaluate ⟨: ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con, which gives +⟨: ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con +=1 +2 +� +η,η′,s,s′ +� +a,a′∈{3,4} +Bµν +aηs,a′η′s′δξ,(−1)a+1ηδξ′,(−1)a′+1η′⟨: c† +k′,aηs(τ1)ck,a′η′s′(τ1) :: c† +k2,a2η2s2(τ2)ck2,a′ +2η2s2(τ2) :⟩0,con += − 1 +2 +� +η,η′,s,s′ +� +a,a′∈{1,2} +Bµν +aηs,a′η′s′δξ,(−1)a+1ηδξ′,(−1)a′+1η′δη,η2δη′,η2δs,s2δs′,s2δa+2,a′ +2δa′+2,a2δa,a′ +(−1)δk2,kδk2,k′|g2,η(k2, τ1 − τ2)|2 +=δµ,0δν,0(−1 +2)δξ,(−1)a2+1η2δa2,a′ +2δk2,k′δk2,k|g2,η(k, τ1 − τ2)|2 +(S222) + +53 +where we find only µ = 0, ν = 0 components can be non-zero, due to structures of Bµν (Eq. S14). Then combine Eq. S221 and +Eq. S222, we find +⟨SJScc⟩0,con = +� � β +0 +� β +0 +� +R,k +� +a2,η +−J +NM +[Hη +cc]a2,a2|g2,η(k, τ1 − τ2)|2(−1 +2)τ1dτ2 +� +νf +which is a constant. +B. +−⟨SKSJ⟩0,con +We then calculate −⟨SKSJ⟩0,con: +− ⟨SKSJ⟩0,con += +� β/2 +−β/2 +� β/2 +−β/2 +� +R,R2,r,r′ +J +Dνc,νf +δr,Rδr2,R : ˆΣ(f,ξ2ξ2) +µ2ν2 +(R2, τ2) : +� +µν,ξξ′ +� +µ2ν2,ξ2ξ2 +� +γ2 : ˆΣ(f,ξξ′) +µν +(R, τ) : +γv′ +⋆(i∂r′x + ξ′∂r′y) : ˆΣ(f,ξ−ξ′) +µν +(R, τ) : +γv′ +⋆(−i∂rx + ξ∂ry) : ˆΣ(f,−ξξ′) +µν +(R, τ) : +� +⟨: ˆΣ(c′,ξ′ξ) +µ2ν2 +(r, r′, τ) :: ˆΣ(c′′,ξ2ξ2) +µν +(R2, τ2) :⟩0,condτdτ2 +As we derived in Sec. S11, Eq. S579, we represent ⟨: ˆΣ(c′,ξ′ξ) +µ2ν2 +(r, r′, τ) :: ˆΣ(c′′,ξ2ξ2) +µν +(R2, τ2) :⟩0,con with single-particle +Green’s function and find +− ⟨SKSJ⟩0,con += +� β/2 +−β/2 +� β/2 +−β/2 +� +R,R2,r,r′ +J +Dνc,νf +� +µν,ξ +� +γ2 : ˆΣ(f,ξξ) +µν +(R, τ) : +(v′ +⋆)2(−i∂rx + ξ∂ry)(i∂r′x + ξ′∂r′y) : ˆΣ(f,−ξ−ξ) +µν +(R, τ) : ++ γv′ +⋆(i∂r′x + ξ′∂r′y) : ˆΣ(f,ξ−ξ) +µν +(R, τ) : +γv′ +⋆(−i∂rx + ξ∂ry) : ˆΣ(f,−ξξ) +µν +(R, τ) : +� +: ˆΣ(f,ξξ) +µν +(R2, τ2) : +(−1)g∗ +2,ξ(r′ − R2, τ − τ2)g2,ξ(r − R2, τ − τ2)dτdτ2 += +� β/2 +−β/2 +� β/2 +−β/2 +� +R,R2,r,r′ +(−J) +� +− +1 +D1,νc,νf ++ +1 +D2,νc,νf +� � +µν,ξ +� +γ2 : ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : g∗ +2,ξ(R − R2, τ − τ2)g2,ξ(R − R2, τ − τ2) ++ (v′ +⋆)γ : ˆΣ(f,ξ−ξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : [(i∂Rx + ξ∂Ry)g∗ +2,ξ(R − R2, τ − τ2)][g2,ξ(R − R2, τ − τ2)] ++ (v′ +⋆)γ : ˆΣ(f,−ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : [g∗ +2,ξ(R − R2, τ − τ2)][(−i∂Rx + ξ∂Ry)g2,ξ(R − R2, τ − τ2) +� +dτdτ2 +where the Green’s function is defined in Sec. S9, Eq. S542 and Eq. S544. +Using the analytical expression of single-particle Green’s function derived in Sec. S9, Eq. S553 and Eq. S554 at zero temper- +ature and infinite momentum cutoff, we have +− ⟨SKSJ⟩0,con += +� ∞ +−∞ +� ∞ +−∞ +� +R,R2 +(−J) +� +− +1 +D1,νc,νf ++ +1 +D2,νc,νf +� � +µν,ξ +� +γ2 : ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : +π2|R − R2|2 +A2 +MBZ(|v⋆(τ − τ2)|2 + |R − R2|2)3 ++ (v′ +⋆)γ : ˆΣ(f,ξ−ξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : +−3π2|R − R2|2 +A2 +MBZ(|v⋆(τ − τ2)|2 + |R − R2|2)4 +� +ξ(Ry − R2,y) + i(Rx − R2,x) +� ++ (v′ +⋆)γ : ˆΣ(f,−ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : +−3π2|R − R2|2 +A2 +MBZ(|v⋆(τ − τ2)|2 + |R − R2|2)4 +� +ξ(Ry − R2,y) − i(Rx − R2,x +�� +dτdτ2 +(S223) + +54 +C. +− 1 +2⟨S2 +K⟩0,con +We now calculate − 1 +2⟨S2 +K⟩0,con: +− 1 +2⟨S2 +K⟩0,con += − +� β/2 +−β/2 +� β/2 +−β/2 +� +R,r,r′ +� +R2,r2,r′ +2 +1 +2D2νc,νf +δr,Rδr′,Rδr2,R2δr′ +2,R2 +� +µν,ξξ′ +� +γ2 : ˆΣ(f,ξξ′) +µν +(R, τ) : +γv′ +⋆(i∂r′x + ξ′∂r′y) : ˆΣ(f,ξ−ξ′) +µν +(R, τ) : +γv′ +⋆(−i∂rx + ξ∂ry) : ˆΣ(f,−ξξ′) +µν +(R, τ) : +� +� +µ2ν2,ξ2ξ′ +2 +� +γ2 : ˆΣ(f,ξ2ξ′ +2) +µ2ν2 +(R2, τ2) : +γv′ +⋆(i∂r′ +2,x + ξ′ +2∂r′ +2,y) : ˆΣ(f,ξ2−ξ′ +2) +µ2ν2 +(R2, τ2) : +γv′ +⋆(−i∂r2,x + ξ2∂r2,y) : ˆΣ(f,−ξ2ξ′ +2) +µ2ν2 +(R2, τ2) : +� +⟨: ˆΣ(c′,ξ′,ξ) +µν +(r, r′, τ) :: ˆΣ(c′,ξ′ +2,ξ2) +µ2ν2 +(r2, r′ +2, τ2) :⟩0,condτdτ2 +We rewrite the above equations with single-particle Green’s function as derived in Sec. S11, Eq. S572 +− 1 +2⟨S2 +K⟩0,con += − +� β/2 +−β/2 +� β/2 +−β/2 +� +R,r,r′ +� +R2,r2,r′ +2 +� +ξ,ξ′ +1 +2D2νc,νf +δr,Rδr′,Rδr2,R2δr′ +2,R2 +� +µν +� +γ2 : ˆΣ(f,ξξ′) +µν +(R, τ) : +γv′ +⋆(i∂r′x + ξ′∂r′y) : ˆΣ(f,ξ−ξ′) +µν +(R, τ) : +γv′ +⋆(−i∂rx + ξ∂ry) : ˆΣ(f,−ξξ′) +µν +(R, τ) : +� +� +γ2 : ˆΣ(f,ξ′ξ) +µν +(R2, τ2) : +γv′ +⋆(i∂r′ +2,x + ξ∂r′ +2,y) : ˆΣ(f,ξ′−ξ) +µν +(R2, τ2) : +γv′ +⋆(−i∂r2,x + ξ′∂r2,y) : ˆΣ(f,−ξ′ξ) +µ2ν2 +(R2, τ2) : +� +(−1)g0(r′ +2 − r, τ2 − τ)g0(r′ − r2, τ − τ2)dτdτ2 += +� β/2 +−β/2 +� β/2 +−β/2 +� +R,r,r′ +� +R2,r2,r′ +2 +� +ξ,ξ′ +1 +2D2νc,νf +� +µν +� +: ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,ξ′ξ) +µν +(R2, τ2) : γ4g0(R2 − R, τ2 − τ)g0(R − R2, τ − τ2) ++ : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,−ξ′−ξ) +µν +(R2, τ2) : 2γ2(v′ +⋆)2[(i∂x + ξ∂y)g0(R2 − R, τ2 − τ)][(i∂x − ξ′∂y)g0(R − R2, τ − τ2)] ++ : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,ξ′−ξ) +µν +(R2, τ2) : 2γ3v′ +⋆(i∂Rx + ξ∂Ry)g0(R2 − R, τ2 − τ)]g0(R − R2, τ − τ2) ++ : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,−ξ′ξ) +µν +(R2, τ2) : 2γ3v′ +⋆g0(R2 − R, τ2 − τ)(i∂Rx − ξ∂Ry)g0(R − R2, τ − τ2)] +� +dτdτ2 +(S224) +where the Green’s function is defined in Sec. S9, Eq. S542 and Eq. S544. Here, we mention that we expand Kondo interaction +in powers of k (note that the hybridization matrix has been expanded in powers of k in Eq. S198). Thus, γ4, γ3v′ +⋆ correspond +to zeroth and linear-order terms and are kept. In addition, we notice that γ2(v′ +⋆)2 term in Eq. S224 provide leading-order +contributions in the interaction channel : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,−ξ′−ξ) +µν +(R2, τ2) :. In other words, even if we expand to Kondo +interaction to higher orders in k, the current γ2(v′ +⋆)2 term still gives the leading order contributions and should be kept. +At zero temperature, we utilize the analytical expression of single-particle Green’s function derived in Sec. S9, Eq. S553 and + +55 +obtain +− 1 +2⟨S2 +K⟩0,con += +� ∞ +−∞ +� ∞ +−∞ +� +R,R2 +� +ξ,ξ′ +1 +2D2νc,νf +� +µν +� +− : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,ξ′ξ) +µν +(R2, τ2) : γ4 +π2 +A2 +MBZ +v2 +⋆|τ − τ2|2 +� +|v⋆|2τ 2 + |R − R2|2 +�3 +− : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,−ξ′−ξ) +µν +(R2, τ2) : +9γ2(v′ +⋆)2π2|v⋆(τ − τ2)|2 +A2 +MBZ +� +v2⋆τ 2 + |R − R2|2 +�5 (i(Rx − R2,x) + ξ(Ry − R2,y)) +(S225) +(−i(Rx − R2,x) + ξ′(Ry − R2,y)) ++ : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,ξ′−ξ) +µν +(R2, τ2) : +6γ3v′ +⋆π2|v⋆(τ − τ)|2 +� +i(R2,x − Rx) + ξ(R2,y − Ry) +� +A2 +MBZ +� +v2⋆τ 2 + |R − R2|2 +�4 ++ : ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,−ξ′ξ) +µν +(R2, τ2) : +6γ3v′ +⋆π2|v⋆(τ − τ)|2 +� +− i(R2,x − Rx) + ξ(R2,y − Ry) +� +A2 +MBZ +� +v2⋆τ 2 + |R − R2|2 +�4 +� +dτdτ2 +(S226) +D. +− 1 +2⟨S2 +J⟩0,con +Finally, we calculate − 1 +2⟨S2 +J⟩0,con. +− 1 +2⟨S2 +J⟩0,con += − J2 +2 +� β/2 +−β/2 +� β/2 +−β/2 +� +µνµ2ν2,ξξ2,R,R2 +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξ2ξ2) +µ2ν2 +(R2, τ2) : ⟨:: ˆΣ(c′′,ξξ) +µν +(R, R, τ) :: ˆΣ(c′′,ξ2ξ2) +µ2ν2 +(R2, R, τ2) : dτdτ2 +Expressing in terms of single-particle Green’s function as we derived in Sec.S11, Eq. S573, we have +− 1 +2⟨S2 +J⟩0,con += − J2 +2 +� β/2 +−β/2 +� β/2 +−β/2 +� +µν,ξ,R,R2 +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : (−1)g0(R2 − R, τ2 − τ)g0(R − R2, τ − τ2)dτdτ2 +where the Green’s function is defined in Sec. S9, Eq. S542 and Eq. S544. Using the analytical expression of single-particle +Green’s function at zero temperature and infinite momentum cutoff as derived in Sec. S9, Eq. S553, we have +− 1 +2⟨S2 +J⟩0,con = −J2 +2 +� ∞ +−∞ +� β/2 +−β/2 +� +µν,ξ,R,R2 +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : +π2|v⋆(τ − τ2)|2 +A2 +MBZ +� +(v⋆(τ − τ2))2 + |R − R2|2 +�3 dτdτ2 +(S227) + +56 +E. +RKKY interactions +We sum over all three terms in Eq. S223, Eq. S226, Eq. S227, and find the analytical formula of SRKKY = −⟨SKSJ⟩0,con − +1 +2⟨S2 +K⟩0,con − 1 +2⟨S2 +J⟩0,con: +SRKKY = +� � +� +R,R2,µν +� � +ξξ′ +: ˆΣ(f,ξξ′) +µν +(R, τ) :: ˆΣ(f,ξ′ξ) +µν +(R2, τ2) : χ0(R − R2, τ − τ2) ++ +� +ξ +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξξ) +µν +(R2, τ2) : χ1(R − R2, τ − τ2) + +� +ξ +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,−ξ−ξ) +µν +(R2, τ2) : χ2(R − R2, τ − τ2) ++ +� +ξ +: ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,−ξξ) +µν +(R2, τ2) : χ3,ξ(R − R2, τ − τ2)+ : ˆΣ(f,ξξ) +µν +(R, τ) :: ˆΣ(f,ξ−ξ) +µν +(R2, τ2) : χ4,ξ(R − R2, τ − τ2) ++ +� +ξ +: ˆΣ(f,ξ−ξ) +µν +(R, τ) :: ˆΣ(f,ξ−ξ) +µν +(R2, τ2) : χ5,ξ(R − R2, τ − τ2) +� +dτdτ2 +(S228) +where +χ0(R, τ) = − +γ4 +2D2νc,νf +π2|v⋆τ|2 +A2 +MBZ +� +|v⋆τ|2 + |R|2 +�3 +χ1(R, τ) = −Jγ2 +Dνc,νf +π2|R|2 +A2 +MBZ(|v⋆τ|2 + |R|2)3 − J2 +2 +π2|v⋆τ|2 +A2 +MBZ +� +|v⋆τ|2 + |R|2 +�3 +χ2(R, τ) = − 9γ2(v′ +⋆)2 +D2νc,νf +π2|v⋆τ|2|R|2 +A2 +MBZ +� +|v⋆τ|2 + |R|2 +�5 +χ3,ξ(R, τ) =−3Jv′ +⋆γ +Dνc,νf +π2|R|2 +A2 +MBZ(|v⋆τ|2 + |R|2)4 +� +− ξRy + iRx +� ++ 3γ3v′ +⋆ +D2νc,νf +π2|v⋆τ|2 +� +− ξRy + iRx +� +A2 +MBZ +� +v2⋆τ 2 + |R|2 +�4 +χ4,ξ(R, τ) =γ 3Jv′ +⋆ +Dνc,νf +π2|R|2 +A2 +MBZ(|v⋆τ|2 + |R|2)4 +� +ξRy + iRx +� ++ 3γ3v′ +⋆ +D2νc,νf +π2|v⋆(τ − τ)|2 +� +− ξRy − iRx +� +A2 +MBZ +� +|v⋆τ|2 + |R|2 +�4 +χ5,ξ(R, τ) = − 9γ2(v′ +⋆)2 +D2νc,νf A +π2|v⋆τ|2 +A2 +MBZ +� +v2⋆τ 2 + |R|2 +�5 +� +[R2 +x − R2 +y] − 2iξRxRy] +� +(S229) +The corresponding RKKY interactions can be derived by taking the zero-frequency contribution of χ in Eq. S229, which leads +to the following RKKY Hamiltonian +HRKKY += +� +R,R2,µν +� +ξξ′ +: ˆΣ(f,ξξ′) +µν +(R) :: ˆΣ(f,ξ′ξ) +µν +(R2) : JRKKY +0 +(R − R2) ++ +� +ξ +: ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,ξξ) +µν +(R2) : JRKKY +1 +(R − R2) + +� +ξ +: ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R2) : JRKKY +2 +(R − R2) ++ +� +ξ +: ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξξ) +µν +(R2) : JRKKY +3,ξ +(R − R2)+ : ˆΣ(f,ξξ) +µν +(R) : ˆΣ(f,ξ−ξ) +µν +(R2) : JRKKY +4,ξ +(R − R2) ++ +� +ξ +: ˆΣ(f,ξ−ξ) +µν +(R) :: ˆΣ(f,ξ−ξ) +µν +(R2) : JRKKY +5,ξ +(R − R2) +(S230) + +57 +where the RKKY interactions are defined as +JRKKY +0 += +� ∞ +−∞ +χ0(R, τ))dτ, +JRKKY +1 += +� ∞ +−∞ +χ1(R, τ))dτ, +JRKKY +2 += +� ∞ +−∞ +χ2(R, τ))dτ +JRKKY +3,ξ += +� ∞ +−∞ +χ3,ξ(R, τ))dτ, +JRKKY +4,ξ += +� ∞ +−∞ +χ4,ξ(R, τ))dτ, +JRKKY +5,ξ += +� ∞ +−∞ +χ5,ξ(R, τ))dτ +(S231) +The analytical expressions of RKKY interactions are given below +JRKKY +0 +(R) = +−π3 +|v⋆|A2 +MBZ +� +γ4 +16D2νc,νf +� +1 +|R|3 +JRKKY +1 +(R) = − +π3 +8A2 +MBZ|v⋆| +� 3Jγ2 +Dνc,νf ++ J2 +2 +� +1 +|R|3 +JRKKY +2 +(R) = − +π3 +A2 +MBZ|v⋆| +45γ2(v′ +⋆)2 +128D2νc,νf +1 +R5 +JRKKY +3,ξ +(R) = − +3π3 +16A2 +MBZ|v⋆| +� 5v′ +⋆γJ +Dνc,νf ++ v′ +⋆γ3 +D2νc,νf +�(ξRy − iRx) +|R|5 +JRKKY +4,ξ +(R) = − +3π3 +16A2 +MBZ|v⋆| +� 5v′ +⋆γJ +Dνc,νf ++ v′ +⋆γ3 +D2νc,νf +�(ξRy + iRx) +|R|5 +JRKKY +5,ξ +(R) = − +π3 +A2 +MBZ|v⋆| +45γ2(v′ +⋆)2 +128D2νc,νf +[R2 +x − R2 +y] − 2iξRxRy +|R|7 +(S232) +We also comment that RKKY interactions diverge at R → 0. During the derivation of RKKY interactions, when we perform +momentum integral, we have set the cutoff of momentum integral to infinity to obtain an analytical expression. This introduces a +short-distance divergence. The divergence can be cured by introducing a short-distance cutoff (aM). In practice, we can replace +|R| in the denominators of Eq. S232 by +� +|R|2 + a2 +M. +Finally, we provide the Fourier transformation of JRKKY +0/1/2/3,ξ/4,ξ(k) = � +R JRKKY +0/1/2/3,ξ/4,ξ(R)e−ik·R using the Fourier trans- +formation derived in Sec. S12, Eq. S581. During the Fourier transformation, we also introduce a short-distance cutoff aM (moir´e +lattice constant) RKKY interactions in the momentum space now take the form of +JRKKY +0 +(k) = +−π3 +|v⋆|aMAMBZ +γ4 +64D2νc,νf +(I0(kaM) − L0(kaM)) +JRKKY +1 +(k) = − +π3 +32AMBZaM|v⋆| +� 3Jγ2 +Dνc,νf ++ J2 +2 +� +(I0(kaM) − L0(kaM)) +JRKKY +2 +(k) = − +π2 +AMBZ|v⋆|a3 +M +45γ2(v′ +⋆)2 +3072D2νc,νf +1 +kaM +� +3π(−2 + k2a2 +M)L1(kaM) + qaM +� +4qaM + 3πI0(qaM) +− 3πqaMI1(qaM) − 3πL2(qaM) +�� +JRKKY +3,ξ +(k) = − +3π2 +32A2 +MBZ|v⋆|a2 +M +� 5v′ +⋆γJ +Dνc,νf ++ v′ +⋆γ3 +D2νc,νf +�1 − e−kaM (1 + kaM) +k2a2 +M +(−iξky − kx) +JRKKY +4,ξ +(k) = − +3π2 +32A2 +MBZ|v⋆|a2 +M +� 5v′ +⋆γJ +Dνc,νf ++ v′ +⋆γ3 +D2νc,νf +�1 − e−kaM (1 + kaM) +k2a2 +M +(−iξky + kx) +JRKKY +5,ξ +(k) = +π2 +A2 +MBZ|v⋆|aM +45γ2(v′ +⋆)2 +256D2νc,νf +kaM +� +− 1 +30 + +π +4k3a3 +M +� +I2(kaM) + kaMI3(kaM) +� +− +π +4k3a3 +M +� +L2(kaM) + kaML3(kaM) +�� +(k2 +x − k2 +y + 2iξkxky) +(S233) +where In(x) is the modified Bessel function of the second kind [139], Ln(x) is the modified Struve function [139]. We also + +58 +provide the long-wavelength behavior of RKKY by expanding in powers of k to second order +JRKKY +0 +(k) ≈ +−π3 +|v⋆|AMBZaM +� +γ4 +64D2νc,νf +� +(1 − 2 +π aMk + 1 +4a2 +Mk2) +JRKKY +1 +(k) ≈ − +� +3π3 +4AMBZ|v⋆| +Jγ2 +2Dνc,νf ++ +π3 +16|v⋆|AMBZ +J2 +� +1 +4aM +(1 − 2 +π aMk + 1 +4a2 +Mk2) +JRKKY +2 +(k) ≈ − +π3 +8AMBZ|v⋆|a3 +M +�45γ2(v′ +⋆)2 +128D2νc,νf +� +(1 − 1 +4(kaM)2) +JRKKY +3,ξ +(k) ≈ +π3 +AMBZ|v⋆|aM +� 15v′ +⋆γJ +16Dνc,νf ++ +3v′ +⋆γ3 +16D2νc,νf +� +(1 +2 − kaM +3 +)(iξky + kx) +JRKKY +4,ξ +(k) ≈ +π3 +AMBZ|v⋆|aM +� 15v′ +⋆γJ +16Dνc,νf ++ +3v′ +⋆γ3 +16D2νc,νf +� +(1 +2 − kaM +3 +)(iξky − kx) +JRKKY +5,ξ +(k) ≈ +π3 +AMBZ|v⋆|aM +45γ2(v′ +⋆)2 +8192D2νc,νf +(k2 +x − k2 +y + 2iξkxky) +(S234) +S6. +GROUND STATE OF f-MOMENTS +In this section, we solve the RKKY Hamiltonian (Eq. S230) and find the corresponding ground state at ν = 0, −1, −2, M = 0. +We will consider both v′ +⋆ = 0 where the system has a U(4) × U(4) symmetry, and v′ +⋆ ̸= 0 where the system only has a flat U(4) +symmetry. +A. +Ground state at v′ +⋆ = 0, M = 0 +Based on the RKKY Hamiltonian we derived in Eq. S110, we calculate the ground state. We first consider the case of v′ +⋆ = 0 +where we have U(4) × U(4) symmetry. The RKKY interactions at v′ +⋆ = 0 are +JRKKY +0 +(R) = +−π3 +|v⋆|A2 +MBZ +� +γ4 +16D2νc,νf +� +1 +|R|3 ≤ 0 +JRKKY +1 +(R) = − +π3 +8A2 +MBZ|v⋆| +� 3Jγ2 +Dνc,νf ++ J2 +2 +� +1 +|R|3 ≤ 0 +JRKKY +2 +(R) = JRKKY +3,ξ +(R) = JRKKY +4,ξ +(R) = JRKKY +5,ξ +(R) = 0 +(S235) +The RKKY Hamiltonian (Eq. S230) now becomes +ˆHv′ +⋆=0,M=0 +RKKY += +� +R,R2,µν,ξξ′ +JRKKY +0 +(R − R2) : Σ(f,ξξ′) +µν +(R) :: Σ(f,ξ′ξ) +µν +(R2) : ++ +� +R,R2,µν,ξ +JRKKY +1 +(R − R2) : Σ(f,ξξ) +µν +(R) :: Σ(f,ξ′ξ′) +µν +(R2) : +(S236) +where the first term has U(8) symmetry and the second term only has flat U(4) symmetry. We then introduce the bond operator +Aξ,ξ′ +R,R2 = +� +i,ξ +(ψf,ξ +R,i)†ψf,ξ′ +R2,i +(S237) + +59 +where we use the ψ basis defined in Eq. S18. Using bond operators and the definition of U(8) moments (Eq. S20), we find the +following relation +� +µν +: Σ(f,ξξ′) +µν +(R) :: Σ(f,ξ′ξ) +µν +(R2) : += +� +i,j,l,m +T µν +ij T µν +lm +4 +: ψf,ξ,† +R,i ψf,ξ′ +R,j :: ψf,ξ′,† +R2,l ψf,ξ +R2,m := +� +i,j +: ψf,ξ,† +R,i ψf,ξ′ +R,j :: ψf,ξ′,† +R2,j ψf,ξ +R2,i : += +� +i,j +(ψf,ξ,† +R,i ψf,ξ′ +R,j − δξ,ξ′δi,j +2 +)(ψf,ξ′,† +R2,j ψf,ξ +R2,i − δξ,ξ′δi,j +2 +) = +� +i,j +ψf,ξ,† +R,i ψf,ξ′ +R,j ψf,ξ′,† +R2,j ψf,ξ +R2,i − +� +i +δξ,ξ′ +2 (ψf,ξ,† +R,i ψf,ξ′ +R,i + ψf,ξ′,† +R2,i ψf,ξ +R2,i) + δξ,ξ′ += +� +i,j +ψf,ξ,† +R,i (δR,R2 − ψf,ξ′,† +R2,j ψf,ξ′ +R,j )ψf,ξ +R2,i − +� +i +δξ,ξ′ +2 (ψf,ξ,† +R,i ψf,ξ′ +R,i + ψf,ξ′,† +R2,i ψf,ξ +R2,i) + δξ,ξ′ += +� +i,j +ψf,ξ,† +R,i (δi,jδξ,ξ′ − ψf,ξ +R2,iψf,ξ′,† +R2,j )ψf,ξ′ +R,j + 4δR,R2 +� +i +ψf,ξ,† +R,i ψf,ξ +R,i − +� +i +δξ,ξ′ +2 (ψf,ξ,† +R,i ψf,ξ′ +R,i + ψf,ξ′,† +R2,i ψf,ξ +R2,i) + δξ,ξ′ += − +� +i,j +ψf,ξ,† +R,i ψf,ξ +R2,iψf,ξ′,† +R2,j ψf,ξ′ +R,j + 4δR,R2 +� +i +ψf,ξ,† +R,i ψf,ξ +R,i + +� +i +δξ,ξ′ +2 (ψf,ξ,† +R,i ψf,ξ′ +R,i − ψξ′,† +f,R2,iψf,ξ +R2,i) + δξ,ξ′ += − Aξ,ξ +R,R2Aξ′ξ′ +R2,R + 4δR,R2 +� +i +ψξ,† +R,iψξ +R,i + +� +i +δξ,ξ′ +2 (ψξ,† +R,iψξ′ +R,i − ψξ′,† +R2,iψξ +R2,i) + δξ,ξ′ +(S238) +Using Eq. S238, the first term in RKKY Hamiltonian (Eq. S240) can then be written as +� +R,R2,µν +� +ξξ′ +JRKKY +0 +(R − R2) : Σ(f,ξξ′) +µν +(R) :: Σ(f,ξ′ξ) +µν +(R2) : += − +� +R,R2,ξξ′ +JRKKY +0 +(R − R2)Aξ,ξ +R,R2Aξ′,ξ′ +R,R2 + 4 +� +R +JRKKY +0 +(0)(νf + 4) + +� +R,R2 +JRKKY +0 +(R − R2)(νf − νf) ++ +� +R,R2 +2JRKKY +0 +(R − R2) += − +� +R̸=R2,ξ,ξ′ +JRKKY +0 +(R − R2)Aξ,ξ,† +R,R2Aξ′,ξ′ +R,R2 − +� +R +JRKKY +0 +(0)(νf + 4)2 + +� +R +JRKKY +0 +(0)4(νf + 4) ++ +� +R,R2 +2JRKKY +0 +(R − R2) +(S239) +Using Eq. S238, the second term in Eq. S236 can be written as +� +R,R2,µν +� +ξ +JRKKY +1 +(R − R2) : Σ(f,ξξ) +µν +(R) :: Σ(f,ξξ) +µν +(R2) : += − +� +R̸=R2,ξ +JRKKY +1 +(R − R2)Aξ,ξ,† +R,R2Aξ,ξ +R,R2 − +� +R,R2,ξ +JRKKY +1 +(R = 0)(νξ +f + 2)2 + 4 +� +R +JRKKY +1 +(0)(νf + 4) ++ +� +R,R2 +2JRKKY +1 +(R − R2) . +(S240) +The Hamiltonian reads +ˆHv′ +⋆=0,M=0 +RKKY += − +� +R̸=R2,ξ,ξ′ +JRKKY +0 +(R − R2)Aξ,ξ,† +R,R2Aξ′,ξ′ +R,R2 − +� +R̸=R2,ξ +JRKKY +1 +(R − R2)Aξ,ξ,† +R,R2Aξ,ξ +R,R2 +(S241) +− +� +R +JRKKY +1 +(0) +� +(ν+1 +f ++ 2)2 + (ν−1 +f ++ 2)2 +� ++ E0 +(S242) +where E0 = − � +R JRKKR +0 +(0)(νf + 4)2 + � +R JRKKY +0 +(0)4(νf + 4) + � +R,R2 2JRKKY +0 +(R − R2) is a constant at fixed νf. + +60 +We now solve the Hamiltonian and find the ground states. We first note that, for given state |ψ⟩, its energy is +⟨ψ| ˆHv′ +⋆=0,M=0 +RKKY +|ψ⟩ += − +� +R̸=R2 +JRKKY +0 +(R − R2)⟨ψ| +� +ξ,ξ′ +Aξ,ξ,† +R,R2Aξ′,ξ′ +R,R2|ψ⟩ − +� +R̸=R2,ξ +JRKKY +1 +(R − R2)⟨ψ|Aξ,ξ,† +R,R2Aξ,ξ +R,R2|ψ⟩ +− +� +R +JRKKY +1 +(0) +� +⟨ψ|(ν+1 +f ++ 2)2 + (ν−1 +f ++ 2)2|ψ⟩ +� ++ E0 +≥ − +� +R +JRKKY +1 +(0) +� +⟨ψ|(ν+1 +f ++ 2)2 + (ν−1 +f ++ 2)2|ψ⟩ +� ++ E0 +(S243) +where the equality holds when ⟨ψ|Aξ,ξ,† +R,R2Aξ,ξ +R,R2|ψ⟩ = 0 for all R ̸= R2, ξ, and we use the fact that JRKKY +0 +(R − R2) ≤ +0, JRKKY +1 +(R − R2) ≤ 0. At fixed νf, we then note that +Eν = − +� +R +JRKKY +1 +(0) +� +⟨ψ|(ν+1 +f ++ 2)2 + (ν−1 +f ++ 2)2|ψ⟩ +� +(S244) +is minimized by (similar calculations have been provided in Eq. S132) +νf = −2 +: +(ν+1 +f , ν−1 +f ) = (−1, −1) +νf = −1 +: +(ν+1 +f , ν−1 +f ) = (0, −1), (ν+1 +f , ν−1 +f ) = (−1, 0) +νf = 0 +: +(ν+1 +f , ν−1 +f ) = (0, 0) . +(S245) +We let Emin +ν +denote the minimum value of Eν with Emin +ν += −2NMJRKKY +1 +(0), −5NMJRKKY +1 +(0), −8NMJRKKY +1 +(0) at +νf = −2, −1, 0 respectively. Then +⟨ψ| ˆHv′ +⋆=0,M=0 +RKKY +|ψ⟩ ≥ Emin +ν ++ E0 +(S246) +Therefore, |ψ⟩ with energy Emin +ν ++ E0 must be the ground state. We now prove the following states have energy Emin +ν ++ E0 +and are the ground states: +|ψ0⟩ = +� +R +� ν+1 +f ++2 +� +n=1 +ψ+,† +R,in +νf +4 +� +n=ν+1 +f ++3 +ψ−,† +R,in|0⟩ +� +(S247) +where {in} are chosen arbitrarily and (ν+1 +f , ν−1 +f += νf − ν+1 +f ) satisfy Eq. S245. We first prove Aξ,ξ +R,R2|ψ0⟩ = 0 with R ̸= R2. +Since Aξ,ξ +R,R2 = � +i ψξ,† +R,iψξ +R2,i will move an electrons at site R with flavor ξ, i to another site R2 with flavor ξ, i, then, for the +states in Eq. S247, there is either zero electron at (R, ξ, i) or one electrons at both (R, ξ, i), (R2, ξ, i). Therefore, Aξ,ξ +R,R2 must +annihilate |ψ0⟩. Thus +� +R̸=R2 +JRKKY +0 +(R − R2)⟨ψ0| +� +ξ,ξ′ +Aξ,ξ,† +R,R2Aξ′,ξ′ +R,R2|ψ0⟩ − +� +R̸=R2,ξ +JRKKY +1 +(R − R2)⟨ψ0|Aξ,ξ,† +R,R2Aξ,ξ +R,R2|ψ0⟩ = 0 . +(S248) +In addition, from the constructions of |ψ0⟩ +− +� +R +JRKKY +1 +(0) +� +⟨ψ|(ν+1 +f ++ 2)2 + (ν−1 +f ++ 2)2|ψ⟩ +� += Emin +ν +(S249) +Combining Eq. S248 and Eq. S249, we have +⟨ψ0| ˆHv′ +⋆=0,M=0 +RKKY +|ψ0⟩ = Emin +ν ++ E0 +(S250) +and |ψ0⟩ in Eq. S247 are the ground states. +Equivalently, we could also rewrite Eq. S247 with f-electron operator (Eq. S18) +|ψ0⟩ = +� +R +� ν+1 +f ++2 +� +i=1 +f † +R,αiηisi +νf +4 +� +i=ν+1 +f ++3 +f † +R,αiηisi +� +|0⟩ +(S251) +where (−1)αi+1ηi = 1 for i = 1, ..., ν+1 +f ++ 2 and (−1)αi+1ηi = −1 for i = ν+1 +f ++ 3, ..., ν−1 +f ++ 2, and ν+ +f , ν− +f satisfy Eq. S245. + +61 +B. +Ground state at v′ +⋆ ̸= 0, M = 0 +We next consider the ground state at v′ +⋆ ̸= 0, M = 0. The Hamiltonian now takes the form of +ˆHv′ +⋆̸=0,M=0 +RKKY += ˆHv′ +⋆=0,M=0 +RKKY ++ ˆHv′ +⋆ +1 +(S252) +where ˆHv′ +⋆=0,M=0 +RKKY +is defined in Eq. S236 and +ˆHv′ +⋆ +1 += +� +ξ,R,R2,µν +JRKKY +2 +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R2) : ++ +� +ξ,R,R2,µν +JRKKY +3,ξ +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξξ) +µν +(R2) : ++ +� +ξ,R,R2,µν +JRKKY +4,ξ +(R − R2) : ˆΣ(f,ξξ) +µν +(R) : ˆΣ(f,ξ−ξ) +µν +(R2) : ++ +� +ξ,R,R2,µν +JRKKY +5,ξ +(R − R2) :: ˆΣ(f,ξ−ξ) +µν +(R) : ˆΣ(f,ξ−ξ) +µν +(R2) : . +We next treat ˆHv′ +⋆ +1 +as perturbations and use degenerate perturbation theory to determine the ground states of ˆHv′ +⋆̸=0,M=0 +RKKY +. We +let {|ψ0,i⟩} be the ground states defined in Eq. S251, and construct the following matrix. Then we can construct the following +matrix +[H1]ij = ⟨ψ0,i| ˆHv′ +⋆ +1 |ψ0,j⟩ . +We then diagonalize H1 and the states with the lowest eigenvalues are the grounds states. We first note that |ψ0,i⟩ (given in +Eq. S251) can be written as the following product states +|ψ0,i⟩ = +� +R +|φ0,i(R)⟩ +(S253) +where |φ0,i(R)⟩ is the corresponding state at R. From the definition (Eq. S251), we find +⟨φ0,i(R)|φ0,j(R)⟩ = δi,j +(S254) +We now evaluate the off-diagonal terms [ ˆH1]ij with i ̸= j. We first consider the effect of JRKKY +2,ξ +. For i ̸= j +⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +2 +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R2) : |ψ0,j⟩ += +� +R,ξ,µν +JRKKY +2 +(0)⟨φ0,i(R)| : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R) : |φ0,j(R)⟩ +� +R2,R2̸=R +⟨φ0,i(R2)||φ0,j(R2)⟩ ++ +� +R,R2,ξ,µν +JRKKY +2 +(R − R2)⟨φ0,i(R)| : ˆΣ(f,ξξ) +µν +(R) : |φ0,j(R)⟩ +⟨φ0,i(R2)| : ˆΣ(f,−ξ−ξ) +µν +(R2) : |φ0,j(R2)⟩ +� +R3,R3̸=R,R3̸=R2 +⟨φ0,i(R3)||φ0,j(R3)⟩ = 0 +(S255) +where we use Eq. S254. Similarly, the contributions of JRKKY +3,ξ +, JRKKY +4,ξ +, JRKKY +5,ξ +also vanishes when i ̸= j. Then [ ˆH1]ij = 0 +when i ̸= j. +We only need to consider the diagonal components [ ˆH1]ii, and the states that minimize [ ˆH1]ii are the ground +states. +We first consider the effect of JRKKY +3,ξ +⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +3,ξ +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξξ) +µν +(R2) : |ψ0,i⟩ += +� +R,R2,ξ,µν +JRKKY +3,ξ +(R − R2)⟨φ0,i(R)| : ˆΣ(f,ξξ) +µν +(R) : |φ0,i(R)⟩ +⟨φ0,i(R2)| : ˆΣ(f,−ξξ) +µν +(R2) : |φ0,i(R2)⟩ +� +R3,R3̸=R,R3̸=R2 +⟨φ0,i(R3)||φ0,i(R3)⟩ +(S256) + +62 +where we use the fact that JRKKY +3,ξ +(0) = 0 (Eq. S232). For the ferromagnetic states (in Eq. S251), ⟨φ0,i(R)| : ˆΣ(f,ξξ′) +µν +(R) : +|φ0,i(R)⟩ does not depend on R. We let +σξξ′ +µν = ⟨φ0,i(R)| : ˆΣ(f,ξξ′) +µν +(R) : |φ0,i(R)⟩ +(S257) +Then +⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +3,ξ +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξξ) +µν +(R2) : |ψ0,i⟩ = +� +R,R2,ξ,µν +σξξ′ +µν σξξ′ +µν JRKKY +3,ξ +(R − R2) = 0 (S258) +where we use the fact that � +R JRKKY +3,ξ +(R − R2) = 0 (Eq. S232). Then we find JRKKY +3,ξ +will not contribute to [ ˆH1]ii For the +same reason, we also have +⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +4,ξ +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,ξ−ξ) +µν +(R2) : |ψ0,i⟩ = 0 +⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +5,ξ +(R − R2) : ˆΣ(f,ξ−ξ) +µν +(R) :: ˆΣ(f,ξ−ξ) +µν +(R2) : |ψ0,i⟩ = 0 +(S259) +(Note that, from Eq. S232, JRKKY +4,ξ +(0) = JRKKY +5,ξ +(0) = 0 and � +R JRKKY +4,ξ +(R) = � +R JRKKY +5,ξ +(R) = 0 ). +Therefore, the only non-zero contributions come from JRKKY +2 +(since JRKKY +2 +(0) ̸= 0, � +R JRKKY +2 +(R) ̸= 0). We find +[H1]ii =⟨ψ0,i| +� +ξ,R,R2,µν +JRKKY +2 +(R − R2) : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R2) : |ψ0,i⟩ += +� +R,ξ,µν +JRKKY +2 +(0)⟨φ0,i(R)| : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(f,−ξ−ξ) +µν +(R) : |φ0,i(R)⟩ ++ +� +R̸=R2,ξ,µν +JRKKY +2 +(R − R2)σξξ +µνσ−ξ−ξ +µν +(S260) +We emphasize that JRKKY +2 +plays the rule of picking the true ground state in the nonchiral-flat limit. +We direct evaluate [ ˆH1]ii (Eq. S260) and the ground states are the states |ψ0,i⟩ with lowest [ ˆH1]ii. We use {αηs} to charac- +terize the states in Eq. S251, where {αηs} denote the set of flavors with one f-electron per site. +At νf = 0, we find the following states have [ ˆH1]ii = � +RR2 2JRKKY +2 +(R − R2) +{1+ ↓, 2− ↓, 2+ ↓, 1− ↓}, {2− ↑, 2− ↓, 1− ↑, 1− ↓}, {1+ ↓, 2− ↑, 2+ ↓, 1− ↑}, {1+ ↑, 2− ↓, 2+ ↑, 1− ↓}, +{1+ ↑, 1+ ↓, 2+ ↑, 2+ ↓}, {1+ ↑, 2− ↑, 2+ ↑, 1− ↑}. +(S261) +The following states have [ ˆH1]ii = − � +RR2 2JRKKY +2 +(R − R2) +{1+ ↑, 2− ↑, 2+ ↑, 1− ↑}, {1+ ↑, 1+ ↓, 2+ ↑, 2+ ↓}, {1+ ↑, 2− ↓, 2+ ↑, 1− ↓}, +{2− ↑, 2− ↓, 1− ↑, 1− ↓}, {1+ ↓, 2− ↓, 1− ↓, 2+ ↓} +(S262) +All other states (of Eq. S251) at νf = 0 have [ ˆH1]ii = 0. Then Eq. S261 gives the ground state at νf = 0. +At νf = −1, we find the following states have [ ˆH1]ii = − � +RR2 JRKKY +2 +(R − R2) +{1+ ↑, 1+ ↓, 2+ ↑}, {1+ ↑, 1+ ↓, 2+ ↓}, {1+ ↑, 2− ↑, 2+ ↓}, {1+ ↑, 2− ↑, 1− ↓}, {1+ ↑, 2− ↓, 2+ ↓}, {1+ ↑, 2− ↓, 1− ↑}, +{1+ ↓, 2− ↑, 2+ ↑}, {1+ ↓, 2− ↑, 1− ↓}, {1+ ↓, 2− ↓, 2+ ↑}, {1+ ↓, 2− ↓, 1− ↑}, {2− ↑, 2− ↓, 2+ ↑}, {2− ↑, 2− ↓, 2+ ↓} +{1+ ↑, 2+ ↑, 2+ ↓}, {1+ ↓, 2+ ↑, 2+ ↓}, {1+ ↓, 2+ ↑, 1− ↑}, {2− ↓, 2+ ↑, 1− ↑}, {1+ ↓, 2+ ↑, 1− ↓}, {2− ↑, 2+ ↑, 1− ↓}, +{1+ ↑, 2+ ↓, 1− ↑}, {2− ↓ 2+ ↓, 1− ↑}, {1+ ↑, 2+ ↓, 1− ↓}, {2− ↑, 2+ ↓, 1− ↓}, {1+ ↑, 1− ↑, 1− ↓}, {1+ ↓, 1− ↑, 1− ↓} +(S263) +and all other states at νf = −1 have [ ˆH1]ii = � +RR2 JRKKY +2 +(R − R2). Then, Eq. S263 gives the ground state at νf = −1. +At νf = −2, we find the following states have [ ˆH1]ii = 2 � +RR2 JRKKY +2 +(R − R2) +{1+ ↑, 2+ ↑}, {1+ ↓, 2+ ↓}, {2− ↑, 1− ↑}, {2− ↓, 1− ↓} +(S264) + +63 +and all other states at νf = −2 have [ ˆH1]ii = 0. Then, Eq. S264 gives the ground state at νf = −2. +In summary, Eq. S261, Eq. S263, Eq. S264 give ground states at νf = 0, −1, −2 respectively. +In a more compact form, +ground states at νf = 0, −2 are +|ψ0⟩ = +� +R +(νf +4)/2 +� +i=1 +f † +R,1ηisifR,2ηisi|0⟩ +(S265) +and the ground states at νf = −1 are +|ψ0⟩ = +� +R +f † +R,ξη′s′ +2 +� +i=1 +f † +R,1ηisifR,2ηisi|0⟩ +(S266) +where (η′, s′) ̸= (ηi, si). We also provide the ground states in the ψ basis (Eq. S18). At νf = 0, −2, the ground states are +|ψ0⟩ = +� +R +(νf +4)/2 +� +n=1 +ψf,+,† +R,in ψf,+,† +R,in |0⟩ +(S267) +where in are chosen arbitrarily. At νf = −1, the ground states are +|ψ0⟩ = ψf,ξ,† +R,jn +� +R +ψf,ξ,† +R,j ψf,+,† +R,i ψf,−,† +R,i |0⟩ +(S268) +where i, j, ξ are chosen arbitrarily. The ground states given in Eq. S265 and Eq. S266 also give rise to the same ground states as +derived from projected Coulomb Hamiltonian [85] in nonchiral-flat limit. +In terms of the Young tableaux, the ground state at νf = 0 corresponds to +· · · +· · · +2NM +There are 2NM boxes in the first and second rows. NM columns correspond to the ξ = +1 and NM columns correspond to +the ξ = −1 fermions (note that we have NM site). The ferromagnetic nature of the ground state indicates the symmetry under +the permutation of the position indices. In addition, the ground state in Eq. S265 is also symmetric under the permutation of ξ +indices. This in total leads to 2NM = 2 × NM boxes for each row. Similarly, at νf = −2, the corresponding Young tableaxu +of the ground states are +· · · +2NM +There are 2NM boxes in the first row. NM columns correspond to the ξ = +1 and NM columns correspond to the ξ = −1 +fermions (note that we have NM site). The ferromagnetic nature of the ground state indicates the symmetry under the permutation +of the position indices. In addition, the ground state in Eq. S265 is also symmetric under the permutation of ξ indices. This in +total leads to 2NM = 2 × NM boxes for each row. +At νf = −1, the corresponding Young tableaux of the ground states are +· · · +· · · +· · · +NM +NM +There are 2NM boxes in the first row and NM boxes in the second row. In the case of ν+1 +f += 0, ν−1 +f += −1. The first NM +columns correspond to the ξ = +1 and N columns correspond to the ξ = −1 fermions. The ferromagnetic nature of the ground +state indicates the symmetry under the permutation of the position indices. In addition, the ground state in Eq. S266 tends to be +as symmetric as possible under the permutation of ξ indices. Consequently, there are 2NM boxes in the first row which indicates +the symmetric properties between ξ = +1, ξ = −1 and also between different sites. The remaining NM boxes in the second +row capture the symmetric feature under the permutation of site indices of the remaining ξ = +1 fermions. + +64 +C. +Comparisons of the ground states at different limits +We now compare the ground states at different limits. In general, we find three types of ground states depending on the values +of γ, v′ +⋆, J. +• Type I: Defined in Eq. S128 or Eq. S251 for v′ +⋆ = 0, M = 0. +� +R +� ν+1 +f ++2 +� +i=1 +f † +R,αiηisi +νf +4 +� +i=ν+1 +f ++3 +f † +R,αiηisi +� +|0⟩ +(S269) +where (−1)αi+1ηi = 1 for i = 1, ..., ν+1 +f ++ 2 and (−1)αi+1ηi = −1 for i = ν+1 +f ++ 3, ..., ν−1 +f ++ 2. and +νf = −2 +: +(ν+1 +f , ν−1 +f ) = (−1, −1) +νf = −1 +: +(ν+1 +f , ν−1 +f ) = (0, −1), (ν+1 +f , ν−1 +f ) = (−1, 0) +νf = 0 +: +(ν+1 +f , ν−1 +f ) = (2, 2) +(S270) +• Type II: Defined in Eq. S265 and Eq. S266 for v′ +⋆ ̸= 0, M = 0. +At ν = 0, −2, +|ψ⟩ = +� +R +(νf +4)/2 +� +i=1 +f † +R,1ηisifR,2ηisi|0⟩ +(S271) +At νf = −1: +|ψ⟩ = +� +R +f † +R,ξη′s′ +2 +� +i=1 +f † +R,1ηisifR,2ηisi|0⟩ +(S272) +where (η′, s′) ̸= (ηi, si). +• Type III: Defined in Eq. S149 for v′ +⋆ = 0, M ̸= 0. Type I ground states (Eq. S269) with additional requirement (ηi, si) ̸= +(ηj, sj) +for +i ̸= j. +The symmetry and the ground states at different limits are +Hybridization γ = 0, v′ +⋆ = 0 γ = 0, v′ +⋆ = 0 γ ̸= 0, v′ +⋆ = 0 γ ̸= 0, v′ +⋆ ̸= 0 +M +M = 0 +M ̸= 0 +M = 0 +M = 0 +Symmetry +U(4) × U(4) +Chiral U(4) +U(4) × U(4) +Flat U(4) +Ground states +Type I +Type III +Type I +Type II +For the type I ground states, we have a degenerate ferromagnetic ground state corresponding to U(4) × U(4) symmetry. For +the type II ground states, it is stabilized by a non-zero v′ +⋆. Due to the ferromagnetic interactions between two flat U(4) moments +with opposite ξ indices, the ground state tends to align two flat U(4) moments. For the type III ground states, it is stabilized by +a non-zero M in the zero-hybridization limit. Due to the antiferromagnetic interactions between two chiral U(4) moments with +opposite ξ indices, the ground state tends to align two chiral U(4) moments. +S7. +FLUCTUATION SPECTRUM OF f-MOMENTS BASED ON RKKY INTERACTIONS +In this section, we derive the fluctuation spectrum of f-moments at M = 0 and νf = 0, −1, −2 from the RKKY Hamiltonian +in Eq. S230. Here, we point out that in this approach, the polarization effect of c-electrons has not been included. In Sec. S8, we +will obtain a more accurate description of the fluctuations of f-moments by including the polarization effect of c-electrons. +To find the fluctuation spectrum we first rewrite the RKKY Hamiltonian (Eq. S230) with bond operators Aξ,ξ2 +R,R2 (Eq. S237). +Then we take the ground states (Eq. S265 and Eq. S266) and obtain the excitation spectrum of f-moments on top of the ground +state by calculating the commutator between ψf,ξ,† +R,i ψf,ξ′,† +R,j +and RKKY Hamiltonian (Eq. S230) [86]. + +65 +A. +RKKY Hamiltonian +We first rewrite the RKKY Hamiltonian in Eq. S230 in a more general formula +ˆHRKKY = +� +R,R2,µν +� +ξ1,ξ2,ξ3,ξ4 +: ˆΣ(f,ξ1ξ2) +µν +(R) :: ˆΣ(f,ξ3ξ4) +µν +(R2) : JRKKY (R − R2, ξ1ξ2ξ3ξ4) +(S273) +The new RKKY interactions are connected with RKKY interactions in Eq. S230 via the following relations +JRKKY (R, ξξξξ) = JRKKY +0 +(R) + JRKKY +1 +(R), +JRKKY (R, ξξ − ξ − ξ) = JRKKY +2 +(R), +JRKKY (R, ξ − ξ − ξξ) = JRKKY +0 +(R), +JRKKY (R, ξ − ξξ − ξ) = JRKKY +5 +(R) +JRKKY (R, ξξξ − ξ) = JRKKY +4ξ +(R), +JRKKY (R, ξξ − ξξ) = JRKKY +3ξ +(R) +(S274) +with all other components of JRKKY (R, ξ1ξ2ξ3ξ4) vanish. +In order to obtain the fluctuation spectrum, we first rewrite all the terms in ˆHRKKY (Eq. S273) via the bond operator +Aξ,ξ2 +R,R2 defined in Eq. S237 which is also given below (We note that in the previous calculation in Sec S6, we only rewrite +JRKKY +0 +, JRKKY +1 +term with bond operators) +Aξ,ξ2 +R,R2 = +� +i +ψf,ξ,† +R,i ψf,ξ2 +R2,i . +(S275) +We introduce the following relations +: ˆΣ(f,ξξ2) +µν +(R) :: ˆΣ(f,ξ3ξ4) +µν +(R2) := +� +i,j +: ψf,ξ,† +R,i ψf,ξ2 +R,j :: ψf,ξ3,† +R2,j ψf,ξ4 +R2,i : +=δξ,ξ2δξ3,ξ4 − 1 +2δξ,ξ2Aξ3,ξ4 +R2,R2 − 1 +2δξ3,ξ4Aξ,ξ2 +R,R + +� +i,j +ψf,ξ,† +R,i ψf,ξ2 +R,j ψf,ξ3,† +R2,j ψf,ξ4 +R2,i +=δξ,ξ2δξ3,ξ4 − 1 +2δξ,ξ2Aξ3,ξ4 +R2,R2 + 1 +2δξ3,ξ4Aξ,ξ2 +R,R + 4δR,R2δξ2,ξ3Aξ,ξ4 +R,R − Aξ,ξ4 +R,R2Aξ3,ξ2 +R2,R +(S276) +Using Eq. S276, we rewrite Hamiltonian in Eq. S273 as +ˆHRKKY = +� +R,R2 +� +− 1 +2 +� +ξ1ξ3ξ4 +JRKKY (R − R2, ξ1ξ1ξ3ξ4)Aξ3,ξ4 +R2,R2 + 1 +2 +� +ξ1ξ2ξ3 +JRKKY (R − R2, ξ1ξ2ξ3ξ3)Aξ1,ξ2 +R,R +� ++ +� +R,ξ1ξ2ξ4 +4JRKKY (0, ξ1ξ2ξ2ξ4)Aξ1,ξ4 +R,R − +� +R,R2,ξ1ξ2ξ4ξ4 +JRKKY (R − R2, ξ1ξ2ξ3ξ4)Aξ1,ξ4 +R,R2Aξ3,ξ2 +R2,R +(S277) +where we drop the constant contribution and use the fact that Aξ,ξ +R,R = ˆνξ +f(R) + 2. +We next simplify the RKKY Hamiltonian in Eq. S277. From Eq. S274 and Eq. S232, we have JRKKY (R = 0, ξ1ξ2ξ3ξ1) ∝ +δξ2,ξ3 and JRKKY (R = 0, ξ1ξ2ξ2ξ4) ∝ δξ1,ξ4. We separate then Hamiltonian in Eq. S277 into two parts and have +ˆHRKKY = ˆHRKKY,0 + ˆHRKKY,1 +ˆHRKKY,0 = +� +R,ξ +f1(ξ)ˆνξ +f(R) + +� +R,ξ1̸=ξ2 +f2(ξ1ξ2)Aξ1,ξ2 +R,R + +� +R,ξ1,ξ2 +f3(ξ1ξ2)ˆνξ1 +f (R)ˆνξ2 +f (R) +ˆHRKKY,1 = +� +R,R2,ξ1,ξ2,ξ3,ξ4 +(R,ξ1)̸=(R2,ξ2) +(R2,ξ3)̸=(R,ξ4) +J(R − R2, ξ1ξ2ξ3ξ4)Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R +(S278) +where we have dropped the constant contribution and define +f1(ξ) = −1 +2 +� +R2,ξ1 +JRKKY (R2 − R, ξ1ξ1ξξ) + 1 +2 +� +R2,ξ3 +JRKKY (R − R2, ξξξ3ξ3) + 2 +� +ξ2 +(JRKKY (0, ξξ2ξ2ξ) − JRKKY (0, ξ2ξξξ2)) +f2(ξ1ξ2) = −1 +2 +� +R2,ξ +JRKKY (R2 − R, ξξξ1ξ2) + 1 +2 +� +R2,ξ +JRKKY (R − R2, ξ1ξ2ξξ) + 4 +� +ξ +JRKKY (0, ξ1ξξξ2) +f3(ξ1ξ2) = −JRKKY (0, ξ1ξ2ξ2ξ1) +J(R − R2, ξ1ξ2ξ3ξ4) = −JRKKY (R − R2, ξ1ξ4ξ3ξ2) +(S279) + +66 +Combining Eq. S279 and the definition of RKKY interaction (Eq. S274), we find +f1(ξ) = 0 +f2(ξ − ξ) = 0 +, +(f2(ξ1ξ2) is only defined for ξ1 ̸= ξ2) +, +f3(ξξ) = −JRKKY +0 +(R = 0) − JRKKY +1 +(R = 0) +, +f3(ξ − ξ) = −JRKKY +0 +(R = 0) +Then, ˆHRKKY,0 (Eq. S278) can be written as +ˆHRKKY,0 = −JRKKY +0 +(R2 = 0) +� +R +ˆνf(R)ˆνf(R) − JRKKY +1 +(R2 = 0) +� +R,ξ +ˆνξ +f(R)ˆνξ +f(R) +(S280) +Since we fix the filling of f at each site, we can replace ˆνf(R) with an integer number νf. The first term is only a constant and +it is sufficient to only keep the second term: +ˆHRKKY,0 = −JRKKY +1 +(R2 = 0) +� +R,ξ +ˆνξ +f(R)ˆνξ +f(R) +(S281) +We next go to momentum space. +We consider the Fourier transformation of RKKY interaction J(q, ξ1ξ2ξ3ξ4) = +� +R J(R, ξ1ξ2ξ3ξ4)e−iq·R. Using Eq. S274 and Eq. S279, we find +J(q, ξξξξ) = −(JRKKY +0 +(q) − JRKKY +0 +(R = 0) + JRKKY +1 +(q) − JRKKY +1 +(R = 0)) +J(q, ξξ, −ξ − ξ) = −(JRKKY +0 +(q) − JRKKY +0 +(R = 0)) +J(q, ξ − ξ − ξξ) = −(JRKKY +2 +(q)) +J(q, ξξ − ξξ) = −JRKKY +3,ξ +(q) +J(q, ξ − ξξξ) = −JRKKY +4,ξ +(q) +J(q, ξ − ξξ − ξ) = −JRKKY +5,ξ +(q) +(S282) +In summary, combining Eq. S281 and Eq. S278, the final RKKY Hamiltonian in terms of the bond operators can be written as +ˆHRKKY = ˆHRKKY,0 + ˆHRKKY,1 +ˆHRKKY,0 = − JRKKY +1 +(R2 = 0) +� +R,ξ +ˆνξ +f(R)ˆνξ +f(R) +ˆHRKKY,1 = +� +R,R2,ξ1,ξ2,ξ3,ξ4 +(R,ξ1)̸=(R2,ξ2) +(R2,ξ3)̸=(R,ξ4) +J(R − R2, ξ1ξ2ξ3ξ4)Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R +(S283) +where the constant term in ˆHRKKY,0 has been dropped and the RKKY interactions are given in Eq. S282. We next use Eq. S283 +to calculate the fluctuation of f-moments on top of the ground states at M = 0, ν = νf = 0, −1, −2 given in Eq. S265 and +Eq. S266. +B. +Excitation spectrum from RKKY interaction at νf = 0, −2 +We take the ground states |ψ0⟩ at νf = 0, −2 that are given in Eq. S265 and are also shown below +|ψ0⟩ = +� +R +(νf +4)/2 +� +n=1 +ψf,+,† +R,in ψf,−,† +R,in |0⟩ +(S284) +where in are chosen arbitrarily. We then have +Aξ,ξ2 +R,R2|ψ0⟩ = 0 +, +(R, ξ) ̸= (R2, ξ2) +(S285) + +67 +We now prove Eq. S285. Aξ,ξ2 +R,R2 = � +i ψf,ξ,† +R,i ψf,ξ2,† +R2,i +describes the procedure of moving one f-electron from (R2, ξ2, i) to +(R, ξ, i) with (R2, ξ2) ̸= (R, ξ). However, for the ground states in Eq. S284, (R, ξ, i) and (R2, ξ2, i) are both filled with +either one f-electron or zero f-electron, because they have same valley-spin flavor i. Thus the procedure described by Aξ,ξ2 +R,R2 +is forbidden and as a consequence Aξ,ξ2 +R,R2|ψ0⟩ = 0 when (R, ξ) ̸= (R2, ξ2). This feature allows us to derive the excitation of +RKKY Hamiltonian exactly by calculating the commutator between Hamiltonian and fermion bilinear operators [86]. +We first introduce the following commutation relations +[Aξ,ξ2 +R,R2, ψf,ξ′ +R′,i] = −δR,R′δξ,ξ′ψf,ξ2 +R2,i +, +[Aξ,ξ2 +R,R2, ψf,ξ′,† +R′,i ] = δR2,R′δξ2,ξ′ψf,ξ,† +R,i +(S286) +We then calculate the commutator +[Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′ +1,† +R3,i ψξ′ +2 +f,R3,j] =δR2,R3δξ2,ξ′ +1ψf,ξ1,† +R,i +ψξ′ +2 +R3,jAξ3,ξ4 +R2,R − δR,R3δξ1,ξ′ +2ψf,ξ′ +1,† +R3,i ψf,ξ2 +R2,jAξ3,ξ4 +R2,R ++ δR,R3δξ′ +1,ξ4Aξ1,ξ2 +R,R2ψf,ξ3,† +R2,i ψf,ξ′ +2 +R3,j − δR2,R3δξ′ +2,ξ3Aξ1,ξ2 +R,R2ψf,ξ′ +1,† +R3,i ψf,ξ4 +R,j +(S287) +Using Eq. S285, Eq. S286 and Eq. S287, for (R, ξ1) ̸= (R2, ξ2) and (R2, ξ3) ̸= (R, ξ4), we find +[Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′,† +R′,i ψf,ξ′ +2 +R′,i2]|ψ0⟩ = [Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′,† +R′,i ]ψf,ξ′ +2 +R′,i2|ψ0⟩ + ψf,ξ′,† +R′,i [Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′ +2 +R′,i2]|ψ0⟩ +=δR,R′δξ2,ξ3δξ4,ξ′ψf,ξ1,† +R,i +ψf,ξ′ +2 +R,i2|ψ0⟩ − δR,R′δξ4,ξ′δξ1,ξ′ +2ψf,ξ3,† +R2,i ψf,ξ2 +R2,i2|ψ0⟩ +− δR2,R′δξ3,ξ′ +2δξ2,ξ′ψf,ξ1,† +R,i +ψf,ξ4 +R,i2|ψ0⟩ + δR2,R′δξ3,ξ′ +2δξ1,ξ4ψf,ξ′,† +R′,i ψξ2 +f,R′,i2|ψ0⟩ +(S288) +We next consider the charge-0 excitation created by the following operators [86]: +OI +q,ii2 = 1 +NM +� +R,ξ′ξ′ +2 +ψf,ξ′,† +R,i +ψf,ξ′ +2 +R,i2uI +ii2,ξ′ξ′ +2(q)eiq·R +(S289) +with uI +ii2,ξ′ξ′ +2(q) a complex number. We comment that, we could also introduce bosonic mode created by ψξ′,† +R,i ψξ′ +2 +R2,i2 with +R2 ̸= R. However, the uniform charge distribution of f will be violated after acting ψξ′,† +R,i ψξ′ +2 +R2,i2 on the ground state. To find +the excitation spectrum, we calculate [ ˆHRKKY , OI +q,ii2]|ψ0⟩ = [ ˆHRKKY,0, OI +q,ii2]|ψ0⟩ + [ ˆHRKKY,1, OI +q,ii2]|ψ0⟩; ( ˆHRKKY and +|ψ0⟩ given in Eq. S283 and Eq. S285 respectively). +We first consider [ ˆHRKKY,1, OI +q,ii2]|ψ0⟩. Using Eq. S283, Eq. S288 and Eq. S289, we have +[ ˆHRKKY,1, OI +q,ii2]|ψ0⟩ += +� +R,R2 +� +ξ1,ξ2,ξ3,ξ4 +J(R − R2, ξ1, ξ2, ξ3, ξ4) 1 +NM +� +R′,ξ′,ξ′ +2 +uI +ii2,ξ′ξ′ +2(q)eiq·R′ +� +δR,R′δξ2,ξ3δξ4,ξ′ψf,ξ1,† +R,i +ψf,ξ′ +2 +R,i2 − δR,R′δξ4,ξ′δξ1,ξ′ +2ψf,ξ3,† +R2,i ψf,ξ2 +R2,i2 +− δR2,R′δξ3,ξ′ +2δξ2,ξ′ψf,ξ1,† +R,i +ψf,ξ4 +R,i2 + δR2,R′δξ3,ξ′ +2δξ1,ξ4ψf,ξ′,† +R′,i ψf,ξ2 +R′,i2 +� +|ψ0⟩ += 1 +NM +� +q +� +J(q2 = 0, ξ1, ξ′, ξ′, ξ3)uI +ii2,ξ3ξ2(q)ψf,ξ1,† +R,i +ψf,ξ2 +R,i2 + J(q2 = 0, ξ′, ξ2, ξ4, ξ′)uI +ii2,ξ1ξ4(q)ψf,ξ1,† +R,i +ψf,ξ2 +R,i2 +− J(−q, ξ4, ξ2, ξ1, ξ3)uI +ii2,ξ3ξ4(q)ψf,ξ1,† +R,i +ψf,ξ2 +R,i2 − J(q, ξ1, ξ3, ξ4, ξ2)uii2,ξ3ξ4(q)ψf,ξ1,† +R,i +ψf,ξ2 +R,i2 +� +eiq·R +(S290) +As for the ˆHRKKY,0(Eq. S283), we first introduce the following commutation relation +[ +� +ξ1 +� +R +ˆνf +ξ1(R) ˆνf +ξ1(R), OI +q,ii2]|ψ0⟩ = 1 +NM +� +R′,ξ′,ξ′ +2,ξ1 +uI +ii2,ξ′ξ′ +2(q)eiq·R′[ +� +R +ˆνf +ξ1(R) ˆνf +ξ1(R), ψf,ξ′,† +R′,i ψf,ξ′ +2 +R′,i2]|ψ0⟩ += 1 +NM +� +R,i,i2,ξ′,ξ′ +2 +uI +ii2,ξ′ξ′ +2(q)eiq·R[ ˆνf +ξ1(R) ˆνf +ξ1(R), ψf,ξ′,† +R,i +ψf,ξ′ +2 +R,i2]|ψ0⟩ += 1 +NM +� +R,i,i2,ξ′,ξ′ +2 +uI +ii2,ξ′ξ′ +2(q)eiq·Rψf,ξ′,† +R,i +ψf,ξ′ +2 +R,i22(1 − δξ′,ξ′ +2)|ψ0⟩ +(S291) + +68 +where we use the fact that +ˆνξ +f(R)|ψ0⟩ = νf/2|ψ0⟩ +with νf = 0, −2. +Combining Eq. S283, Eq. S290 and Eq. S291, we have we have +[ ˆHRKKY , OI +q,ii2]|ψ0⟩ = +� +k,ξ1,ξ2 +� +k′,ξ′ +1,ξ′ +2 +M(q)ξ1ξ2,ξ′ +1ξ′ +2uii2,ξ′ +1ξ′ +2(q)ψf,ξ1,† +R,i +ψf,ξ2 +R,i2eiq·R|ψ0⟩ +(S292) +with +M(q)ξ1ξ2,ξ′ +1ξ′ +2 = +� +ξ +� +J(q2 = 0, ξ1, ξ, ξ, ξ′ +1)δξ′ +2,ξ2 + J(q2 = 0, ξ, ξ2, ξ′ +2, ξ)δξ1,ξ′ +1 +� +− +� +J(q, ξ1, ξ′ +1, ξ′ +2, ξ2) + J(−q, ξ′ +2, ξ2, ξ1, ξ′ +1) +� +− 2JRKKY +1 +(R2 = 0)δξ1,−ξ2δξ1,ξ′ +1δξ2,ξ′ +2 +(S293) +Here we comment that M(q) matrix takes the same form at νf = 0, −2. However, the filling νf will affect the choice of ii2 +(because not all choice of ii2 in uii2,ξξ2 produces valid fluctuations) and the values of RKKY interactions J(q, ξ1, ξ2, ξ3, ξ4) +Using Eq. S319, dispersion of f-moment fluctuation (EI +q) and the corresponding wavefunctions (uI +ii2,ξ1ξ2(k, q)) can be derived +by solving the following eigen-equations +� +ξ′ +1,ξ′ +2 +M(q)ξ1ξ2,ξ′ +1ξ′ +2uI +ii2,ξ′ +1ξ′ +2(q) = EI +quI +ii2,ξ1ξ2(q) +(S294) +where M(q)ξ1ξ2,ξ′ +1ξ2 can be treated as 4 × 4 matrix with row and column indices ++, −−, +−, −+. More explicitly, M(q) +can be written as +M(q)++,++ = − 2JRKKY +0 +(q2 = 0) + JRKKY +0 +(q) + JRKKY +0 +(−q) +− 2JRKKY +1 +(q2 = 0) + JRKKY +1 +(q) + JRKKY +1 +(−q) − 2JRKKY +2 +(q = 0) +M(q)−−,−− = − 2JRKKY +0 +(q2 = 0) + JRKKY +0 +(q) + JRKKY +0 +(−q) +− 2JRKKY +1 +(q2 = 0) + JRKKY +1 +(q) + JRKKY +1 +(−q) − 2JRKKY +2 +(q = 0) +M(q)+−,+− =JRKKY +0 +(q) + JRKKY +0 +(−q) − 2JRKKY +0 +(q2 = 0) − 2JRKKY +1 +(R = 0) − 2JRKKY +2 +(q = 0) +M(q)−+,−+ =JRKKY +0 +(q) + JRKKY +0 +(−q) − 2JRKKY +0 +(q2 = 0) − 2JRKKY +1 +(R = 0) − 2JRKKY +2 +(q = 0) +M(q)+−,−+ =JRKKY +5,+ +(q) + JRKKY +5,+ +(−q) +, +M(q)−+,+− = JRKKY +5,− +(q) + JRKKY +5,− +(−q) +M(q)++,−− =JRKKY +2 +(q) + JRKKY +2 +(−q) +, +M(q)−−,++ = JRKKY +2 +(q) + JRKKY +2 +(−q) +M(q)++,+− = − JRKKY +3,+ +(q) +, +M(q)+−,++ = −JRKKY +4,+ +(−q) +M(q)++,−+ = − JRKKY +4,+ +(q) +, +M(q)−+,++ = −JRKKY +3,+ +(−q) +M(q)−−,+− = − JRKKY +4,− +(q) +, +M(q)+−,−− = −JRKKY +3,− +(−q) +M(q)−−,−+ = − JRKKY +3,− +(q) +, +M(q)−+,−− = −JRKKY +4,− +(−q) +(S295) +We define the following functions +h0(q) = − 2JRKKY +0 +(q2 = 0) + JRKKY +0 +(q) + JRKKY +0 +(−q) − 2JRKKY +1 +(q2 = 0) +− JRKKY +1 +(q) − JRKKY +1 +(−q) − 2JRKKY +2 +(q = 0) +h1(q) =JRKKY +0 +(q) + JRKKY +0 +(−q) − 2JRKKY +0 +(q2 = 0) − 2JRKKY +1 +(R = 0) − 2JRKKY +2 +(q = 0) +h2(q) =JRKKY +2 +(q) + JRKKY +2 +(−q) +h3(q) = − JRKKY +3,+ +(q) +h4(q) =2J5,+(q) + 2J5,+(−q) +(S296) +where h0(q), h1(q), h2(q) are real numbers h3(q) is a complex number Then we express M I(q) in a matrix form +M I(q) = +� +�� +h0(q) +h2(q) +h3(q) +−h∗ +3(q) +h2(q) +h0(q) +−h3(q) +h∗ +3(q) +h∗ +3(q) +−h∗ +3(q) +h1(q) +h4(q) +−h3(q) +h3(q) +h∗ +4(q) +h1(q) +� +�� +(S297) + +69 +We now discuss the choice of ii2 of uii2,ξξ2(q). For the purpose of discussion, we pick the following ground state given in +Eq. S284 (at νf = 0, −2) +νf = 0 +: +|ψ0⟩ = +� +R +ψf,+,† +R,1 ψf,−,† +R,1 ψf,+,† +R,2 ψf,−,† +R,2 |0⟩ +(S298) +νf = −2 +: +|ψ0⟩ = +� +R +ψf,+,† +R,1 ψf,−,† +R,1 |0⟩ +(S299) +We first consider the diagonal component. We first consider the diagonal components uI +ii,ξξ2(q). From Eq. S289 and Eq. S298, +we find the corresponding ”excitation” state is +OI +q,ii|ψ0⟩ = +1 +NM +� +ξ,ξ2,R +uii,ξξ2(R)ψf,ξ,† +R,i ψf,ξ2 +R,i eiq·R|ψ0⟩ = +1 +NM +� +ξ,R +uii,ξξ(R)n(ξ, i)|ψ0⟩ +(S300) +where n(ξ, i) is the filling of ξ, i flavors defined as ψξ,† +R,iψξ +R,i|ψ0⟩ = n(ξ, i)|ψ0⟩, and we have used the fact that ψf,+,† +R,i ψf,− +R,i |ψ0⟩ = +0 (as proved near Eq. S285). From Eq. S300, OI +q,ii2|ψ0⟩ = A|ψ0⟩, with A = +1 +NM +� +ξ,R uii,ξξ(R)n(ξ, i) which is a complex +number. Therefore OI +q,ii|ψ0⟩ is the same state (up to a phase and normalization factor) as |ψ0⟩ when A ̸= 0. If A = 0, OI +q,ii|ψ0⟩ +vanish. For both A ̸= 0 and A = 0, OI +q,ii does not create an excitation state and will not be considered. We next consider the +off-diagonal term +OI +q,ii2|ψ0⟩ = +1 +NM +� +ξ,ξ2,R +uii,ξξ2(R)ψf,ξ,† +R,i ψf,ξ2 +R,i2eiq·R|ψ0⟩ +(S301) +A valid mode should have OI +q,ii2|ψ0⟩ ̸= 0. Since ψξ,† +R,iψξ2 +R,i2 in Eq. S301 describes the procedure of moving one electron +from (R, i2, ξ2) to (R, i1, ξ1). This procedure can only be valid when there are zero electrons at (R, i, ξ) and one electron at +(R, i2, ξ2). We next find the valid procedure at νf = 0, −2 corresponding to the ground states given in Eq. S298, Eq. S299. +From Eq. S298 and Eq. Eq. S299, the valid choices are +νf = 0 +: +(i1, i2) ∈ {(1, 3), (1, 4), (2, 3), (2, 4)} +νf = −2 +: +(i1, i2) ∈ {(1, 2), (1, 3), (1, 4)} +(S302) +Combining Eq. S294 and Eq. S302, the excitation modes are derived by solving the following equations +� +ξ′ +1,ξ′ +2 +M(q)(ξ1ξ2),(ξ′ +1ξ′ +2)uI +ii2,ξ′ +1ξ′ +2(q) = EI +quI +ii2,ξ1ξ2(q) +(i1, i2) ∈ {(1, 3), (1, 4), (2, 3), (2, 4)}, +for νf = 0 +(i1, i2) ∈ {(1, 2), (1, 3), (1, 4)}, +for νf = −2 +(S303) +By diagonalizing M(q) (Eq. S297), we plot the spin spectrum at ν = 0, −2 in Fig. S4. +We next discuss the Goldstone modes. At q = 0, we have h3(q = 0) = h4(q = 0) = 0 (Eq. S296). The eigenvalues of +M(q) (Eq. S297) are +E1 +q=0 = −JRKKY +1 +(R = 0) − 2JRKKY +2 +(q = 0) +E2 +q=0 = 0 +E3 +q=0 = −4JRKKY +2 +(q = 0) +E4 +q=0 = −JRKKY +1 +(R = 0) − 2JRKKY +2 +(q = 0) +(S304) +where E2 +q branch gives a Goldstone mode. However, we have four choices of i, i2 at νf = 0 and three choices of i, i2 at +νf = −2. Then we find 4 Goldstone modes at νf = 0 and three Goldstone modes at νf = −2. We plot the spin fluctuations at +νf = 0, νf = −2 as shown in Fig. S4. + +70 +FIG. S4. Charge 0 excitations at νf = 0, −2, −3. +C. +Excitation spectrum from RKKY interaction at νf = −1 +We next calculate the excitation spectrum on top of the ground state at νf = −1 (Eq. S266). We first point out the differences +between νf = 0, −2 and νf = −1. At νf = 0, −2, all the valley-spin flavors are filled with either zero or two f-electrons. +However, at νf = −1, there is one valley-spin flavor filled with two f-electron, one valley-spin flavor filled with one f-electron +and two valley-spin flavors filled with zero f-electrons (Eq. S266). Thus the condition in Eq. S285 is not satisfied by the ground +state at νf = −1. In other words, not all bond operators Aξ,ξ2 +R,R (Eq. S275) annihilate the ground state at νf = −1. +To obtain the excitation spectrum at νf = −1, we first rewrite [Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψξ′ +1,† +R3,iψξ′ +2 +R3,j] (Eq. S287) with normal order. +From Eq. S287 and Wick’s theorem, we have (consider the case of (R, ξ1) ̸= (R2, ξ2)) +[Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′ +1,† +R3,i ψf,ξ′ +2 +R3,j] =δR2,R3δξ2,ξ′ +1 : ψf,ξ1,† +R,i +ψf,ξ′ +2 +R3,jAξ3,ξ4 +R2,R : −δR,R3δξ1,ξ′ +2 : ψf,ξ′ +1,† +R3,i ψf,ξ2 +R2,jAξ3,ξ4 +R2,R : ++ δR,R3δξ′ +1,ξ4 : Af,ξ1,ξ2 +R,R2 ψf,ξ3,† +R2,i ψf,ξ′ +2 +R3,j : −δR2,R3δξ′ +2,ξ3 : Aξ1,ξ2 +R,R2ψf,ξ′ +1,† +R3,i ψf,ξ4 +R,j : ++ δR2,R3δξ2,ξ′ +1δξ′ +2,ξ3(1 − n(ξ′ +2, j))ψf,ξ1,† +R,i +ψf,ξ4 +R,j + δR2,R3δξ2,ξ′ +1δξ1,ξ4n(ξ1, i)ψf,ξ′ +2 +R3,jψf,ξ3,† +R3,i +− δR,R3δξ1,ξ′ +2δξ2,ξ3(1 − n(ξ3, j))ψf,ξ′ +1,† +R,i +ψf,ξ4 +R,j − δR,R3δξ1,ξ′ +2δξ′ +1,ξ4n(ξ′ +1, i)ψξ2 +f,R2,jψf,ξ3,† +R2,i ++ δR,R3δξ′ +1,ξ4δξ2,ξ3(1 − n(ξ2, i))ψf.ξ1,† +R,i +ψf.ξ′ +2 +R3,j + δR,R3δξ′ +1,ξ4δξ1,ξ′ +2n(ξ1, j)ψf,ξ2 +R2,jψf,ξ3,† +R2,i +− δR2,R3δξ3,ξ′ +2δξ2,ξ′ +1(1 − n(ξ2, i))ψf,ξ1,† +R,i +ψf,ξ4 +R,j − δR2,R3δξ3,ξ′ +2δξ1,ξ4n(ξ1, j)ψf,ξ2 +R2,jψf,ξ′ +1,† +R3,i ++ δi,jδR,R2δR,R3(n(ξ′ +2, i) − n(ξ′ +1, i)) +� +δξ′ +1,ξ2δξ1,ξ′ +2Aξ3,ξ4 +R2,R + δξ′ +1,ξ4δξ3,ξ′ +2Aξ1,ξ2 +R,R2 +� +− δi,jC0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2) +(S305) +where : O : represents the normal ordered form of the operator O with respect to the ground state |ψ0⟩ in Eq. S266, and +n(ξ, i) = ⟨ψ0|ψvξ,† +R,i ψf,ξ +R,i|ψ0⟩ +, +1 − n(ξ, i) = ⟨ψ0|ψf,ξ +R,iψf,ξ,† +R,i |ψ0⟩ . +(S306) +The constant C0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2) is defined as +C0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2) += − δR2,R3δξ2,ξ′ +1δξ3,ξ′ +2δξ1,ξ4n(ξ1, i)[−n(ξ3, i) + n(ξ2, i)] +− δR,R3δξ1,ξ′ +2δξ4,ξ′ +1δξ2,ξ3[−n(ξ4, i) + n(ξ1, i) + n(ξ2, i)n(ξ4, i) − n(ξ2, i)n(ξ1, i)] +− δR,R2δR,R3[n(ξ′ +2, i) − n(ξ′ +1, i)] +� +δξ′,ξ2δξ1,ξ′ +2( +� +m +δξ3,ξ4n(ξ3, m) + δξ′ +1,ξ4δξ′ +2,ξ3δξ1,ξ2 +� +m +n(ξ1, m) +� +We also mention the difference between νf = −1 and νf = 0, −2 again. At νf = 0, −2, when acting the commutator +[Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψξ′ +1,† +R3,iψξ′ +2 +R3,j] (Eq. S305) on the ground state (Eq. S284), the four-fermion normal ordered term vanishes because +of Eq. S285. However, at νf = −1, when acting the commutator [Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψξ′ +1,† +R3,iψξ′ +2 +R3,j] on the ground state (Eq. S266), +the four-fermion normal ordered term will not vanish, because Eq. S285 no longer holds at νf = −1. + +71 +At νf = −1, we approximate the commutator by dropping the four-fermion normal-ordering term +[Aξ1,ξ2 +R,R2Aξ3,ξ4 +R2,R, ψf,ξ′ +1,† +R3,i ψf,ξ′ +2 +R3,j] ≈δR2,R3δξ2,ξ′ +1δξ′ +2,ξ3(1 − n(ξ′ +2, j))ψf,ξ1,† +R,i +ψf,ξ4 +R,j + δR2,R3δξ2,ξ′ +1δξ1,ξ4n(ξ1, i)ψf,ξ′ +2 +R3,jψf,ξ3,† +R3,i +− δR,R3δξ1,ξ′ +2δξ2,ξ3(1 − n(ξ3, j))ψf,ξ′ +1,† +R,i +ψf,ξ4 +R,j − δR,R3δξ1,ξ′ +2δξ′ +1,ξ4n(ξ′ +1, i)ψf,ξ2 +R2,jψf,ξ3,† +R2,i ++ δR,R3δξ′ +1,ξ4δξ2,ξ3(1 − n(ξ2, i))ψf,ξ1,† +R,i +ψf,ξ′ +2 +R3,j + δR,R3δξ′ +1,ξ4δξ1,ξ′ +2n(ξ1, j)ψf,ξ2 +R2,jψf,ξ3,† +R2,i +− δR2,R3δξ3,ξ′ +2δξ2,ξ′ +1(1 − n(ξ2, i))ψf,ξ1,† +R,i +ψf,ξ4 +R,j ++ δi,jδR,R2δR,R3(n(ξ′ +2, i) − n(ξ′ +1, i)) +� +δξ′ +1,ξ2δξ1,ξ′ +2Aξ3,ξ4 +R2,R + δξ′ +1,ξ4δξ3,ξ′ +2Aξ1,ξ2 +R,R2 +� ++ δi,jC0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2) +(S307) +We now calculate the commutator between ˆHRKKY,1 + ˆHRKKY,0 (Eq. S283) and OI +q,ij (Eq. S289). Using Eq. S307, we find +[ ˆHRKKY,1, OI +q,ij]|ψ0⟩ +≈ +� +R,R2 +� +ξ1,ξ2,ξ3,ξ4 +J(R − R2, ξ1, ξ2, ξ3, ξ4) 1 +NM +� +R3,ξ′,ξ′ +2 +uI +ij,ξ′ +1ξ′ +2(q)eiq·R3 +� +δR2,R3δξ2,ξ′ +1δξ′ +2,ξ3(1 − n(ξ′ +2, j))ψf,ξ1,† +R,i +ψf,ξ4 +R,j + δR2,R3δξ2,ξ′ +1δξ1,ξ4n(ξ1, i)ψf,ξ′ +2 +R3,jψf,ξ3,† +R3,i +− δR,R3δξ1,ξ′ +2δξ2,ξ3(1 − n(ξ3, j))ψf,ξ′ +1,† +R,i +ψf,ξ4 +R,j − δR,R3δξ1,ξ′ +2δξ′ +1,ξ4n(ξ′ +1, i)ψf,ξ2 +R2,jψf,ξ3,† +R2,i ++ δR,R3δξ′ +1,ξ4δξ2,ξ3(1 − n(ξ2, i))ψf,ξ1,† +R,i +ψf,ξ′ +2 +R3,j + δR,R3δξ′ +1,ξ4δξ1,ξ′ +2n(ξ1, j)ψf,ξ2 +R2,jψf,ξ3,† +R2,i +− δR2,R3δξ3,ξ′ +2δξ2,ξ′ +1(1 − n(ξ2, i))ψf,ξ1,† +R,i +ψf,ξ4 +R,j − δR2,R3δξ3,ξ′ +2δξ1,ξ4n(ξ1, j)ψf,ξ2 +R2,jψf,ξ′ +1,† +R3,i ++ δi,jδR,R2δR,R3(n(ξ′ +2, i) − n(ξ′ +1, i)) +� +δξ′ +1,ξ2δξ1,ξ′ +2Aξ3,ξ4 +R2,R + δξ′ +1,ξ4δξ3,ξ′ +2Aξ1,ξ2 +R,R2 +� +− δi,jC0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2) +� +|ψ0⟩ += +� +R,q +� +ξ1,ξ2,ξ3,ξ4 +eiq·R +NM +� +J(q, ξ1, ξ2, ξ3, ξ4)(1 − n(ξ3, j))uI +ij,ξ2ξ3(q)ψf,ξ1,† +R,i +ψξ4 +f,R,j + +� +ξ′ +2 +J(q2 = 0, ξ1, ξ2, ξ3, ξ1)n(ξ1, i)ψξ′ +2 +R,juI +ij,ξ2ξ′ +2(q)ψξ3,† +R,i +− +� +ξ′ +1 +J(q2 = 0, ξ1, ξ2, ξ2, ξ4)(1 − n(ξ2, j))uij,ξ′ +1ξ1ψf,ξ′ +1,† +R,i +ψf,ξ4 +R,j − J(−q, ξ1, ξ2, ξ3, ξ4)n(ξ4, i)uI +ij,ξ4ξ1(q)ψf,ξ2 +R,j ψf,ξ3,† +R,i ++ +� +ξ′ +2 +J(q2 = 0, ξ1, ξ2, ξ2, ξ4)(1 − n(ξ2, i))uI +ij,ξ4ξ′ +2(q)ψf,ξ1,† +R,i +ψf,ξ′ +2 +R,j ++ J(−q, ξ1, ξ2, ξ3, ξ4)n(ξ1, j)uI +ij,ξ4ξ1(q)ψξ2 +R,jψξ3,† +R,i − +� +ξ′ +1 +J(q, ξ1, ξ2, ξ3, ξ4)(1 − n(ξ2, i))uI +ij,ξ2ξ3(q)ψf,ξ1,† +R,i +ψf,ξ4 +R,j +− J(q2 = 0, ξ1, ξ2, ξ3, ξ1)n(ξ1, j)uI +ij,ξ′ +1ξ3(q)ψf,ξ2 +R,j ψf,ξ′ +1,† +R,i +� ++ +� +ξ1,ξ2,ξ3,ξ4,q,i +eiq·RJ(R2 = 0, ξ1, ξ2, ξ3, ξ4) +NM +δi,j +� +(n(ξ1, i) − n(ξ2, i))uI +ii,ξ2ξ1(q) +� +m +ψf,ξ3,† +R,m ψf,ξ4 +R,m ++ (n(ξ4, i) − n(ξ3, i))uI +ii,ξ4ξ3(q) +� +m +ψf,ξ1,† +R,m ψf,ξ2 +R,m +� +− +� +R,R2 +� +ξ1,ξ2,ξ3,ξ4 +J(R − R2, ξ1, ξ2, ξ3, ξ4) 1 +NM +� +R3,ξ′,ξ′ +2,i,j +uI +ij,ξ′ +1ξ′ +2(q)eiq·R3δi,j +C0(R, R2, R3, ξ1, ξ2, ξ3, ξ4, ξ′ +1, ξ′ +2)|ψ0⟩ +(S308) + +72 +Using Eq. S307, we find +[ ˆHRKKY,0, OI +q,ij]|ψ0⟩ = − JRKKY +1 +(R2 = 0)[ +� +ξ1 +� +R +ˆνf +ξ1(R) ˆνf +ξ1(R), OI +q,ij]|ψ0⟩ +=−JRKKY +1 +(R2 = 0) +NM +� +R′,ξ′,ξ′ +2,ξ1 +uI +ii2,ξ′ξ′ +2(q)eiq·R′[ +� +R +ˆνf +ξ1(R) ˆνf +ξ1(R), ψf,ξ′,† +R′,i ψf,ξ′ +2 +R′,i2]|ψ0⟩ +=−JRKKY +1 +(R2 = 0) +NM +� +R,ξ′,ξ′ +2,ξ1 +uI +ij,ξ′ξ′ +2(q)eiq·R[ ˆνf +ξ1(R) ˆνf +ξ1(R), ψf,ξ′,† +R,i +ψf,ξ′ +2 +R,j ]|ψ0⟩ +=−JRKKY +1 +(R2 = 0) +NM +� +R,ξ′,ξ′ +2 +uI +ij,ξ′ξ′ +2(q)eiq·R2(νξ′ +f − νξ′ +2 +f + 1)(1 − δξ′,ξ′ +2)ψf,ξ′,† +R,i +ψf,ξ′ +2 +R,j ]|ψ0⟩ +(S309) +Before calculating the spectrum with the commutator, we first discuss the choice of ij in OI +q,ij. We take the following ground +state (from Eq. S266) as an example +|ψ0⟩ = +� +R +ψf,+,† +R,1 ψf,+,† +R,2 ψf,−,† +R,1 |0⟩ . +(S310) +For the same reason given below Eq. S301, the diagonal components OI +q,ii will not contribute to the excitation spectrum. For the +off-diagonal term OI +q,ij, OI +q,ij (when acting on the ground state) will move one electron from valley-spin flavor j to valley-spin +flavor i. Then a valid procedure requires valley-spin flavor j to have at least one filled ξ-orbital (remember we have ξ = ±1 +orbitals at each valley-spin flavor) and valley-spin flavor i has one empty ξ-orbital. Then only the following choices of ij (in +Oq,ij) are valid for the ground state in Eq. S310 +(i, j) ∈ {(3, 1), (4, 1), (3, 2), (4, 2), (2, 1), (2, 2)} +(S311) +(S312) +We further classify them into four sectors according to whether flavor i, j are fully-filled, empty or half-filled +Full-empty sector : (i, j) ∈ {(3, 1), (4, 1)} +Full-half sector : (i, j) = (2, 1) +Half-empty sector : (i, j) ∈ {(3, 2), (4, 2)} +Half-half sector : (i, j) = (2, 2) +(S313) +We next give the explicit form of OI +q,ij|ψ0⟩ (from Eq. S289) +OI +q,ij|ψ0⟩ = 1 +NM +� +R,ξ′ξ′ +2 +ψf,ξ′,† +R,i +ψf,ξ′ +2 +R,i2uI +ij,ξ′ξ′ +2(q)eiq·R|ψ0⟩ +(S314) +For full-half sector with (i, j) = (2, 1). ξ = +, i = 2 flavor has been filled with one electron (Eq. S310). The term associated +with uI +ij,+ξ′ +2 annihilate the ground state (Eq. S314). In other words, +ψf,+,† +R,i=2ψf,ξ′ +2 +R,j=1|ψ0⟩ = 0 +(S315) +Then we can simply set the corresponding uI +ij,+ξ′ +2(q) to zero +uI +ij,+ξ′ +2(q) = 0 +, +(i, j) = (2, 1) +(S316) +since uI +ij,+ξ′ +2(q) does not describe a valid procedure. +For half-empty sector with (i, j) = (3, 2) or (i, j) = (4, 2). ξ′ +2 = −, j = 2 is empty. Then ψ+,† +R,iψξ′ +2=− +R,j=2|ψ0⟩ = 0 (Eq. S314) +and we let +uI +ij,−ξ′(q) = 0 +, +(i, j) ∈ {(3, 2), (4, 2)} +(S317) +For half-half sector, with (i, j) = (2, 2). ξ′ +2 = −, j = 2 flavor is empty and ξ′ = +, i = 2 flavor is filled with one f-electron, +we then set +uI +ii,ξξ(q) = 0 +, +(i, j) = (2, 2), ξ ∈ {±} +(S318) +As for full-empty sector, with (i, j) ∈ {(3, 1), (4, 1)}, all components of uI +ij,ξξ′(q) can be non-zero. + +73 +1. +Full-empty sector +We now consider the full-empty sector with (i, j) ∈ {(3, 1), (4, 1)}. Combining Eq. S308 and Eq. S309, we have +[ ˆHRKKY , OI +ij,q]|ψ0⟩ = +� +R +� +ξ1,ξ2 +� +ξ′ +1,ξ′ +2 +uij,ξ′ +1ξ′ +2(q)M(q)(ξ1ξ2),(ξ′ +1ξ′ +2)ψf,ξ1,† +R,i +ψf,ξ2 +R,j +eiq·R +NM +|ψ0⟩ +(S319) +where +M(q)++,++ =2[JRKKY +0 +(q) − JRKKY +0 +(q = 0) + JRKKY +1 +(q) − JRKKY +1 +(q = 0)] − 2JRKKY +2 +(q2 = 0) +M(q)−−,−− =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] − 2JRKKY +2 +(q2 = 0) +M(q)++,−− = − 2JRKKY +2 +(q)(−1) +, +M(q)−−,++ = −2JRKKY +2 +(q)(−1) +M(q)+−,+− =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0) − JRKKY +2 +(q2 = 0)] +− 2JRKKY +1 +(R2 = 0) +M(q)−+,−+ =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0) − JRKKY +2 +(q2 = 0)] +M(q)++,+− =JRKKY +3+ +(q) +, +M(q)+−,++ = JRKKY +4+ +(−q) +M(q)++,−+ =JRKKY +4+ +(q) +, +M(q)−+,++ = JRKKY +3+ +(−q) +M(q)−−,+− =JRKKY +4− +(q) +, +M(q)+−,−− = JRKKY +3− +(−q) +M(q)−−,−+ =JRKKY +3− +(q) +, +M(q)−+,−− = JRKKY +4− +(−q) +M(q)+−,−+ =J5,+(q) + J5,+(−q) +, +M(ij, q)−+,+− = J∗ +5,+(q) + J∗ +5,+(q) +(S320) +There are four modes for each choice of (i, j), and eight modes in total. +At q = 0, we can derive the analytical expressions of four eigenvalues of M(q = 0): +E1 +q=0 = 0 +, +E2 +q=0 = −4JRKKY +2 +(q2 = 0) +E3 +q=0 = −2JRKKY +2 +(q2 = 0) − 2JRKKY +1 +(R2 = 0) +, +E4 +q=0 = −2JRKKY +2 +(q2 = 0) +(S321) +where E1(q) corresponds to Goldstone modes. +2. +Full-half sector +We consider the full-half sector with (i, j) = (2, 1). Combining Eq. S308, Eq. S309 and also the constraints S318, we +construct the following eigenequations with eigenvalue EI +q +[ ˆHRKKY , OI +21,q]|ψ0⟩ = +� +R +� +ξ2,ξ′ +2 +uI +21,−ξ′ +2(q)M(q)ξ2,ξ′ +2ψf,−,† +R,2 ψf,ξ2 +R,j +eiq·R +NM +|ψ0⟩ +=EI +q +� +R +uI +21,−ξ2(q)ψf,−,† +R,2 ψf,ξ2 +R,j +eiq·R +NM +|ψ0⟩ +(S322) +where the 2 × 2 matrix M(q)ξ2,ξ′ +2 is defined as +M(q)−,− =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] +M(q)+,+ =2[JRKKY +0 +(q) − JRKKY +0 +(q = 0) + JRKKY +1 +(q) − JRKKY +1 +(q = 0) − JRKKY +2 +(q = 0)] +M(q)−,+ =JRKKY +3,− +(q) +M(q)+,− =JRKKY +4,− +(−q) +(S323) +Equivalently, Eq. S322 can be written as +� +ξ′ +2 +M(q)ξ2,ξ′ +2uI +21,−ξ′ +2(q) = EI +quI +21,−ξ2(q) +(S324) + +74 +There are two excitation modes with dispersions +E1,2 +q +=2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] − JRKKY +2 +(q = 0) +± +� +(JRKKY +2 +(q = 0))2 + |JRKKY +3,+ +(q)|2 +(S325) +At q = 0, two excitation modes are +E1 +q=0 = −2JRKKY +2 +(q = 0) +, +E2 +q=0 = 0 +(S326) +with E2 +q=0 gives the Goldstone modes. +3. +Half-empty sector +We next discuss the half-empty sector with (i, j) ∈ {(3, 2), (4, 2)}. Combining Eq. S308, Eq. S309 and also the con- +straints S318, we construct following eigenequations (where i = 3, 4) with eigenvalue EI +q +[ ˆHRKKY , OI +i2,q]|ψ0⟩ = +� +R +� +ξ1,ξ′ +1 +uI +i2,ξ′ +1+(q)M(q)ξ1,ξ′ +1ψf,ξ1,† +R,i +ψf,+ +R,2 +eiq·R +NM +|ψ0⟩ +=EI +q +� +R +uI +i2,ξ1+(q)ψf,ξ2,† +R,i +ψf,+ +R,2 +eiq·R +NM +|ψ0⟩ +(S327) +where the 2 × 2 matrix M(q)ξ1,ξ′ +1 is defined as +M(q)+,+ =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] +M(q)−,− =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0) − JRKKY +2 +(q2 = 0)] +M(q)+,− =JRKKY +4+ +(q) +, +M(q)−,+ = JRKKY +3+ +(−q) +(S328) +Equivalently, Eq. S327 can be written as +� +ξ′ +1 +M(q)ξ1,ξ′ +1uI +I2,ξ′ +1+(q) = EI +quI +i2,ξ1+(q) +(S329) +The two eigenvalues of M(q)ξ1,ξ′ +1 are +E1 +q =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] − JRKKY +2 +(q2 = 0) ++ +� +JRKKY +2 +(q2 = 0)2 + |JRKKY +3+ +(−q)|2 +E2 +q =2[JRKKY +0 +(q) − JRKKY +0 +(q2 = 0) + JRKKY +1 +(q) − JRKKY +1 +(q2 = 0)] − JRKKY +2 +(q2 = 0) +− +� +JRKKY +2 +(q2 = 0)2 + |JRKKY +3+ +(−q)|2 +(S330) +At q = 0, we have +E1 +q=0 = −2JRKKY +2 +(q2 = 0) +, +E2 +q=0 = 0 +(S331) +where E2 +q corresponds to the Goldstone mode + +75 +4. +Half-half sector +We next discuss the half-empty sector with (i, i) = (2, 2). Combining Eq. S308, Eq. S309 and also the constraints S318, we +have +[ ˆHRKKY , OI +q,22]|ψ0⟩ +≈ +� +R +eiq·R +NM +� +− J(q2 = 0, +, −, −, +)uI +22,−+(q)ψf,−,† +R,2 ψ+ +f,R,2 − J(q2 = 0, +, −, −, +)u22,−+ψf,−,† +R,2 ψf,+ +R,2 ++ J(q2 = 0, −, −, −, −)uI +22,−+(q)ψf,−,† +R,2 ψf,+ +R,2 + J(−q, +, +, −, −)uI +22,−+(q)(−1)ψf,−,† +R,2 ψf,+ +R,2 +− J(q, −, −, +, +)uI +22,−+(q)ψf,−,† +R,2 ψf,+ +R,2 − J(q2 = 0, +, +, +, +)uI +22,−+(q)(−1)ψf,−,† +R,2 ψf,+ +R,2 ++ J(R2 = 0, +, −, −, +)uI +22,−+(q)ψf,−,† +R,2 ψf,+ +R,2 + J(R2 = 0, −, +, +, −)uI +22,−+(q)ψf,−,† +f,R,2ψf,+ +f,R,2 +� +|ψ0⟩ +=2 +� +JRKKY +2 +(q2 = 0) − JRKKY +2 +(R = 0) − 2JRKKY +1 +(q2 = 0) + JRKKY +1 +(R = 0) + JRKKY +0 +(q) − JRKKY +0 +(q = 0) +� +OI +q,22|ψ0⟩ +=EI +qOI +q,22|ψ0⟩ +where the dispersion is +EI +q = 2 +� +JRKKY +2 +(q2 = 0) − JRKKY +2 +(R = 0) − JRKKY +1 +(q2 = 0) + JRKKY +1 +(R = 0) + JRKKY +0 +(q) − JRKKY +0 +(q = 0) +� +(S332) +5. +Number of Goldstone modes +We now count the number of Goldstone modes of ground states in Eq. S310. According to Eq. S321, there are one Goldstone +modes for each (i, j) ∈ {(3, 1), (4, 1)} choice in half-empty sector. Then, there are in total two Goldstone modes in the full- +empty sector. According to Eq. S326, there are one Goldstone modes for each (i, j) ∈ {(2, 1)} choices in the full-half sector. +Then, there is one Goldstone mode in the full-half sector. According to Eq. S331, there are one Goldstone modes for each +(i, j) ∈ {(3, 2), (4, 2)} choices in the half-empty sector. Then, there are in total two Goldstone modes in the half-empty sector. +Therefore, we have 5 Goldstone modes in total, which is consistent with Ref. [86]. +D. +Discussion +First, we mention that our spectrum is calculated from RKKY Hamiltonian (Eq. S110). Thus the polarization of conduction +electrons is ignored and the RKKY interactions are long-range (power-law decay). Consequently, we observe a linear dispersion +of the Goldstone modes. Second, we derive the RKKY interaction in the momentum space by performing Fourier transformation +integral and introducing a short distance cutoff (as described below Eq. S234 and also in Sec. S12). Therefore, the short-distance +information (large momentum) is lost in the spectrum. In the next section (Sec. S8, we will include the polarization of conduction +electrons and produce the excitation spectrum that will more accurately describe the fluctuations of f-moments. +S8. +EFFECTIVE THEORY OF f-MOMENTS +After finding the ground state from RKKY interactions, we now derive the effective theory of f-moments in the nonchiral-flat +limit (v′ +⋆ = 0, M ̸= 0) and at integer filling ν = νf = 0, −1, −2. +We first define the coherent state |u⟩ of f-moments as +|u⟩ = +� +R +|u(R)⟩R +(S333) +|u(R)⟩R = ˆR[u(R)]|ψ0⟩R . +(S334) + +76 +|u(R)⟩R is the coherent state of the f-moment at site R where |ψ0⟩R is a basis state that corresponds to the ground state given +in Eq. S265 or Eq. S266. Without loss of generality, at each filling, we let +νf = 0 +: +|ψ0⟩R = ψf,+,† +R,1 ψf,−,† +R,1 ψf,+,† +R,2 ψf,−,† +R,2 |0⟩ +νf = −1 +: +|ψ0⟩R = ψf,+,† +R,2 ψf,+,† +R,1 ψf,−,† +R,1 |0⟩ +νf = −2 +: +|ψ0⟩R = ψf,+,† +R,1 ψf,−,† +R,1 |0⟩ +(S335) +The corresponding ground states are then +|ψ0⟩ = +� +R +|ψ0⟩R +(S336) +ˆR[u(R)] is a SU(8) rotation defined as +ˆR[u(R)] = +� +R +exp +� +− i +� +ij,ξξ′ +uiξ,jξ′(R)ψf,ξ,† +R,i ψf,ξ′ +R,j +� +. +(S337) +(Here we do not include the U(1) charge rotation. Because we fix the filling of f to be integer, and the U(1) charge is a good +quantum number). If we treat iξ as row indices and jξ′ as column indices, then u(R) can be understood as a 8 × 8 matric. +Since ˆRR[u(R)] generates a SU(8) rotation of f-fermions, u(R) is a traceless Hermitian matrix. To observe the nature of the +transformation, we act the transformation operator on the f electrons which gives +� +ˆR† +R[u(R)] +� +ψf,ξ +R,i +� +ˆRR[u(R)] +� += [e−iu(R)]iξ,jξ′ψf,ξ′ +R,j +(S338) +where exp(−iu(R)) is a matrix exponential. We let +Riξ,jξ′(R) = [e−iu(R)]iξ,jξ′ +(S339) +then +� +ˆR† +R[u(R)] +� +ψf,ξ +R,i +� +ˆRR[u(R)] +� += Riξ,jξ′(R)ψf,ξ′ +R,j +(S340) +We now derive the path integral of the following Hamiltonian +ˆH = ˆH′ +c + ˆHint . +(S341) +ˆH′ +c contains the fermion bilinear terms and ˆHint denotes the interactions between c-electrons and f-moments. ˆHc′ takes the +form of +ˆH′ +c = +� +k,i +� +v⋆(kx + iξky)ψξ,c′,† +k,i +ψξ,c′′ +k,i + h.c. +� ++ (−µ + Wνf + V0 +Ω0 +νc) +� +k,i +[ψξ,c′,† +k,i +ψξ,c′ +k,i + ψξ,c′′,† +k,i +ψξ,c′′ +k,i ] +− +� +k,ξ,i +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +e−λ2|k|2�γ2 +2 ψξ,c′,† +k,i +ψξ,c′ +k,i + v′ +⋆γ(kx − iξky)ψc′,ξ,† +k,i +ψc′,−ξ +k,i +� +where the contributions from ˆHW (Eq. S7), ˆHV (Eq. S8), as well as the one-body scattering term from SW transformation +ˆHcc(Eq. S203), have been included, and ˆHV has been treated by mean-field approximation. In addition, we have dropped the +constant terms. The interaction term (Eq. S202) is +ˆHint = +� +R,k,k′,ξ,ξ′ +� +µν +e−i(k′−k)·R)F(|k)|F(|k′) +NMDνc,νf +� +γ2 : ˆΣ(f,ξξ′) +µν +(R) : ++ γv′ +⋆ : ˆΣ(f,ξ−ξ′) +µν +(R) : (k′ +x − iξ′k′ +y) + γv′ +⋆ : ˆΣ(f,−ξξ′) +µν +(R) : (kx + iξky) +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k) : +− J +� +R,k,k′,µν,ξ +e−i(k′−k)·R : ˆΣ(f,ξξ) +µν +(R) :: ˆΣ(c′′,ξξ) +µν +(k, k′ − k) : +(S342) + +77 +We now derive the path integral formula of the partition function corresponding to the Hamiltonian ˆH = ˆH′ +c + ˆHint. The +partition function can be written as +Z = lim +N→∞ +N +� +n=1 +Tr[e−∆τ ˆ +H], +where ∆τ = β/N +(S343) +For each time slice τn = n∆τ, we insert the identity operator (with Haar measure [140]) and have +Z = lim +N→∞ +� +D[u(R, τn), c† +k,αηs(τn), c† +k,αηs(τn)]Tr +� +N +� +n=1 +e−∆τH |c(τn)⟩|u(τn)⟩⟨u(τn)|⟨c(τn)| +����⟨u(τn)|⟨c(τn)| +���� +� += lim +N→∞ +� +D[u(R, τn), c† +k,αηs(τn), c† +k,αηs(τn)]Tr +� +N +� +n=1 +⟨ ⟨u(τn)⟨c(τn+1)|e−∆τH|c(τn+1)⟩|u(τn+1)⟩ +�����⟨u(τn + ∆τ)|⟨c(τn + ∆τ) +���� +����⟨u(τn)|⟨c(τn) +���� +� +(S344) +where |u(τn)⟩ = � +R |u(R, τn)⟩, |c(τn)⟩ are coherent states of c and u fields, respectively, at time slice τn = nβ/N. For each +time slice and large N (small ∆τ), we have +⟨u(τn + ∆τ)|⟨c(τn + ∆τ)|e−∆τ ˆ +H|c(τn)⟩|u(τn)⟩ +�����⟨u(τn + ∆τ)|⟨c(τn + ∆τ) +���� +����⟨u(τn)|⟨c(τn) +���� +≈ +⟨c(τn + ∆τ)||c(τn)⟩ +�����|⟨c(τn + ∆τ) +���� +����|⟨c(τn) +���� +⟨u(τn + ∆τ)||u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +− ∆τ ⟨u(τn + ∆τ)|⟨c(τn + ∆τ)| ˆH|c(τn)⟩|u(τn)⟩ +�����⟨u(τn + ∆τ)|⟨c(τn + ∆τ) +���� +����⟨u(τn)|⟨c(τn) +���� +≈ +� +1 − +� +k,aηs +[c† +k,aηs(τn + ∆τ) − c† +k,aηs(τn)]ck,aηs(τ) +�� +1 − +� +⟨u(τn + ∆τ)|u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +− 1 +�� +− H(τn)∆τ +≈1 − +� +k,aηs +c† +k,aηs(τn)∂τck,aηs(τn)∆τ + +� +⟨u(τn + ∆τ)|u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +− 1 +� +− H(τn)∆τ +≈ exp +� +− +� +k,aηs +c† +k,aηs(τn)∂τck,aηs(τn)∆τ + +� +⟨u(τn + ∆τ)|u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +− 1 +� +− H(τn)∆τ +� +(S345) +where H(τn) is the original Hamiltonian ˆH where each ck,aηs is replaced by ck,aηs(τn), and each ˆΣ(f,ξξ′) +µν +(R) is replaced by +Σ(f,ξξ′) +µν +(R, τn) = ⟨u(τn)| : ˆΣ(f,ξξ′) +µν +(R) : |u(τ)⟩. And we use the fact that |u(τn)⟩ is normalized with ||u(τn)⟩| = 1. +We then define the action as +S = +� β +0 +� +k,aηs +c† +k,aηs(τ)∂τck,aηs(τ)dτ − lim +∆τ→0 +� +n +� +⟨u(τn + ∆τ)|u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +− 1 +� ++ +� β +0 +H(τ)dτ . +(S346) +We next evaluate each term in the action. The first term is the dynamical term of c electrons. The second one corresponds to + +78 +the Berry phase term of f-moments. We note that +⟨u|˜u⟩ ≈ +� +R +⟨ψ0| +� +1 + i +� +ij,ξξ′ +uiξ,jξ′(R)ψξ,† +R,iψξ′ +R,j − 1 +2 +� � +ij,ξξ′ +uiξ,jξ′(R)ψξ,† +R,iψξ′ +R,j +�2� +� +1 − i +� +i2j2,ξ2ξ′ +2 +˜ui2ξ2,j2ξ′ +2(R)ψξ2,† +R,i2ψξ′ +2 +R,j2 − 1 +2 +� +� +i2j2,ξ2ξ′ +2 +˜ui2ξ2,j2ξ′ +2(R)ψξ2,† +R,i2ψξ′ +2 +R,j2 +�2� +|ψ0⟩ +≈ +� +R +� +1 + +� +iξ +iuiξ,iξn(ξ, i)(R) − i˜uiξ,iξ(R)n(ξ, i) ++ +� +i,ξ,j,ξ′,i2,ξ2,j2,ξ′ +2 +[δi,jδξ,ξ′δi2,j2δξ2,ξ′ +2n(ξ, i)n(ξ2, i2) + δi,j2δξ,ξ′ +2δj,i2δξ′,ξ2δj,i2n(ξ, i)(1 − n(ξ′, j))] +[uiξ,jξ′(R)˜ui2ξ2,j2ξ′ +2(R) − 1 +2uiξ,jξ′(R)ui2ξ2,j2ξ′ +2(R) − 1 +2 ˜uiξ,jξ′(R)˜ui2ξ2,j2ξ′ +2(R)] +� +(S347) +where we use the fact that ⟨ψ0|ψξ,† +R,iψξ′ +R,j|ψ0⟩ ∝ δi,jδξ,ξ′, with |ψ0⟩ the ground state given in Eq. S265 or Eq. S266. In addition, +we let n(ξ, i) = ⟨ψ0|ψξ,† +R,iψξ +R,i|ψ0⟩. We aim to rewrite the above equation in a more compact form. We introduce the following +matrix +Λiξ,jξ′ = 1 +2⟨ψ0| : ψf,ξ,† +R,i ψf,ξ′ +R,j : |ψ0⟩ = δi,jδξ,ξ′ 1 +2(n(ξ, i) − 1 +2) +(S348) +For |ψ0⟩ defined in Eq. S335 (which are ground states at v′ +⋆ ̸= 0, M = 0), the values of Λiξ,iξ are +νf = 0 +: +Λ1+,1+ = Λ2+,2+ = Λ1−,1− = Λ2−,2− = 1 +4 +, +Λ3+,3+ = Λ4+,4+ = Λ3−,3− = Λ4−,4− = −1 +4 +νf = −1 +: +Λ1+,1+ = Λ2+,2+ = Λ1−,1− = 1 +4 +, +Λ2−,2− = Λ3+,3+ = Λ4+,4+ = Λ3−,3− = Λ4−,4− = 1 +4 +νf = −2 +: +Λ1+,1+ = Λ1−,1− = 1 +4 +, +Λ2+,2+ = Λ2−,2− = Λ3+,3+ = Λ4+,4+ = Λ3−,3− = Λ4−,4− = 1 +4 +(S349) +Here we comment that no matter what types of ferromagnetic order/ground state we considered, we can always make a basis +transformation to make Λ a diagonal matrix. To observe this, we assume Λ is an arbitrary Hermitian matrix. We perform an +eigendecomposition Λ = V ˜ΛV † where ˜Λ is a diagonal matrix that characterizes the eigenvalues of Λ and V is the matrix formed +by eigenvectors of Λ. Clearly, V can be understood as a SU(8) rotation and we can make a basis change accordingly ψf → ˜ψf, +where ˜ψf,ξ +R,i = � +j,ξ′ ψf,ξ′ +R,j Vjξ′,iξ, such that 1 +2⟨ψ0| : ˜ψf,ξ,† +R,i ˜ψf,ξ′ +R,j : |ψ0⟩ = ˜Λ which is a diagonal matrix. Then in the new basis, we +have a new model with a diagonal Λ matrix. However, there are two situations. In the first case, the SU(8) rotation characterized +by V is also a symmetry transformation of the flat U(4) group. Since the Hamiltonian is invariant under a flat U(4) group, so +the Hamiltonian is invariant under the basis change which is just a flat U(4) transformation (Note that we also need to perform +the same V -transformation on c-electron). Then our effective theory remains the same compared to the effective theory we built +for the ground state in Eq. S335. This is also a consequence of the flat U(4) symmetry of the system, namely all the ground +states (characterized by Λ) that are connected by flat U(4) transformation giving rise to the same effective theory. In the second +case, the SU(8) rotation characterized by V does not belong to the flat U(4) group. Then we have different types of ground +state compared to the ground state in Eq. S335, and we have a different effective theory. Then we have +⟨u|˜u⟩ = +� +R +1 + 2iTr[(u − ˜u)Λ] − 2 +� +Tr[(u − ˜u)Λ] +�2 ++ 2Tr +� +[(u − ˜u)Λ]2 +� +− 1 +8Tr +� +(u − ˜u)2 +� +− Tr[uΛ˜u] + Tr[˜uΛu] +(S350) +Using Eq. S350, the Berry phase term of f-moments becomes +⟨u(τn + ∆τ)|u(τn)⟩ +�����⟨u(τn + ∆τ) +���� +����⟨u(τn) +���� +≈ +� +R +� +1 + 2iTr[Λ∂τu(R, τ)]∆τ − Tr[(Λu(R, τn) − u(R, τn)Λ)∂τu(R, τn)]∆τ +� +(S351) +Finally, to the formula of H(τ)(Eq. S345), we represent the U(8) moments in the path integral with u, +Σ(f,ξξ′) +µν +(R, τn) = ⟨u(τn)| : ˆΣ(f,ξξ′) +µν +(R) : |u(τn)⟩ = 1 +2 +� +i,j +T µν +ij ⟨ψ0| ˆR†[u(R, τ)] : ψξ,† +R,iψξ′ +R,j : ˆR[u(R, τ)]|ψ0⟩ + +79 +Using Eq. S339, Eq. S340 and Eq. S348, we can now express f-moments with u fields +Σ(f,ξξ′) +µν +(R, τ) =1 +2 +� +i,j +T µν +ij Rjξ′,j2ξ′ +2(R, τn)R∗ +iξ,i2ξ2(R, τn)⟨ψ0| : ψf,ξ2,† +R,i2 ψf,ξ′ +2 +R,j2 : |ψ0⟩ += +� +i,j,i2,j2,ξ2,ξ′ +2 +T µν +ij Rjξ′,j2ξ′ +2(R, τ)R∗ +iξ,i2ξ2(R, τ)Λi2ξ2,j2ξ′ +2 += +� +ij +T µν +ij [R†(R, τ)ΛR(R, τ)]iξ,jξ′ = +� +ij +T µν +ij [eiu(R,τ)Λe−iu(R,τ)]iξ,jξ′ +(S352) +where we use R(R, τ) = exp[−iu(R, τ)] (Eq. S339). We next expand in powers of u, which gives +Σ(f,ξξ′) +µν +(R, τ) = +� +ij +T µν +ij [R†(R, τ)ΛR(R, τ)]iξ,jξ′ +≈ +� +ij +T µν +ij +� +Λ + iu(R, τ)Λ − iΛu(R, τ) − u(R, τ)u(R, τ)Λ + Λu(R, τ)u(R, τ) − 2u(R, τ)Λu(R, τ) +2 +� +iξ,jξ′ +(S353) +We can introduce the following two matrices +A(R, τ)iξ,jξ′ = [iu(R, τ)Λ − iΛu(R, τ)]iξ,jξ′ = i[u(R, τ), Λ]iξ,jξ′ +B(R, τ)iξ,jξ′ = i +2 +� +u(R, τ)A(R, τ) − A(R, τ)u(R, τ) +� +iξ,jξ′ = i +2[u(R, τ), A(R, τ)]iξ,jξ′ . +(S354) +We also let +[δΣ(R, τ)]iξ,jξ′ = A(R, τ)iξ,jξ′ + B(R, τ)iξ,jξ′ +(S355) +such that +Σ(f,ξξ′) +µν +(R, τ) ≈ +� +ij +T µν +ij +� +Λiξ,jξ′ + [δΣ(R, τ)]iξ,jξ′ +� +(S356) +Using Eq. S353, we can rewrite the interaction term (Eq. S342) as +ˆHint +≈ +� +R,ξ,ξ′,k,k′ +� +µν +γ2e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +[ +� +i,j +Λiξ,jξ′T µν +ij ] : ˆΣ(c′,ξ′ξ) +µν +(k, k′ − k) : +− J +� +R,µν,ξ,k,k′ +e−i(k′−k)·R +NM +[ +� +ij +Λiξ,jξT µν +ij ] : ˆΣ(c′′,ξξ) +µν +(k, k′ − k) : ++ +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·RF(|k|)F(|k′|) +NM +� +γv′ +⋆T µν +ij Λξi,−ξ′j(k′ +x − iξ′k′ +y) + γv′ +⋆T µν +ij Λ−ξi,ξ′j(kx + iξky) +� +: Σ(c′,ξ′ξ) +µν +(k, k′ − k) : ++ +� +R,k,k′,ξ,ξ′,µν,i,j +e−i(k′−k)·RF(|k|)F(|k′|)T µν +ij +NMDνc,νf +� +γ2[δΣ(R)]iξ,jξ′ + γv′ +⋆ : [δΣ(R)]iξ,j−ξ′ : (k′ +x − iξ′k′ +y) ++ γv′ +⋆[δΣ(R)]i−ξ,jξ′(kx + iξky) +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k) : −J +� +R,k,k′,µν,ξ,i,j +e−i(k′−k)·R +NM +T µν +ij [δΣ(R)]iξ,jξ : ˆΣ(c′′,ξξ) +µν +(k, k′ − k) : +where the first two lines describe the polarization of conduction electrons due to the ferromagnetic ordering of f-moments and +the last three lines describe the interactions between conduction electrons and the fluctuations of f-moments. F(|k|) (Eq. S189) +is the damping factor. + +80 +Now we are able to write down the full action. Combining Eq. S341, Eq. S346, Eq. S351 and Eq. S357, we have +S =Su + Sc,order + Sint +Su = − +� β +0 +� +R +� +2iTr[Λ∂τu(R, τ)] − Tr[[Λ, u(R, τ)]∂τu(R, τ)] +� +dτ +Sc,order = +� β +0 +� � +k,aηs +c† +k,aηs(τ)∂τck,aηs + Hc,order(τ) +� +dτ +Sint = +� β +0 +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +� +µν,i,j +T µν +ij +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′k′ +y) ++ γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) : +− J +� +R,k,k′,µν,ξ,i,j +e−i(k′−k)·R +NM +T µν +ij A(R, τ)iξ,jξ : ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : dτ +Sint,2 = +� β +0 +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +� +µν,i,j +T µν +ij +� +γ2B(R, τ)iξ,jξ′ + γv′ +⋆B(R, τ)iξ,j−ξ′(k′ +x − iξ′ky]) ++ γv′ +⋆B(R, τ)i−ξ,jξ′(kx + iξky) +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′, τ) : +− J +� +R,k,k′,µν,ξ,i,j +e−i(k′−k)·R +NM +T µν +ij B(R, τ)iξ,jξ : ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : dτ +(S357) +where Su describes the Berry phase of the f moments, Sc,order contains all the fermion bilinear and is characterized by the +Hamiltonian ˆHc,order. ˆHc,order takes the form of +ˆHc,order = +� +k,i +� +v⋆(kx + iξky)ψξ,c′,† +k,i +ψξ,c′′ +k,i + h.c. +� ++ (−µ + W + V0 +Ω0 +νc)νf +� +k,i +[ψξ,c′,† +k,i +ψξ,c′ +k,i + ψξ,c′′,† +k,i +ψξ,c′′ +k,i ] +− +� +k,ξ,i +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +|F(|k|)|2 +�γ2 +2 ψξ,c′,† +k,i +ψξ,c′ +k,i + v′ +⋆γ(kx − iξky)ψc′,ξ,† +k,i +ψc′,−ξ +k,i +� ++ +� +k,ξ,i +2γ2 +Dνc,νf +Λiξ,iξψc′,ξ,† +k,i +ψc′,ξ +k,i − +� +k,ξ,i +2JΛiξ,iξψc′′,ξ,† +k,i +ψc′′,ξ +k,i + +� +k,ξ,i +2γv′ +⋆ +Dνc,νf +(Λi−ξ,i−ξ ++ Λiξ,iξ)(kx − ikyξ)ψξ,c′,† +k,i +ψ−ξ,c′ +k,i +(S358) +where we rewrite the Hamiltonian with ψc′ +k,i, ψc′′ +k,i basis and drop the constant term. +In Eq. S358, The first line contains the contribution from ˆHc, ˆHW and ˆHV (mean-field level). The second line comes from +ˆHcc (Eq. S203). The third line describes the polarization of conduction electrons due to the long-range order of the f-moments. +In addition, we utilize the fact that Λiξ,jξ′ is a diagonal matrix. We also rewrite the ˆHc,order in a more compact form +ˆHc,order += +� +k,i +� +ψξ=+,c′,† +k,i +ψξ=+,c′′,† +k,i +ψξ=−,c′,† +k,i +ψξ=−,c′′,† +k,i +� +� +���� +E+,i +0,k + E+,i +3,k +v⋆(kx + iky) vi +k(kx − iky) +0 +v⋆(kx − iky) E+,i +0,k − E+,i +3,k +0 +0 +vi +k(kx + iky) +0 +E−,i +0,k + E−,i +3,k +v⋆(kx − iky) +0 +0 +v⋆(kx + iky) E−,i +0,k − E−,i +3,k +� +���� +� +����� +ψξ=+,c′ +k,i +ψξ=+,c′′ +k,i +ψξ=−,c′ +k,i +ψξ=−,c′′ +k,i +� +����� +(S359) + +81 +where +Eξ,i +0,k = −µ + W + V0 +Ω0 +νc − +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +�e−λ2|k|2γ2 +4 ++ γ2Λiξ,iξ +Dνc,νf +− JΛiξ,iξ +Eξ,i +3,k = − +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +�e−λ2|k|2γ2 +4 ++ γ2Λiξ,iξ +Dνc,νf ++ JΛiξ,iξ +vi +k = − +� +k,ξ,i +� +1 +D1,νc,νf ++ +1 +D2,νc,νf +� +e−λ2|k|2v′ +⋆ + 2γv′ +⋆ +Dνc,νf +(Λi−,i− + Λi+,i+) +(S360) +Sint and Sint,2 contain the interaction between u and c. We separate Sint into two parts S′ +K, S′ +J +Sint =S′ +K + S′ +J +S′ +K = +� β +0 +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +� +µν,i,j +T µν +ij +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′ky]) ++ γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) : +S′ +J = − J +� +R,k,k′,µν,ξ,i,j +e−i(k′−k)·R +NM +T µν +ij A(R, τ)iξ,jξ : ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) : dτ +(S361) +We now integrate out conduction electrons to derive the effective action of the f-moments described by u(R, τ). The partition +function can be written as +Z = +� +D[c† +k,aηs(τ), ck,aηs(τ), u(R, τ)] exp +� +− Su − Sc,order − Sint − Sint,2 +� +∝ +� +D[u(R, τ)]e−Su⟨e−Sint−Sint,2⟩0 +≈ +� +D[u(R, τ)]e−Su exp +� +− ⟨Sint⟩0 − ⟨Sint,2⟩0 + 1 +2[⟨S2 +int⟩0 − (⟨Sint⟩0)2] +� +≈ +� +D[u(R, τ)] exp +� +− +� +Su + ⟨Sint⟩0 + ⟨Sint,2⟩0 − 1 +2[⟨S2 +int⟩0 − (⟨Sint⟩0)2] +�� +(S362) +where we integrate out c-electrons in the second line and introduce +⟨O⟩0 = 1 +Z0 +� +D[c† +k,aηs(τ), ck,aηs(τ)]Oe−Sc,order +, +Z0 = +� +D[c† +k,aηs(τ), ck,aηs(τ)]e−Sc,order . +(S363) +Sint(Eq. S357) is linear in u(R, τ), so we keep its first-order and second-order contributions. Sint,2(Eq. S357) contains bilinear +term of u(R, τ), so we only keep its first-order contribution. Overall, we keep the zeroth order, first order and second-order +terms of u(R, τ) in the effective action. From Eq. S362. We then define the effective action of f-moments as +Seff =Su + ⟨Sint⟩0+⟨Sint,2⟩0 − 1 +2 +� +⟨SintSint⟩0,con +� +(S364) +where +−1 +2⟨TτSintSint⟩0,con = −1 +2⟨TτS′ +KS′ +K⟩0,con − 1 +2⟨TτS′ +JS′ +J⟩0,con − ⟨S′ +JSK⟩0,con +(S365) +A. +Single-particle Green’s function of conduction electrons +In order to find the effective theory of f-moments, it is useful to first consider the following single-particle Green’s function +of conduction electrons +gξξ′,i +c′c′ (k, τ) = −⟨Tτψc′,ξ +k,i (τ)ψc′,ξ′,† +k,i +(0)⟩0 +, +gξξ′,i +c′′c′′(R, τ) = −⟨Tτψc′′,ξ +k,i (τ)ψc′′,ξ′,† +k,i +(0)⟩0 +gξξ′,i +c′c′′ (k, τ) = −⟨Tτψc′,ξ +k,i (τ)ψc′′,ξ′,† +k,i +(0)⟩0 +, +gξξ′,i +c′′c′ (k, τ) = −⟨Tτψc′′,ξ +k,i (τ)ψc′,ξ′,† +k,i +(0)⟩0 +(S366) + +82 +where the expectation value is calculated with respect to the Hamiltonian ˆHc,order (Eq. S359). Using Wick’s theorem we have +⟨Tτ : Σ(c′,ξ′ξ) +µν +(k, k′ − k, τ) :: Σ(c′,ξ2ξ′ +2) +µ2ν2 +(k′ +2, k2 − k′ +2, 0) :⟩ = − +� +ij +δk,k2δk,k′ +2T µν +ij T µ2ν2 +ji +gξ′ +2ξ′,i +c′c′ +(k′, −τ)gξξ2,j +c′c′ (k, τ)/4 +⟨Tτ : Σ(c′′,ξ′ξ) +µν +(k, k′ − k, τ) :: Σ(c′′,ξ2ξ′ +2) +µ2ν2 +(k′ +2, k2 − k′ +2, 0) :⟩ = − +� +ij +δk,k2δk,k′ +2T µν +ij T µ2ν2 +ji +gξ′ +2ξ′,i +c′′c′′ (k′, −τ)gξξ2,j +c′′c′′ (k, τ)/4 +⟨Tτ : Σ(c′,ξ′ξ) +µν +(k, k′ − k, τ) :: Σ(c′′,ξ2ξ′ +2) +µ2ν2 +(k′ +2, k2 − k′ +2, 0) :⟩ = − +� +ij +δk,k2δk,k′ +2T µν +ij T µ2ν2 +ji +gξ′ +2ξ′,i +c′′c′ (k′, −τ)gξξ2,j +c′c′′ (k, τ)/4 +(S367) +Now we are in the position to calculate each term in the effective action. +B. +⟨Sint⟩0 +We have +⟨TτSint⟩ = +� β +0 +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +� +µν,i,j +T µν +ij +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′k′ +y) ++ γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +⟨Tτ : ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :⟩0 +− J +� +R,k,k′,µν,ξ,i,j +e−i(k′−k)·R +NM +T µν +ij A(R, τ)iξ,jξ⟨Tτ : ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) :⟩0dτ +(S368) +where F(|k|) (Eq. S189) is the damping factor. For the given Hamiltonian of conduction electrons (Eq. S359), we have +⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :⟩ = 1 +2 +� +s,t +T µν +st ⟨: ψc′,ξ′,† +k,s +(τ)ψc′,ξ +k′,t(τ) :⟩0 = δk,k′ +2 +� +s +T µν +ss ⟨: ψc′,ξ′,† +k,s +(τ)ψc′,ξ +k,s (τ) :⟩0 +⟨: ˆΣ(c′′,ξ′,ξ) +µν +(k, k′, τ) :⟩ = 1 +2 +� +s,t +T µν +st ⟨: ψc′′,ξ′,† +k,s +(τ)ψc′′,ξ +k′,s (τ) :⟩0 = δk,k′ +2 +� +s +T µν +ss ⟨: ψc′′,ξ′,† +k,s +(τ)ψc′′,ξ +k,s (τ) :⟩0 +(S369) +Combining above equation with Eq. S368, we find +⟨TτSint⟩0 += +� β +0 +� +R,k,ξ,ξ′ +|F(|k|)|2 +Dνc,νf NM +� +i +2 +� +γ2A(R, τ)iξ,iξ′ + γv′ +⋆A(R, τ)iξ,i−ξ′(kx − iξ′ky) + γv′ +⋆A(R, τ)i−ξ,iξ′(kx + iξky) +� +⟨: ψc′,ξ′,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − J +� +R,k,ξ,i +2 +NM +A(R, τ)iξ,iξ⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0dτ +(S370) +According to the definition of A(R, τ)iξ,jξ′ (Eq. S354), we have +A(R, τ)iξ,iξ = iu(R, τ)(Λiξ,iξ − Λiξ,iξ) = 0 +(S371) +Then only A(R, τ)iξ,i−ξ(R, τ) can give non-zero contribution +⟨TτSint⟩0 += +� β +0 +� +R,k,ξ +|F(|k|)|2 +Dνc,νf NM +� +i +2 +� +γ2A(R, τ)iξ,i−ξ⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 ++ +� +γv′ +⋆A(R, τ)iξ,i−ξ(kx − iξky) + γv′ +⋆A(R, τ)i−ξ,iξ(kx + iξky) +� +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 +� +(S372) + +83 +In terms of Green’s function +⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 = −gξ,−ξ +c′c′,i(k, −0+) +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 = −gξ,ξ +c′c′,i(k, −0+) − 1/2 +(S373) +where the −1/2 comes from the normal ordering and minus sign comes from the fermion anti-commutation relation. By direct +solving the ˆHc,order, we find gξ,−ξ,i +c′c′ +(k, τ) = (kx − iξky)f4(|k|, τ, i) (see Sec. S13, Eq. S584) where f4(|k|, τ, i) is a function +that only depends on |k|. Then the following k summation vanishes due to the (kx ± iξky) contribution +� +k +⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 = +� +k +−(kx − iξky)f4(|k|, 0+, i) = 0 +(S374) +As for gξ,ξ,i +c′c′ (k, τ), by direct solving ˆHc,order, we find it only depends on |k| (see Sec. S13, Eq. S584). Then the following k +summation vanishes due to the (kx ± iξky) term +� +k +(kx ± iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 = +� +k +(kx ± iξky)[−gξ,ξ +c′c′(k, −0+) − 1/2] = 0 +(S375) +From Eq. S374 and Eq. S375, all k summation in Eq. S372 goes to zero and then +⟨TτSint⟩0 = 0 +(S376) +C. +⟨Sint,2⟩0 +Expanding Σc′/c′′,ξ,ξ′) +µν +(k, k′, τ) in Sint,2, we have +⟨TτSint,2⟩0 += +� +� +R,k,k′ξ,ξ′ +� +µν,i,j +e−i(k−k′)·RF(|k|)F(|k′|) +NM +� +lm +T µν +lm +T µν +ij +2 +� +γ2 +Dνc,νf +B(R, τ)lξ,mξ′⟨: ψc′,ξ′,† +k,i +(τ)ψc′,ξ +k′,j(τ) :⟩0 +− Jδξ,ξ′B(R, τ)iξ,jξ⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k′,j (τ)⟩0 + γv′ +⋆B(R, τ)lξ,mξ′(k′ +x + iξ′ky)⟨: ψc′,−ξ′,† +k,i +(τ)ψc′,ξ +k′,j(τ) :⟩0 ++ γv′ +⋆B(R, τ)lξ,mξ′(kx − iξky)⟨: ψc′,ξ′,† +k,i +(τ)ψc′′,−ξ +k′,j +(τ) :⟩0 +� +dτ += +� +� +R,k,ξ,ξ′,i +B(R, τ)iξ,iξ′ |F(|k|)|2 +NM +� +k +� 2γ2 +Dνc,νf +⟨: ψc′,ξ′,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − 2J2δξ,ξ′⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0 ++ 2γv′ +⋆ +Dνc,νf +(kx + iξ′ky)⟨: ψc′,−ξ′,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 + 2γv′ +⋆ +Dνc,νf +(kx − iξky)⟨: ψc′,ξ′,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩0 +� +dτ +(S377) +where F(|k|) (Eq. S189) is the damping factor. Using Eq. S374 and Eq. S375, we find +⟨TτSint,2⟩0 = +� +� +R,k,ξ,i +B(R, τ)iξ,iξ +|F(|k|)|2 +NM +� +k +� 2γ2 +Dνc,νf +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − 2J2⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0 ++ 2γv′ +⋆ +Dνc,νf +(kx + iξky)⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 + 2γv′ +⋆ +Dνc,νf +(kx − iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩0 +� +dτ +(S378) +We then have +⟨TτSint,2⟩0 = +� � +R,ξ,i +B(R, τ)iξ,iξNiξdτ +(S379) +where we have defined Niξ as +Niξ = 1 +NM +� +k +�2|F(|k|)|2γ2 +Dνc,νf +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − 2J⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0 ++ 2γv′ +⋆|F(|k|)|2 +Dνc,νf +(kx + iξky)⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 + 2γv′ +⋆|F(|k|)|2 +Dνc,νf +(kx − iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩0 +� +(S380) + +84 +D. +− 1 +2⟨S′ +KS′ +K⟩0,con +From Eq. S362, we obtain +− 1 +2⟨S′ +KS′ +K⟩0 += − 1 +2 +� β +0 +� β +0 +� +R,k,k′,ξ,ξ′ +� +R2,k2,k′ +2,ξ2,ξ′ +2 +e−i(k′−k)·RF(|k|)F(|k′|) +NMDνc,νf +e−i(k2−k′ +2)·R2F(|k|)F(|k′|) +NMDνc,νf +� +µν,i,j +T µν +ij +� +µ2ν2,i2,j2 +T µ2ν2 +i2j2 +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′k′ +y) + γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +� +γ2A(R2, τ2)i2ξ2,j2ξ′ +2 + γv′ +⋆A(R2, τ2)i2ξ2,j2−ξ′ +2(k2,x − iξ′ +2k2,y) + γv′ +⋆A(R2, τ2)i2−ξ2,j2ξ′ +2(k′ +2,x + iξ2k′ +2,y) +� +⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :: ˆΣ(c′,ξ′ +2,ξ2) +µ2ν2 +(k′ +2, k′ +2 − k2, τ2) :⟩0,condτdτ2 += − 1 +2 +� β +0 +� β +0 +� +R,k,k′,ξ,ξ′ +� +R2,ξ2,ξ′ +2 +e−i(k′−k)·(R−R2)|F(|k|)F(|k′|)|2 +NMDνc,νf +1 +NMDνc,νf +� +µν,i,j +T µν +ij +� +µ2ν2,i2,j2 +T µ2ν2 +i2j2 +� +s,t +T µν +st T µ2ν2 +ts +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′k′ +y) + γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +� +γ2A(R2, τ2)i2ξ2,j2ξ′ +2 + γv′ +⋆A(R2, τ2)i2ξ2,j2−ξ′ +2(kx − iξ′ +2ky) + γv′ +⋆A(R2, τ2)i2−ξ2,j2ξ′ +2(k′ +x + iξ2k′ +y) +� +(−1)gξ2ξ′,s +c′c′ +(k′, τ2 − τ)gξξ′ +2,t +c′c′ (k, τ − τ2)/4dτdτ2 +(S381) +where we use Eq. S367 in the final step and the factor 1/4 comes from the two 1/2 factors in the definition of Σ(c′,ξ′ξ) +µν +and +Σ(c′,ξ′ +2,ξ2) +µ2ν2 +(Eq. S20). Using � +µν T µν +ij T µν +st = 4δi,tδi,s, we find +− 1 +2⟨S′ +KS′ +K⟩0 +=2 +� β +0 +� β +0 +� +R,R2,k,k′,ξ,ξ′,ξ2,ξ′ +2,i,j +e−i(k′−k)·(R−R2)|F(|k|)F(|k|)|2 +(NMDνc,νf )2 +A(R, τ)iξ,jξ′Ajξ2,iξ′ +2(R2, τ2) +� +γ4gξ2ξ′,j +c′c′ +(k′, τ2 − τ)gξξ′ +2,i +c′c′ (k, τ − τ2) ++ γ3v′ +⋆ +� +(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ2 − τ)gξ−ξ′ +2,i +c′c′ +(k, τ − τ2) + (k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ2 − τ)gξξ′ +2,i +c′c′ (k, τ − τ2) ++ (k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′c′ +(k′, τ2 − τ)gξξ′ +2,i +c′c′ (k, τ − τ2) + (kx − iξky)gξ2ξ′,j +c′c′ +(k′, τ2 − τ)g−ξξ′ +2,i +c′c′ +(k, τ − τ2) +� ++ γ2(v′ +⋆)2 +� +(k′ +x + iξ′k′ +y)(kx + iξ′ +2ky)gξ2−ξ′,j +c′c′ +(k′, τ2 − τ)gξ−ξ′ +2,i +c′c′ +(k, τ − τ2) ++ (kx − iξky)(k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ2 − τ)g−ξξ′ +2,i +c′c′ +(k, τ − τ2) ++ (k′ +x + iξ′ky)(k′ +x − iξ2k′ +y)g−ξ2−ξ′,j +c′c′ +(k′, τ2 − τ)gξξ′ +2,i +c′c′ (k, τ − τ2) ++ (kx − iξky)(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ2 − τ)g−ξ−ξ′ +2,i +c′c′ +(k, τ − τ2) +�� +dτdτ2 +(S382) +Here, we comment that, all terms that are at the second order of S′ +K (γ4, γ2(v′ +⋆)2, γ3v′ +⋆ terms) need to be kept in order to recover +the Goldstone mode in the ordered ground state. This is because all terms we kept in the ˆHK, (γ2, γv′ +⋆ terms) introduce a +polarization effect to the conduction electrons. This can be observed from Eq. S359 and Eq. S360 where both γ2 and γv′ +⋆ terms +affect the single-particle Hamiltonian of conduction electrons. Therefore, when we derive the effective action, all the second- +order terms induced by γ2, γv′ +⋆ terms in ˆHK, that are γ4, γ2(v′ +⋆)2, γ3v′ +⋆ terms, need to be kept, in order to be consistent with the +single-particle Hamiltonian of conduction electrons. + +85 +E. +− 1 +2⟨S′ +JS′ +J⟩0,con +Using Eq. S367, we find +− 1 +2⟨S′ +JS′ +J⟩0,con += − +� β +0 +� β +0 +J2 +2N 2 +M +� +R,R2,k,k′,k2,k′ +2 +� +µν,µ2ν2 +� +ξ,ξ2 +� +ij,i2j2 +e−i(k′−k)·R−i(k2−k′ +2)·R2 +N 2 +M +T µν +ij T µ2ν2 +i2j2 A(R, τ)iξ,jξA(R2, τ2)i2ξ2,j2ξ2 +⟨: ˆΣ(c′′,ξξ) +µν +(k, k′ − k, τ) :: ˆΣ(c′′,ξ2ξ2) +µ2ν2 +(k′ +2, k2 − k′ +2, τ2) :⟩0,condτdτ2 += − +� β +0 +� β +0 +J2 +2 +� +R,R2,k,k′ +� +µν,µ2ν2 +� +ξ,ξ2 +� +ij,i2j2 +e−i(k′−k)·(R−R2)T µν +ij T µ2ν2 +i2j2 A(R, τ)iξ,jξA(R2, τ2)i2ξ2,j2ξ2 +(−1) +� +i′j′ +T µν +i′j′T µ2ν2 +j′i′ gξ2ξ,i′ +c′′c′′ (k′, −τ + τ2)gξξ2,j′ +c′′c′′ (k, τ − τ2)/4dτdτ2 +(S383) +Using � +µν T µν +ij T µν +st = 4δi,tδi,s, we find +− 1 +2⟨S′ +JS′ +J⟩0,con = +� β +0 +� β +0 +2J2e−i(k′−k)·(R−R2) +N 2 +M +� +R,R2 +� +ξ,ξ2,i,j +A(R, τ)iξ,jξA(R2, τ2)jξ2,iξ2gξ2ξ,j +c′′c′′ (k′, −τ + τ2)gξξ2,i +c′′c′′(k, τ − τ2)dτdτ2 +(S384) +F. +−⟨S′ +JS′ +K⟩0,con +Using Eq. S367, we find +− ⟨S′ +JS′ +K⟩0,con += +� β +0 +� β +0 +J +� +R,k,k′,k2,k′ +2,ξ,ξ′ +e−i(k′−k)·R−i(k2−k′ +2)·R2F(|k|)F(|k′|) +N 2 +MDνc,νf +� +µν,i,j +T µν +ij +� +R2,µ2ν2,i,j,ξ2 +T µ2ν2 +i2j2 +� +i′j′ +T µν +i′j′T µ2ν2 +j′i′ +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξk′ +y) + γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +A(R2, τ2)i2ξ2,j2ξ2 +⟨: ˆΣ(c′,ξ′,ξ) +µν +(k, k′ − k, τ) :: ˆΣ(c′′,ξ2,ξ2) +µ2ν2 +(k′ +2, k2 − k′ +2, τ2) :⟩0,condτdτ2 += +� β +0 +� β +0 +J +� +R,k,k′,ξ,ξ′ +e−i(k′−k)·(R−R2)F(|k|)F(|k′|) +N 2 +MDνc,νf +� +µν,i,j +T µν +ij +� +R2,µ2ν2,i,j,ξ2 +T µ2ν2 +i2j2 +� +i′j′ +T µν +i′j′T µ2ν2 +j′i′ +� +γ2A(R, τ)iξ,jξ′ + γv′ +⋆A(R, τ)iξ,j−ξ′(k′ +x − iξ′k′ +y) + γv′ +⋆A(R, τ)i−ξ,jξ′(kx + iξky) +� +A(R2, τ2)i2ξ2,j2ξ2(−1)gξ2ξ′,i′ +c′′c′ +(k′, −τ + τ2)gξξ2,j′ +c′c′′ (k, τ − τ2)/4dτdτ2 +(S385) +Using � +µν T µν +ij T µν +st = 4δi,tδi,s, we find +− ⟨S′ +JS′ +K⟩0,con += − 4 +� β +0 +� β +0 +J +� +R,R2,k,k′,ξ,ξ′,ξ2 +e−i(k′−k)·(R−R2)F(|k|)F(|k|) +N 2 +MDνc,νf +A(R, τ)iξ,jξ′A(R2, τ2)jξ2,iξ2 +� +γ2gξ2ξ′,j +c′′c′ (k′, −τ + τ2)gξξ2,i +c′c′′ (k, τ − τ2) + γv′ +⋆(k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′′c′ +(k′, −τ + τ2)gξξ2,i +c′c′′ (k, τ − τ2) ++ γv′ +⋆(kx − iξky)gξ2ξ′,j +c′′c′ (k′, −τ + τ2)g−ξξ2,i +c′c′′ +(k, τ − τ2) +� +dτdτ2 +(S386) + +86 +G. +Effective action +Combining Eq. S364, Eq. S379, Eq. S382, Eq. S384 and Eq. S386, we find the following effective action of the f-moments. +Seff =Su + +� � +ξ,i,R +B(R, τ)iξ,iξNiξdτ ++ 1 +2π +� � +� +R,R2,q,ξ,ξ′ +A(R, τ)iξ,jξ′A(R2, τ2)jξ2,iξ′ +2 +eiq·(R2−R)−iω(τ2−τ) +NM +χA +ξξ′,ξ′ +2ξ2(q, iω, i, j)dτdτ2 +(S387) +where we consider zero temperature and define +χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) += +� � +k,k′ +δk′−k,qeiωτ +NM +�2γ4[F(|k|)F(|k′|)]2 +D2νc,νf +gξ2ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ 2γ3v′ +⋆[F(|k|)F(|k′|)]2 +D2νc,νf +� +(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ)gξ−ξ′ +2,i +c′c′ +(k, −τ) + (k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ (k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) + (kx − iξky)gξ2ξ′,j +c′c′ +(k′, τ)g−ξξ′ +2,i +c′c′ +(k, −τ) +� ++ 2γ2(v′ +⋆)2[F(|k|)F(|k′|)]2 +D2νc,νf +� +(k′ +x + iξ′k′ +y)(kx + iξ′ +2ky)gξ2−ξ′,j +c′c′ +(k′, τ)gξ−ξ′ +2,i +c′c′ +(k, −τ) ++ (kx − iξky)(k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ)g−ξξ′ +2,i +c′c′ +(k, −τ) + (k′ +x + iξ′ky)(k′ +x − iξ2k′ +y)g−ξ2−ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ (kx − iξky)(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ)g−ξ−ξ′ +2,i +c′c′ +(k, −τ) +� ++ 2J2δξ,ξ′δξ′ +2,ξ2gξ2ξ,j +c′′c′′ (k′, τ)gξξ2,i +c′′c′′(k, −τ) +− 4δξ2,ξ′ +2JF(|k|)F(|k′|) +Dνc,νf +� +γ2gξ2ξ′,j +c′′c′ (k′, τ)gξξ′ +2,i +c′c′′ (k, −τ) ++ γv′ +⋆(k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′′c′ +(k′, τ)gξξ2,i +c′c′′ (k, −τ) + γv′ +⋆(kx − iξky)gξ2ξ′,j +c′′c′ (k′, τ)g−ξξ2,i +c′c′′ +(k, −τ) +�� +dτ +(S388) +where ω denotes the Matsubara frequency (w ∈ 2π/βZ at finite temperature and can be treated as a continuous variable at zero +temperature). +We next rewrite A(R, τ)iξ,jξ′, B(R, τ)iξ,iξ with u(R, τ)iξ,jξ′ +A(R, τ)iξ,jξ′ = i(Λjξ′,jξ′ − Λiξ,iξ)u(R, τ)iξ,jξ′ +, +B(R, τ)iξ,iξ = u(R, τ)iξ,jξ′u(R, τ)jξ′,iξ +� +Λjξ′,jξ′ − Λiξ,iξ +� +(S389) +and and introduce the Fourier transformation of u(R, τ)iξ,jξ′ +u(q, iω)iξ,jξ′ = +� � +R +e−iq·R+iωτu(R, τ)iξ,jξ′dτ +Then +Seff = − +� � +R +2i +� +iξ +Λiξ,iξ∂τu(R, τ)iξ,iξ + +1 +2πNM +� � +q +� +iξ,jξ′ +iω(Λiξ,iξ − Λjξ′,jξ′)u(q, iω)iξ,jξ′u(−q, −iω)jξ′,iξdω ++ +1 +2πNM +� � +q +� +iξ,jξ′ +Niξ +� +Λjξ′,jξ′ − Λiξ,iξ +� +u(q, iω)iξ,jξ′u(−q, −iω)jξ′,iξdω ++ +1 +2πNM +� � +q +� +i,j,ξ,ξ′,ξ2,ξ′ +2 +� +Λjξ′,jξ′ − Λiξ,iξ +�� +Λjξ2,jξ2 − Λiξ′ +2,iξ′ +2 +� +χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2dω +(S390) +We next to separate u(R, τ) (or u(q, iω)) into two-parts, the diagonal components u(R, τ)iξ,iξ and the off-diagonal components +u(R, τ)iξ,jξ′ with iξ ̸= jξ′. Correspondingly, we can separate the effective action into two parts Seff = Seff,diag + Seff,off, + +87 +where Seff,diag(Seff,off) describes the behaviors of diagonal (off-diagonal) components of u(R, τ). Seff,diag can be written +as +Seff,diag = − +� � +R +2i +� +iξ +Λiξ,iξ∂τu(R, τ)iξ,iξdτ +(S391) +where we write the action in the position and imaginary-time spaces to illustrate its behaviors. Clearly, Seff,diag is a total +derivative and only matters when u(R, τ)iξ,iξ develops topologically non-trivial configurations which give a non-zero winding +number Qiξ(R) ̸= 0 where Qiξ(R) = +� +∂τu(R, τ)iξ,iξdτ. Then, Seff,diag = −2i � +R,iξ Λiξ,iξQiξ(R) is just a phase factor. +Therefore, we will focus on the off-diagonal components of u. +The off-diagonal components give +Seff,off = +1 +2πNM +� � +q +� +i,j,ξ,ξ′,ξ2,ξ′ +2 +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +− iω(Λiξ,iξ − Λjξ′,jξ′)δξ2,ξ′δξ′ +2,ξ + Niξ(Λjξ′,jξ′ − Λiξ,iξ)δξ2,ξ′δξ′ +2,ξ ++ +� +Λjξ′,jξ′ − Λiξ,iξ +�� +Λjξ2,jξ2 − Λiξ′ +2,iξ′ +2 +� +χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) +� +dω +(S392) +It is worth mentioning that not all the off-diagonal terms appear in the effective action Seff,off. To observe that, we first +introduce the following two sets of indices: +Sfill = {iξ|Λiξ,iξ = 1/4} +, +Semp = {iξ|Λiξ,iξ = −1/4} +(S393) +where Sfill (Semp) denotes the set of flavors, that are filled with one (zero) f electron. For the ground states in Eq. S335, we +have +νf = 0 : +Sfill = {1+, 1−, 2+, 2−}, +Semp = {3+, 3−, 4+, 4−} +νf = −1 : +Sfill = {1+, 1−, 2+}, +Semp = {2−, 3+, 3−, 4+, 4−} +νf = −2 : +Sfill = {1+, 1−}, +Semp = {2+, 2−, 3+, 3−, 4+, 4−} +(S394) +and +Λiξ,iξ − Λjξ′,jξ′ = 0 +, +iξ, jξ′ ∈ Sfill +Λiξ,iξ − Λjξ′,jξ′ = 0 +, +iξ, jξ′ ∈ Semp +Λiξ,iξ − Λjξ′,jξ′ = 1/2 +, +iξ ∈ Sfill +, +jξ′ ∈ Semp +Λiξ,iξ − Λjξ′,jξ′ = −1/2 +, +iξ ∈ Semp +, +jξ′ ∈ Sfill . +(S395) +All terms in Eq. S392 that contain u(q, iω)iξ,jξ′ will have a prefactor Λiξ,iξ − Λjξ′,jξ′. Then, only u(q, iω)iξ,jξ′ with iξ ∈ +Sfill, jξ′ ∈ Semp or iξ ∈ Semp, jξ′ ∈ Sfill produces non-zero contribution. This allows us to write the effective action as +Seff,off = +1 +2πNM +� � +q +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Niξ − Njξ′ +2 +δξ2,ξ′δξ′ +2,ξ + 1 +4 +� +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j) + χA +ξ2ξ′ +2,ξ′ξ(−q, iω = 0, j, i) +�� +dω +(S396) +where we also take the low-frequency approximation of the χA by letting χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) ≈ χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j). For +what follows, we will focus on the Seff,off which describes the spin fluctuations of the system. Here, we also provide the + +88 +formula of Niξ, χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) +Niξ += 1 +NM +� +k +�2γ2[F(|k|)]2 +Dνc,νf +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − 2J⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0 ++ 2γv′ +⋆[F(|k|)]2 +Dνc,νf +(kx + iξky)⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 + 2γv′ +⋆[F(|k|)]2 +Dνc,νf +(kx − iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩0 +� +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j) += +� � +k,k′ +δk′−k,q +NM +�2γ4[F(|k|)F(|k′|)]2 +D2νc,νf +gξ2ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ 2γ3v′ +⋆[F(|k|)F(|k′|)]2 +D2νc,νf +� +(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ)gξ−ξ′ +2,i +c′c′ +(k, −τ) + (k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ (k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) + (kx − iξky)gξ2ξ′,j +c′c′ +(k′, τ)g−ξξ′ +2,i +c′c′ +(k, −τ) +� ++ 2γ2(v′ +⋆)2[F(|k|)F(|k′|)]2 +D2νc,νf +� +(k′ +x + iξ′k′ +y)(kx + iξ′ +2ky)gξ2−ξ′,j +c′c′ +(k′, τ)gξ−ξ′ +2,i +c′c′ +(k, −τ) ++ (kx − iξky)(k′ +x − iξ2k′ +y)g−ξ2ξ′,j +c′c′ +(k′, τ)g−ξξ′ +2,i +c′c′ +(k, −τ) + (k′ +x + iξ′ky)(k′ +x − iξ2k′ +y)g−ξ2−ξ′,j +c′c′ +(k′, τ)gξξ′ +2,i +c′c′ (k, −τ) ++ (kx − iξky)(kx + iξ′ +2ky)gξ2ξ′,j +c′c′ +(k′, τ)g−ξ−ξ′ +2,i +c′c′ +(k, −τ) +� ++ 2J2δξ,ξ′δξ′ +2,ξ2gξ2ξ,j +c′′c′′ (k′, τ)gξξ2,i +c′′c′′(k, −τ) +− 4δξ2,ξ′ +2JF(|k|)F(|k′|) +Dνc,νf +� +γ2gξ2ξ′,j +c′′c′ (k′, τ)gξξ′ +2,i +c′c′′ (k, −τ) ++ γv′ +⋆(k′ +x + iξ′k′ +y)gξ2−ξ′,j +c′′c′ +(k′, τ)gξξ2,i +c′c′′ (k, −τ) + γv′ +⋆(kx − iξky)gξ2ξ′,j +c′′c′ (k′, τ)g−ξξ2,i +c′c′′ +(k, −τ) +�� +dτ +(S397) +We next introduce +F ij +ξξ′,ξ′ +2ξ2(q) = Niξ − Njξ′ +2 +δξ2,ξ′δξ′ +2,ξ − 1 +4 +� +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j) + χA +ξ2ξ′ +2,ξ′ξ(−q, iω = 0, j, i) +� +(S398) +Then Eq. S396 can be written as +Seff,off = +1 +2πNM +� � +q +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − F ij +ξξ′,ξ′ +2ξ2(q) +�� +dω (S399) +We mention that the excitation spectrum is obtained by numerically evaluating Eq. S397 and Eq. S398, and then fining the +eigenvalues of F ij +ξξ′,ξ′ +2ξ2(q). We now prove F ij +ξξ′,ξ′ +2ξ2(q), as a matrix with row and column indices ξξ′, ξ′ +2ξ2, is a Hermitian +matrix, in other words +F ij +ξξ′,ξ′ +2ξ2(q) = [Fξ′ +2ξ2,ξξ′(q)]∗ +(S400) +To prove Eq. S400, we first consider Niξ (Eq. S397) +N ∗ +iξ = 1 +NM +� +k +�2γ2[F(|k|)]2 +Dνc,νf +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩∗ +0 − 2J⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩∗ +0 ++ 2γv′ +⋆[F(|k|)]2 +Dνc,νf +(kx − iξky)⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩∗ +0 + 2γv′ +⋆[F(|k|)|2 +Dνc,νf +(kx + iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩∗ +0 +� += 1 +NM +� +k +�2γ2[F(|k|)]2 +Dνc,νf +⟨: ψc′,ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 − 2J⟨: ψc′′,ξ,† +k,i +(τ)ψc′′,ξ +k,i (τ) :⟩0 ++ 2γv′ +⋆[F(|k|)]2 +Dνc,νf +(kx − iξky)⟨: ψc′,ξ,† +k,i +(τ)ψc′,−ξ +k,i +(τ) :⟩0 + 2γv′ +⋆[F(|k|)|2 +Dνc,νf +(kx + iξky)⟨: ψc′,−ξ,† +k,i +(τ)ψc′,ξ +k,i (τ) :⟩0 +� +=Niξ +(S401) + +89 +As for χA in Eq. S397, we find +[χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j)]∗ += +� � +k,k′ +δk′−k,q +NM +�2γ4[F(|k|)F(|k′|)]2 +D2νc,νf +gξ′ξ2,j +c′c′ +(k′, τ)gξ′ +2ξ,i +c′c′ (k, −τ) ++ 2γ3v′ +⋆[F(|k|)F(|k′|)]2 +D2νc,νf +� +(kx − iξ′ +2ky)gξ′ξ2,j +c′c′ +(k′, τ)g−ξ′ +2ξ,i +c′c′ +(k, −τ) + (k′ +x + iξ2k′ +y)gξ′−ξ2,j +c′c′ +(k′, τ)gξ′ +2ξ,i +c′c′ (k, −τ) ++ (k′ +x − iξ′k′ +y)g−ξ′ξ2,j +c′c′ +(k′, τ)gξ′ +2ξ,i +c′c′ (k, −τ) + (kx + iξky)gξ′ξ2,j +c′c′ +(k′, τ)gξ′ +2−ξ,i +c′c′ +(k, −τ) +� ++ 2γ2(v′ +⋆)2[F(|k|)F(|k′|)]2 +D2νc,νf +� +(k′ +x − iξ′k′ +y)(kx − iξ′ +2ky)g−ξ′ξ2,j +c′c′ +(k′, τ)g−ξ′ +2ξ,i +c′c′ +(k, −τ) ++ (kx + iξky)(k′ +x + iξ2k′ +y)gξ′−ξ2,j +c′c′ +(k′, τ)g−ξξ′ +2−ξ,i +c′c′ +(k, −τ) + (k′ +x − iξ′ky)(k′ +x + iξ2k′ +y)g−ξ′−ξ2,j +c′c′ +(k′, τ)gξ′ +2ξ,i +c′c′ (k, −τ) ++ (kx + iξky)(kx − iξ′ +2ky)gξ′ξ2,j +c′c′ +(k′, τ)g−ξ′ +2−ξ,i +c′c′ +(k, −τ) +� ++ 2J2δξ,ξ′δξ′ +2,ξ2gξξ2,j +c′′c′′ (k′, τ)gξ2ξ,i +c′′c′′(k, −τ) +− 4δξ2,ξ′ +2JF(|k|)F(|k′|) +Dνc,νf +� +γ2gξ′ξ2,j +c′c′′ (k′, τ)gξ′ +2ξ,i +c′′c′ (k, −τ) ++ γv′ +⋆(k′ +x − iξ′k′ +y)g−ξ′ξ2,j +c′c′′ +(k′, τ)gξ2ξ,i +c′′c′ (k, −τ) + γv′ +⋆(kx + iξky)gξ′ξ2,j +c′c′′ (k′, τ)gξ2−ξ,i +c′′c′ +(k, −τ) +�� +dτ +=χA +ξ′ +2ξ2,ξξ′(−q, iω = 0, j, i) +(S402) +Combining Eq. S398, Eq. S401 and Eq. S402, we have +[F ij +ξ′ +2ξ2,ξξ′(q)]∗ = (Niξ − Njξ′)∗ +2 +δξ2,ξ′δξ′ +2,ξ − 1 +4 +� +[χA +ξ′ +2ξ2,ξξ′(q, iω = 0, i, j)]∗ + [χA +ξ′ξ,ξ2ξ′ +2(−q, iω = 0, j, i)]∗ +� +=Niξ − Njξ′ +2 +δξ2,ξ′δξ′ +2,ξ − 1 +4 +� +[χA +ξξ′,ξ′ +2ξ2(−q, iω = 0, j, i)] + [χA +ξ2ξ′ +2,ξ′ξ(q, iω = 0, i, j)] +� += F ij +ξ′ +2ξ2,ξξ′(q) +(S403) +Thus we proved Eq. S400. +H. +Flat U(4) symmetry and Noether’s theorem +We consider the model at M = 0, which has a flat U(4) symmetry. We note that U(4) = SU(4) × U(1)c with U(1)c a U(1) +charge symmetry. We focus on the SU(4) part, which will produce the Goldstone modes (note that the ground states (Eq. S335) +preserve U(1)c symmetry). And we will call it flat SU(4) symmetry. +We first introduce flat SU(4) transformation. We consider a flat SU(4) transformation characterize by real number vµν with +µ, ν ∈ {0, x, y, z} and µν ̸= 00. The SU(4) transformation on the f electron is then defined as +g = e−i � +R,ξ,i,j φijψf,ξ,† +R,i ψf,ξ +R,j +(S404) +where φ = � +µν̸=00 vµνT µν is a 4 × 4 matrix and T µν are the generators of flat U(4) group (Eq. S20). In addition, φ is a +4 × 4 traceless Hermitian matrix. We next discuss how uiξ,jξ′(R) transform under flat SU(4) transformation. We act SU(4) +transformation on the |u⟩ state (using Eq. S333,Eq. S334 and Eq. S337) +g|u⟩ = e−i � +R,ξ,i,j φijψf,ξ,† +R,i ψf,ξ +R,j|u⟩ = +� +R +gR ˆR[u(R)]|ψ0⟩ +(S405) +where we define +gR = e−i � +ξ,i,j φijψf,ξ,† +R,i ψf,ξ +R,j . +(S406) +such that g = � +R gR (note that φij is site R-independent). Since gR generate a SU(4) transformation, ˆR[u(R)] (Eq. S337) +generate a SU(8) transformation (for each site). Then gR ˆR[u(R)] also generate a SU(8) transformation. Thus there exists +˜u(R), such that +gR ˆR[u(R)] = ˆR[˜u(R)] +(S407) + +90 +To find ˜u, we can act the transformation on the ψf fermion. From Eq. S407 +� +gR ˆRR[u(R)] +� +ψf,ξ +R,i +� +gR ˆRR[u(R)] +�−1 += +� +ˆR[˜u(R)] +� +ψf,ξ +R,i +� +ˆR[˜u(R)] +�−1 +(S408) +The left-hand-side (LHS) of Eq. S408 gives +LHS = +� +j,ξ′ +[eiΦeiu(R)]iξ,jξ′ψf,ξ′ +R,j +(S409) +where Φ is a 8 × 8 matrix with column and row indices iξ, jξ′ and [Φ]iξ,jξ′ = φijδξ,ξ′. The right-hand-side (LHS) of Eq. S408 +gives +RHS = +� +j,ξ′ +[ei˜u(R)]iξ,jξ′ψf,ξ′ +R,j +(S410) +Then +LHS = RHS ⇒ +� +j,ξ′ +[eiΦeiu(R)]iξ,jξ′ψf,ξ′ +R,j = +� +j,ξ′ +[ei˜u(R)]iξ,jξ′ψf,ξ′ +R,j ⇒ +� +jξ′ +[e−i˜u(R)eiΦeiu(R)]iξ,jξ′ψf,ξ′ +R,j = ψf,ξ +R,i +⇒e−i˜u(R)eiΦeiu(R) = I ⇒ ei˜u(R) = eiΦeiu(R) +(S411) +where I is an 8 × 8 identity matrix. We can now give the expression of ˜u(R) +˜u(R) = −i log +� +eiΦeiu(R) +� +(S412) +We then take an infinitesimal transformation g with |φij| << 1. From Eq. S412 we have +˜u(R) ≈i log +� +(1 − iΦ)e−iu(R) +� += i log +� +e−iu(R) − iΦe−iu(R) +� +=u(R) + +� 1 +0 +1 +1 − t(1 − e−iu(R))[Φe−iu(R)] +1 +1 − t(1 − e−iu(R))dt +(S413) +where we use the derivative of a matrix logarithm. +From Eq. S413, we introduce a tensor Du which is a function of u(R) +Du[u(R)]mn +iξ,jξ′ = +� +ξ2,j2ξ′ +2 +� 1 +0 +� +1 +1 − t(1 − e−iu(R)) +� +iξ,mξ2 +[e−iu(R)]nξ2,j2ξ′ +2 +� +1 +1 − t(1 − e−iu(R)) +� +j2ξ′ +2,jξ′dt +(S414) +such that Eq. S413 can be written as +˜u(R) = u(R) + +� +mn +Du[u(R)]mnφmn +(S415) +Therefore Du[u(R)] characterize the transformation properties of u(R) under flat U(4) transformation. +We next discuss the consequence of infinitesimal symmetry transformation. We note that we have integrated out conduction +c-electron fields and derived an effective action of u fields by performing expansion to second order in u. Here, we let S[u] +denote the effective action obtained by integrating out conduction c electrons and performing expansion to the infinity order. +Thus, S[u] is an exact theory and has flat U(4) symmetry (Here we comment that the effective action we derived in Eq. S390 +describes the fluctuations around one symmetry breaking state and thus does not have the flat U(4) symmetry. However, by +performing expansion to infinity order, we effectively sum over the contributions from all the possible symmetry-breaking states +and the symmetry is recovered.) Then it must be invariant under a symmetry transformation. In other words +S[˜u] = S[u] +(S416) +where u = {u(R, τ)} denotes the configuration of the original u fields where ˜u = {˜u(R, τ)} denotes the configuration of fields +after acting SU(4) transformation g on each time slice. τ is the imaginary time. Written explicitly, for an infinitesimal flat + +91 +SU(4) transformation, we have +S[u] =S[˜u] = S +� +{u(R, τ) + +� +mn +Du[u(R, τ)]mnφmn} +� +=S[u] + +� +τ +� +R +� +iξ,jξ′,m,n +δS[u] +δuiξ,jξ′(R, τ)Du[u(R, τ)]mn +iξ,jξ′φmn +(S417) +Therefore we have +� +τ +� +R +� +iξ,jξ′,m,n +δS[u] +δuiξ,jξ′(R, τ)Du[u(R, τ)]mn +iξ,jξ′φmn = 0 +(S418) +φmn is an infinitesimal arbitrary 4 × 4 traceless Hermitian matrix. Here we focus on the off-diagonal components and have +� +τ +� +R +� +iξ,jξ′,m,n +δS[u] +δuiξ,jξ′(R, τ)Du[u(R, τ)]mn +iξ,jξ′ = 0 +, m ̸= n +(S419) +We next define the functional C[u]mn +C[u]mn = +� +τ +� +R +� +iξ,jξ′,m,n +δS[u] +δuiξ,jξ′(R, τ)Du[u(R, τ)]mn +iξ,jξ′ +(S420) +such that Eq. S419 can be written as +C[u]mn = 0, +m ̸= n +(S421) +Since Eq. S419 and Eq. S421 , holds for any given configuration of {u(R, τ)}, we must have +δC[u]mn +δu(R, τ)iξ,jξ′ = 0, +for all R, τ, iξ, jξ′ +(S422) +where m ̸= n. And consequently +δC[u]mn +δu(R, τ)iξ,jξ′ +���� +u=0 += 0, +for all R, τ, iξ, jξ′ +(S423) +To evaluate Eq. S423, we perform an expansion in u. We first consider Du[u(R, τ)]. From Eq. S414, we have +Du[u(R, τ)]mn +iξ,jξ′ ≈ Du[0]mn +iξ,jξ′ + O(u) = δi,mδj,nδξ,ξ′ + o(u) +(S424) +We next perform expansion of S. We have already derived the effective action up to the second order Seff (Eq. S390) +S[u] = Seff[u] + O(u3) +(S425) +(for the purpose of discussion, Seff is treated as a as a functional of u and is written as Seff[u]). Then we have +δS[u] +δuiξ,jξ′(R, τ) ≈ +δSeff[u] +δuiξ,jξ′(R, τ) + O(u2) +(S426) +Since we have separated Seff into diagonal (Eq. S391) and off-diagonal part (Eq. S396): Seff = Seff,diag + Seff,off, then +δS[u] +δuiξ,jξ′(R, τ) ≈ δSeff,diag[u] +∂uiξ,jξ′(R, τ) + δSeff,off[u] +δuiξ,jξ′(R, τ) + O(u2) +(S427) +Combining Eq. S420, Eq. S424 and Eq. S427, we find +C[u]mn = +� +τ +� +R +� +iξ,jξ′ +� δSeff,diag[u] +δuiξ,jξ′(R, τ) + δSeff,off[u] +δuiξ,jξ′(R, τ) + o(u2) +�� +δi,mδj,nδξ,ξ′ + o(u) +� += +� +τ +� +R +� +iξ,jξ′ +� δSeff,diag[u] +δuiξ,jξ′(R, τ) + δSeff,off[u] +δuiξ,jξ′(R, τ) + o(u2) +�� +δi,mδj,nδξ,ξ′ + o(u) +� +(S428) + +92 +We next consider Eq. S423 with i2ξ2 ̸= j2ξ′ +2 +0 = +δC[u]mn +δu(R2, τ2)i2ξ2,j2ξ′ +2 +���� +u=0 +, +i2ξ2 ̸= j2ξ′ +2 +(S429) +Combine Eq. S428 and Eq. S429 +0 = +δC[u]mn +δu(R2, τ2)i2ξ2,j2ξ2 +���� +u=0 += +� � +τ +� +R +� +iξ,jξ′ +� +δ δSeff,off [u] +δuiξ,jξ′(R,τ) +δu(R2, τ2)i2ξ2,j2ξ′ +2 ++ o(u2) +�� +δi,mδj,nδξ,ξ′ + o(u) +�� +u=0 +(S430) +since Seff,diag (Eq. S391 only contains the diagonal components of u, its contribution vanishes at leading order. Then +0 = +� +τ +� +R +� +ξ +δ2Seff,off[u] +δu(R2, τ2)iξ2,jξ′ +2δumξ,nξ(R, τ) +���� +u=0 +, +for all R2, τ2, iξ2, jξ′ +2, m, n with iξ2 ̸= jξ′ +2 and m ̸= n +(S431) +To evaluate Eq. S431, we rewrite transform Eq. S399 into imaginary-time and real space +Seff,off[u] = +� +τ +� +R1,R2 +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(R1, τ)iξ,jξ′ +� +∂τδξ2,ξ′δξ′ +2,ξδR1,R2 − F ij +ξξ′,ξ′ +2ξ2(R2 − R1) +� +u(R2, τ)jξ2,iξ′ +2 +(S432) +where F ij +ξξ′,ξ′ +2ξ2(R) = +1 +NM +� +q F ij +ξξ′,ξ′ +2ξ2(q)eiq·R. We take +iξ2 ∈ Sfill, jξ′ +2 ∈ Semp, i ̸= j +m = j, n = i +(S433) +and then +δSeff,off[u] +δu(R2, τ2)iξ2,jξ′ +2 += ∂τ2u(R2, τ2)jξ2,iξ2δξ′ +2,ξ2 − +� +R3 +� +ξ3 +jξ3 ∈ Semp +� +ξ′ +3 +iξ′ +3 ∈ Sfill +F ij +ξ2ξ′ +2,ξ′ +3ξ3(R3 − R2)u(R3, τ)jξ3,iξ′ +3 . +(S434) +(Note that i, j are not summed over) We next take m = j, n = i and find +δ2Seff,off[u] +δu(R2, τ2)iξ2,jξ′ +2δumξ,nξ(R, τ) = ∂τ2δ(R2, R)δ(τ2 − τ) − +� +ξ +jξ ∈ Semp, iξ ∈ Sfill +F ij +ξ2ξ′ +2,ξξ(R − R2) +(S435) +Combining Eq. S431 and Eq. S435, we find +0 = +� +τ +� +R +� +ξ +� +ξ +� +∂τ2δ(R2, R)δ(τ2 − τ) − +� +jξ∈Semp,iξ∈Sfill +F ij +ξ2ξ′ +2,ξξ(R − R2)] +� += − +� +τ +� +R,ξ +� +ξ +jξ ∈ Semp, iξ ∈ Sfill +F ij +ξ2ξ′ +2,ξξ(R − R2) +(S436) +which indicates +0 = +� +R +� +ξ +jξ ∈ Semp, iξ ∈ Sfill +F ij +ξ2ξ′ +2,ξξ(R) +(S437) +(Note that i, j are not summed over) Transforming to the momentum space, we reach the following symmetry constraint on +F ij +ξ2ξ2,ξξ +� +ξ +jξ ∈ Semp, iξ ∈ Sfill +F ij +ξ2ξ′ +2,ξξ(q = 0) = 0 +(S438) +with ξ2, ξ′ +2 satisfies iξ2 ∈ Sfill, jξ′ +2 ∈ Semp and i ̸= j. (Eq. S433). This constraint will lead to exact Goldstone modes. + +93 +I. +Symmetry +We next discuss the symmetry of the ground state given in Eq. S335 and Eq. S336. The effective theory (Eq. S396) has the +same symmetry properties as the ground state since the action describes the fluctuation on top of the ground state. At νf = 0, +we have U(2) × U(2). The first U(2) symmetry corresponds to the flavors i = 1, 2, where we have filled two f-electrons +(ξ = ±1) for each flavor. The second U(2) symmetry corresponds to the flavors i = 3, 4, where both flavors are filled with zero +f electrons. At νf = −2, we have a U(1) × U(3) symmetry. The U(1) symmetry corresponds to the flavor i = 1, where we +have filled two f-electrons (ξ = ±1). The U(3) symmetry comes from the flavors i = 2, 3, 4, where all three flavors are filled +with zero f electrons. At νf = −1, we have U(1) × U(1) × U(2). The first U(1) corresponds to the flavor i = 1, where we +have filled two f-electrons. The second U(1) corresponds to the flavor i = 2, where we have filled one f electron. The U(2) +corresponds to the flavor i = 3, 4, where we have filled two f electrons. +At νf = 0, 2, the ground states (defined in Eq. S335 and Eq. S336) also have C3z, C2x and C2zT symmetry. As for νf = −1, +the ground state (defined in Eq. S335 and Eq. S336) only has C3z symmetry. This is because, at νf = −1, i = 2 spin-valley +only has one f-electrons. +We next define the symmetry transformation of the f-electrons and conduction c-electrons. To begin with, we introduce the +representation matrix Df(g), Dc′(g), Dc′′(g) for a given symmetry operator g as following +gψf,ξ,† +R,i g−1 = +� +j,ξ′ +ψf,ξ′,† +gR,j Df(g)jξ′,iξ +gψc′,ξ,† +R,i g−1 = +� +j,ξ′ +ψc′,ξ′,† +gR,j Dc′(g)jξ′,iξ +, +gψc′′,ξ,† +R,i +g−1 = +� +j,ξ′ +ψc′′,ξ′,† +gR,j Dc′′(g)jξ′,iξ +(S439) +For the symmetries we considered, the corresponding representation matrices [122] are +Df(C3z)jξ′,iξ = δi,jδξ′,ξei 2π +3 ξ +, +Df(C2x)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +, +Df(C2zT)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +Dc′(C3z)jξ′,iξ = δi,jδξ′,ξei 2π +3 ξ +, +Dc′(C2x)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +, +Dc′(C2zT)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +Dc′(C3z)jξ′,iξ = δi,jδξ′,ξ +, +Dc′′(C2x)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +, +Dc′′(C2zT)jξ′,iξ = [ς′ +0ρz]ji[ζx]ξ′ξ +(S440) +where ζ0,x,y,z are identity matrix and Puli matrices for the ξ degrees of freedom, ρ0,x,y,z, ς0,x,y,z are introduced in Eq. S20. +In addition, the system at M = 0 also has a flat-U(4) symmetry. As introduced in Eq. S404, we characterize the flat-U(4) +transformation with a 4 × 4 traceless Hermitian matrices φij. We use gφ to represent the corresponding symmetry operator and +introduce the representation matrices as Df(U(4), φ), Dc′(U(4), φ), Dc′′(U(4), φ). The representation matrices are defined +below +gvψf,ξ,† +R,i gv = +� +j +ψf,ξ,† +R,j vDf(U(4), φ)ji +, +gvψc′,ξ,† +k,i +gv = +� +j +ψc′,ξ,† +k,j +Dc′(U(4), φ)ji +, +gvψc′′,ξ,† +k,i +gv = +� +j +ψc′′,ξ,† +k,j +Dc′′(U(4), φ)ji +, +Df(U(4), φ) = Dc′(U(4), φ) = Dc′′(U(4), φ) = eiφ +(S441) +where eiv denotes the matrix exponential. +We next discuss the effect of symmetry on the single-particle Green’s function. For a given symmetry g of ˆHc,order (Eq. S360) +gψc′,ξ,† +k,i +g−1 = +� +j,ξ′ +ψc′,ξ′,† +gk,j Dc′(g)ξ′j′,ξi +, +gψc′′,ξ,† +k,i +g−1 = +� +j,ξ′ +ψc′′,ξ′,† +gk,j +Dc′′(g)ξ′j′,ξi +(S442) +The corresponding Green’s functions (Eq. S366) satisfy (for the unitary transformation) +⟨Tτψa,ξ +k,i(τ)ψa′,ξ′,† +k,j +(k′, 0)⟩0 = +� +i2,ξ2,j2,ξ′ +2 +Da,∗(g)ξ2i2,ξiDa′(g)ξ′j,ξ′ +2j2⟨Tτψa,ξ,† +gk,i2(τ)ψa′,ξ′ +k,j2 (gk′, 0)⟩0 +⇒gξξ′,i +aa′ (k, τ) = +� +i2,ξ2,ξ′ +2 +Da(g)∗ +ξ2i2,ξiDa′(g)ξ′i,ξ′ +2i2gξ2ξ′ +2,i2 +aa′ +(gk, τ) +(S443) +where a, a′ ∈ {c′, c′′} and we use the fact that ⟨Tτψa,ξ +k,i(τ)ψa,ξ′,† +k,j +(0)⟩ = 0 when i ̸= j (because ˆHc,order is blocked diagonal +with respect to the flavor indices i as shown in Eq. S360). Then the product of two Green’s functions satisfy (for the unitary + +94 +transformation) +gξξ′,i +aa′ (k, τ)gξ2ξ′ +2,j +a2a′ +2 (k′, τ ′) = +� +i2,ξ3,ξ′ +3 +� +j2,ξ4,ξ′ +4 +Da(g)∗ +ξ3i2,ξiDa′(g)ξ′i,ξ′ +3i2Da2(g)∗ +ξ4j2,ξ2jDa′ +2(g)ξ′ +2j,ξ′ +4j2gξ2ξ′ +2,i2 +aa′ +(gk, τ)gξ3ξ′ +3,j2 +a2a′ +2 +(gk′, τ ′) +(S444) +For anti-unitary transformation, we have +gξξ′,i +aa′ (k, τ) = +� +i2,ξ2,ξ′ +2 +Da(g)∗ +ξ2i2,ξiDa′(g)ξ′i,ξ′ +2i2 +� +gξ2ξ′ +2,i2 +aa′ +(gk, τ) +�∗ +gξξ′,i +aa′ (k, τ)gξ2ξ′ +2,j +a2a′ +2 (k′, τ ′) = +� +i2,ξ3,ξ′ +3 +� +j2,ξ4,ξ′ +4 +Da(g)∗ +ξ3i2,ξiDa′(g)ξ′i,ξ′ +3i2Da2(g)∗ +ξ4j2,ξ2jDa′ +2(g)ξ′ +2j,ξ′ +4j2 +� +gξ2ξ′ +2,i2 +aa′ +(gk, τ) +�∗� +gξ3ξ′ +3,j2 +a2a′ +2 +(gk′, τ ′) +�∗ +(S445) +J. +νf = 0 +We now analyze the effective theory at νf = 0 and νf = −2. +At νf = 0, we consider the U(2) × U(2) ∈ U(4) symmetry gφ (in Eq. S441 ,note that U(2) × U(2) is a subgroup of flat +U(4)). From Eq. S443. +[eiφ]ijgξξ′,i +aa′ (k, τ)[e−iφ]ji = gξξ′,j +aa′ (k, τ) +(S446) +For the ground state we considered in Eq. S335, one U(2) corresponding to the rotations of flavor i = 1, 2 and the other U(2) +corresponding to the rotations of flavor i = 3, 4. Thus eiφ is block diagonal +[eiφ]ij = +� +eiφ1 +02×2 +02×2 +eiφ2 +� +ij +(S447) +where φ1, φ2 are two traceless Hermitian matrices that characterize two U(2) symmetries. Therefore, Eq. S446 can only be +satisfied when +gξξ′,i +aa′ (k, τ) = gξξ′,j +aa′ (k, τ) +, +a, a′ ∈ {c′, c′′} +, +i, j ∈ {1, 2} +gξξ′,i +aa′ (k, τ) = gξξ′,j +aa′ (k, τ) +, +a, a′ ∈ {c′, c′′} +, +i, j ∈ {3, 4} +(S448) +Consequently, from Eq. S397 and Eq. S448, we have +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i, j) = χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i′, j′) +, +i, i′ ∈ {1, 2} +, +j, j′ ∈ {3, 4} +Niξ − Njξ′ = Ni′ξ − Nj′ξ′ +, +i, i′ ∈ {1, 2} +, +j, j′ ∈ {3, 4} +(S449) +We can thus define (Eq. S398) +Hνf =0(q)ξξ′,ξ′ +2ξ2 = F ij +ξξ′,ξ′ +2ξ2(q) +i ∈ {1, 2}, j ∈ {3, 4} +(S450) +which does not depend on i, j. Then the effective action in Eq. S399 can be written as +Seff,off = +1 +2πNM +� � +q +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Hνf =0(q)ξξ′,ξ′ +2ξ2 +� +dω +(S451) +We then discuss the discrete symmetries. We first illustrate the symmetry properties of single-particle Green’s function. We +consider C2zT, C2x and C3z. From Eq. S440, Eq. S443 and Eq. S445, we have +gξξ′,i +aa′ (k, τ) = (g−ξ−ξ′,i +aa′ +(C2zTk, τ))∗ +, +a, a′ ∈ {c′, c′′} +gξξ′,i +aa′ (k, τ) = g−ξ−ξ′,i +aa′ +(C2xk, τ) +, +a, a′ ∈ {c′, c′′} +gξξ′,i +c′c′ (k, τ) = gξξ′,i +c′c′ (C3zk, τ)ei 2π +3 (ξ′−ξ) +, +gξξ′,i +c′′c′′(k, τ) = gξξ′,i +c′′c′′(C3zk, τ) +gξξ′,i +c′c′′ (k, τ) = gξξ′,i +c′c′′ (C3zk, τ)ei 2π +3 (−ξ) +, +gξξ′,i +c′′c′ (k, τ) = gξξ′,i +c′′c′ (C3zk, τ)ei 2π +3 (ξ′) . +(S452) + +95 +Thus using Eq. S397, Eq. S451 and Eq. S452, we find +Hνf =0(q)ξξ′,ξ′ +2ξ2 = (Hνf (C2zTq)−ξ−ξ′,−ξ′ +2−ξ2)∗ +Hνf =0(q)ξξ′,ξ2ξ′ +2 = Hνf =0(C2xq)−ξ−ξ′,−ξ′ +2−ξ2 +Hνf =0(q)ξξ′,ξ2ξ′ +2 = Hνf =0(C3zq)ξξ′,ξ′ +2ξ2ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +(S453) +In addition, from Eq. S400 and Eq. S450 indicates ˆHνf (q) is a Hermitian matrix +Hνf (q)ξξ′,ξ′ +2ξ2 = [Hνf (q)ξ2ξ′ +2,ξ′ξ]∗ +(S454) +We next consider the long-wavelength limit (or small q expansion) of Hνf =0(q)ξξ′,ξ2ξ′ +2. We let +M νf (q)ξξ′,ξ′ +2ξ2 ≈ C0,ξξ′,ξ′ +2ξ2 + +� +µ={x,y} +Cµ,ξξ′,ξ′ +2ξ2qµ + +� +µ,ν∈{x,y} +Cµν,ξξ′,ξ′ +2ξ2qµqν +(S455) +where C0,ξξ′,ξ′ +2ξ2, Cµ,ξξ′,ξ′ +2ξ2, Cµν,ξξ′,ξ′ +2ξ2 are the coefficients of the expansion. We next combine Eq. S454, Eq. S454 and +Eq. S455 and derive the symmetry constraints of the coefficients. For C0,ξξ′,ξ′ +2ξ2, we find +C0,ξξ′,ξ′ +2ξ2 = (C0,ξ′ +2ξ2,ξξ′)∗ +C0,ξξ′,ξ2ξ′ +2 = C0,−ξ−ξ′,−ξ′ +2−ξ2 +C0,ξξ′,ξ′ +2ξ2 = C0,ξξ′,ξ′ +2ξ2ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +C0,ξξ′,ξ′ +2ξ2 = C0,ξ2ξ′ +2,ξ′ξ +(S456) +Thus, we can introduce three real numbers C0,0, C0,1, C0,2: +C0,0 = C0,++,++ = C0,−−,−− +, +C0,1 = C0,+−,+− = C0,−+,−+ +C0,2 = C0,++,−− = C0,−−,++ +(S457) +and all other components of C0,ξξ′,ξ′ +2ξ2 are zero. +As for Cµ,ξξ′,ξ2ξ′ +2, we find (using Eq. S454, Eq. S454 and Eq. S455) +Cµ,ξξ′,ξ′ +2ξ2 = C∗ +µ,ξ′ +2ξ2,ξξ′ +Cx,ξξ′,ξ′ +2ξ2 = Cx,−ξ−ξ′,−ξ′ +2−ξ2 +, +Cy,ξξ′,ξ′ +2ξ2 = −Cy,−ξ−ξ′,−ξ′ +2−ξ2 +Cx,ξξ′,ξ′ +2ξ2 = (−1 +2Cx,ξξ′,ξ′ +2ξ2 + +√ +3 +2 +Cy,ξξ′,ξ′ +2ξ2)ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +, +Cy,ξξ′,ξ′ +2ξ2 = (− +√ +3 +2 Cx,ξξ′,ξ′ +2ξ2 − 1 +2Cy,ξξ′,ξ′ +2ξ2)ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +Cµ,ξξ′,ξ′ +2ξ2 = −Cµ,ξ2ξ′ +2,ξ′ξ +(S458) +Then we can introduce a real number C1 +C1,1 =Cx,++,−+ = Cx,−−,+− = Cx,−+,++ = Cx,+−,−− = −Cx,+−,++ = −Cx,−+,−− = −Cx,++,+− = −Cx,−−,−+ +=iCy,++,−+ = −iCy,−−,+− = −iCy,−+,++ = iCy,+−,−− = −iCy,+−,++ = iCy,−+,−− = iCy,++,+− = −iCy,−−,−+ +(S459) +As for the Cµν,ξξ′,ξ′ +2ξ2, Eq. S454, Eq. S454 and Eq. S455 we have +Cµν,ξξ′,ξ′ +2ξ2 = (Cµν,ξ′ +2ξ2,ξξ′)∗ +Cxx,ξξ′,ξ′ +2ξ2 = Cxx,−ξ−ξ′,−ξ′ +2−ξ2 +, +Cyy,ξξ′,ξ′ +2ξ2 = Cyy,−ξ−ξ′,−ξ′ +2−ξ2 +Cxy,ξξ′,ξ′ +2ξ2 = −Cxy,−ξ−ξ′,−ξ′ +2−ξ2 +, +Cyx,ξξ′,ξ′ +2ξ2 = −Cyx,−ξ−ξ′,−ξ′ +2−ξ2 +Cxx,ξξ′,ξ′ +2ξ2 = Cxx,ξξ′,ξ′ +2ξ2 + 3Cyy,ξξ′,ξ′ +2ξ2 − +√ +3Cxy,ξξ′,ξ′ +2ξ2 − +√ +3Cyx,ξξ′,ξ′ +2ξ2 +4 +ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +Cyy,ξξ′,ξ′ +2ξ2 = 3Cxx,ξξ′,ξ′ +2ξ2 + Cyy,ξξ′,ξ′ +2ξ2 + +√ +3Cxy,ξξ′,ξ′ +2ξ2 + +√ +3Cyx,ξξ′,ξ′ +2ξ2 +4 +ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +Cxy,ξξ′,ξ′ +2ξ2 = +√ +3Cxx,ξξ′,ξ′ +2ξ2 − +√ +3Cyy,ξξ′,ξ′ +2ξ2 + Cxy,ξξ′,ξ′ +2ξ2 − 3Cyx,ξξ′,ξ′ +2ξ2 +4 +ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +Cyx,ξξ′,ξ′ +2ξ2 = +√ +3Cxx,ξξ′,ξ′ +2ξ2 − +√ +3Cyy,ξξ′,ξ′ +2ξ2 − 3Cxy,ξξ′,ξ′ +2ξ2 + Cyx,ξξ′,ξ′ +2ξ2 +4 +ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +(S460) + +96 +We can then introduce the following real numbers +C2,0 = Cxx,++,++ = Cyy,++,++ = Cxx,−−,−− = Cyy,−−,−− +C2,1 = Cxx,+−,+− = Cyy,+−,+− = Cxx,−+,−+ = Cyy,−+,−+ +C2,2 = Cxx,++,−− = Cyy,++,−− = Cxx,−−,++ = Cyy,−−,++ +C2,3 = Cxx,+−,−+ = −Cyy,+−,−+ = −iCxy,+−,−+ = −iCyx,+−,−+ = Cxx,−+,+− = −Cyy,−+,+− += iCxy,+−,−+ = iCyx,−+,+− +(S461) +and all other components of Cµν,ξξ′,ξ2ξ′ +2 vanishes. +In summary, we have the following q expansion of Hνf (q) +Hνf (q) = +� +��� +C0,0 + C2,0|q|2 +C0,2 + C2,2|q|2 +C1,1(−qx − iqy) +C1,1(qx − iqy) +C0,2 + C2,2|q|2 +C0,0 + C2,0|q|2 +C1,1(qx + iqy) +C1,1(−qx + iqy) +C1,1(−qx + iqy) +C1,1(qx − iqy) +C0,1 + C2,1|q|2 +C2,3(q2 +x − q2 +y − 2iqxqy) +C1,1(qx + iqy) +C1,1(−qx − iqy) C2,3(q2 +x − q2 +y + 2iqxqy) +C0,1 + C2,1|q|2 +� +��� +(S462) +Finally, we utilize the symmetry constraints introduced in Eq. S438. +We take i, j = (1, 3) (or equivalently (i, j) = +(1, 4), (2, 4), (2, 3)) and ξ2 = ξ′ +2 = ++ in Eq. S438 and find +Hνf (q = 0)++,++ + Hνf (q = 0)++,−− = 0 +⇒C0,0 + C0,2 = 0 +(S463) +Then we have C0,0 = −C0,2 and +Hνf (q) = +� +��� +C0,0 + C2,0|q|2 +−C0,0 + C2,2|q|2 +C1,1(−qx − iqy) +C1,1(qx − iqy) +−C0,0 + C2,2|q|2 +C0,0 + C2,0|q|2 +C1,1(qx + iqy) +C1,1(−qx + iqy) +C1,1(−qx + iqy) +C1,1(qx − iqy) +C0,1 + C2,1|q|2 +C2,3(q2 +x − q2 +y − 2iqxqy) +C1,1(qx + iqy) +C1,1(−qx − iqy) +C2,3(q2 +x − q2 +y + 2iqxqy) +C0,1 + C2,1|q|2 +� +��� +(S464) +where we note that C0,0 = −C0,2 leads to a Goldstone mode at q = 0. +We also provide the expression of the parameters +C0,0 = [Hνf (q = 0)]++,++ +C2,0 = [(∂qx)2Hνf (q = 0)]++,++ +C2,2 = [(∂qx)2Hνf (q = 0)]++,−− +C1,1 = −[(∂qx)2Hνf (q = 0)]++,+− +C0,1 = [Hνf (q = 0)]+−,+− +C2,1 = [∂2 +qxHνf (q = 0)]+−,+− +C2,3 = [∂2 +qxHνf (q = 0)]+−,−+ . +(S465) +In +practice, +we +first +calculate +Niξ, +χA +and +Hνf (q) +(Eq. +S397, +Eq. +S455) +and +then +find +the +values +of +C0,0, C2,0, C2,2, C1,1, C0,1, C2,1, C2,3 using Eq. S465. The values of parameters are discussed in Sec. S8 L. +We now discuss the number of Goldstone modes. The original model has (flat) U(4) symmetry with rank (number of inde- +pendent generators) 16. At νf = 0, the symmetry group of the ground state (Eq. S335) is U(2) × U(2) with rank 4 + 4 = 8. +Then the number of Goldstone modes is (16 − 8)/2 = 4, where 1/2 comes from the fact that each Goldstone mode is a +complex boson [86]. We find M νf =0 +ξξ′,ξ2ξ′ +2(q) has one zero eigenvalues at q = 0, which corresponds to the Goldstone mode. +However, since we have four ways to pick i, j indices (of the bosnoic fields uiξ,jξ′ for the given ground state in Eq. S335), with +(i, j) = (1, 3), (2, 3), (1, 4), (2, 4), we have one Goldstone mode for each i, j choices and 4 Goldstone modes in total. +In the spectrum of spin excitation, we also observe a quasi-degeneracy between gapped modes at ΓM at ν = 0. At q = 0, we +note (Eq. S464) +Hνf (q = 0) = +� +�� +C0,0 +−C0,0 +0 +0 +−C0,0 +C0,0 +0 +0 +0 +0 +C0,1 +0 +0 +0 +0 +C0,1 +� +�� +(S466) + +97 +Four eigenstates are 0, 2C0,0, C0,1, C0,1. Then there is one Goldstone zero mode and three gapped modes. We next estimate the +energy difference between the gapped modes +∆E =C0,1 − 2C0,0 = 1 +4 +� +χξξ,ξξ(q = 0, iω = 0, i, j) − χξξ,−ξ−ξ(q = 0, iω = 0, i, j) − χξ−ξ,ξ−ξ(q = 0, iω = 0, i, j) +� +i ∈ {1, 2}, j ∈ {3, 4} +(S467) +Approximately, we take the non-interacting single-particle Green’s functions (given in Sec. S9) to calculate ∆E. From Eq. S397, +we have From Eq. S388, we find +∆E ≈1 +4 +� � +k +1 +NM +� 2γ4 +D2νc,νf +� +2gξξ,j +c′c′ (k, τ)gξξ,i +c′c′ (k, −τ) − gξξ,j +c′c′ (k, τ)g−ξ−ξ,i +c′c′ +(k, −τ) − gξξ,i +c′c′ (k, τ)g−ξ−ξ,j +c′c′ +(k, −τ) +� ++ 2J2 +� +2gξξ,j +c′′c′′(k, τ)gξξ,i +c′′c′′(k, −τ) +� +− 2Jγ2 +Dνc,νf +� +2gξξ,j +c′′c′(k, τ)gξξ,i +c′c′′(k, −τ) +� ++ 2γ2(v′ +⋆)2 +D2νf ,νc +|k|2 +� +− 2gξξ,j +c′c′ (k, τ)gξξ,i +c′c′ (k, −τ) − 2gξξ,i +c′c′ (k, τ)gξξ,j +c′c′ (k, −τ) +�� +dτ +(S468) +Using non-interacting single-particle Green’s function (Eq. S366, Eq. S544, Eq. S553, Eq. S554), we find +∆E ≈1 +4 +� +k +1 +NM +� +4J2 2 +4 +1 +2|v⋆||k| + 4Jγ2e−|k|2λ2 +Dνc,νf +2 +4 +1 +2|v⋆|k| − 8γ2(v′ +⋆)2e−2|k|2λ2 +D2νc,νf +e−2|k|2λ2|k|2 2 +4 +1 +2|v⋆|k| +� +(S469) +≈ +1 +4AMBZ +� 2π +0 +� Λc +0 +� J2 +|v⋆| + Jγ2e−k2λ2 +Dνc,νf |v⋆| − 2γ2(v′ +⋆)2e−2k2λ2k2 +D2νc,νf |v⋆| +� +dkdθ +≈ +π +2AMBZ|v⋆| +� +J2Λc + +Jγ2 +Dνc,νf +√πErf[λΛc] +2λ +− 2γ2(v′ +⋆)2 +D2νc,νf +1 +16λ3 +� +− 4λΛce−2λ2Λ2 +c + +√ +2πErf[ +√ +2Λcλ] +�� +(S470) +where Erf[x] is the error function. Approximately, we take the momentum cutoff Λc = 1/λ and find the energy difference +between two gapped modes is +∆E = +π +2AMBZ|v⋆|λ +� +J2 + 0.747Jγ2 +Dνc,νf +− 0.231γ2(v′ +⋆)2 +D2νc,νf λ2 +� +(S471) +where λ = 0.3375aM is the damping factor of the hybridization term. The degenerate condition ∆E = 0 is +α = +γ2 +JDνc,νf +� +− 0.374 + +� +0.140 + 0.231(v′⋆)2 +γ2λ2 +� += 1 +(S472) +Using the realistic values of each parameter, we find α = 1.074 ≈ 1. +K. +νf = −2 +At νf = −2, we have U(1) × U(3) symmetry. U(1) acts on the flavor i = 1 and U(3) acts on the flavor i = 2, 3, 4. Thus +Green’s function in flavor i = 2, 3, 4 are equivalent +gξξ′,i +aa′ (k, τ) = gξξ′,j +aa′ (k, τ) +, +a, a′ ∈ {c′, c′′} +, +i, j ∈ {2, 3, 4} +(S473) +Consequently, from Eq. S397 and Eq. S473, we have +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i = 1, j) = χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i = 1, j′) +, +j, j′ ∈ {2, 3, 4} +N1ξ − Njξ′ = N1ξ − Nj′ξ′ +, +j, j′ ∈ {2, 3, 4} +(S474) +We can thus define (Eq. S398) +Hνf =−2(q)ξξ′,ξ′ +2ξ2 = F ij +ξξ′,ξ′ +2ξ2(q) +i = 1, j ∈ {2, 3, 4} +(S475) + +98 +which utilize the symmetry properties of Eq. S474. Then the effective action in Eq. S396 can be written as +Seff,off = +1 +2πNM +� � +q +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Hνf =−2(q)ξξ′,ξ′ +2ξ2 +� +dω +(S476) +We next consider the discrete symmetry. We have C3z, C2zT, C2x. Using Eq. S452, we find +Hνf =−2(q)ξξ′,ξ′ +2ξ2 = (Hνf (C2zTq)−ξ−ξ′,−ξ′ +2−ξ2)∗ +Hνf =−2(q)ξξ′,ξ2ξ′ +2 = Hνf =−2(C2xq)−ξ−ξ′,−ξ′ +2−ξ2 +Hνf =−2(q)ξξ′,ξ2ξ′ +2 = Hνf =−2(C3zq)ξξ′,ξ′ +2ξ2ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +(S477) +In addition, the definition of Hνf =−2(q)ξξ′,ξ′ +2ξ2 (Eq. S450) and Eq. S400, we have +Hνf =−2(q)ξξ′,ξ′ +2ξ2 = [Hνf =−2(q)ξ2ξ′ +2,ξ′ξ]∗ +(S478) +Finally, we utilize the symmetry constraints in Eq. S438. We take i, j = (1, 2) (or equivalently (i, j) = (1, 2), (1, 4)) and +ξ2 = ξ′ +2 = ++ in Eq. S438 and find +Hνf =−2(q = 0)++,++ + Hνf =−2(q = 0)++,−− = 0 +(S479) +Combining Eq. S453, Eq. S477 and Eq. S479), we conclude ˆHνf =−2(q) follows the same constraint under symmetry as +ˆHνf =0(q) (Eq. S453, Eq. S454, Eq. S463). Therefore, the long-wavelength behavior of ˆHνf =−2(q) also takes the form of +Eq. S464. +We now discuss the number of Goldstone modes. At νf = −2, the symmetry group of the ground state (Eq. S335) is +U(1) × U(3) with rank 1 + 9 = 10. The symmetry of the original Hamiltonian at M = 0 has flat U(4) symmetry with rank +16. The number of Goldstone modes is (16 − 10)/2 = 3 [86]. From Eq. S464, Hνf =−2 +ξξ′,ξ2ξ′ +2(q) has one zero eigenvalues at q = 0, +which corresponds to the Goldstone mode. Since we have three ways to pick i, j indices (of the bosonic fields uiξ,jξ′), with +(i, j) = (1, 2), (1, 3), (1, 4) (for the given ground state in Eq. S335), we have one Goldstone mode for each i, j choices and 3 +Goldstone modes in total. +We also comment that, due to the fact that values of γ2/Dνc,νf at νf = 2 is much larger than its value at νf = 0, −1. The +Kondo interaction is much stronger at νf = −2, which leads to a larger bandwidth of the excitation spectrum. +L. +Lagrangian at νf = 0, −2 in the long-wavelength limit +In this section, we provide the Lagrangian of effective theory at νf = 0, −2 in a more compact form. From Eq. S451, Eq S476 +and Eq. S464, we have the following Lagrangian density in the long-wavelength limit at νf = 0, −2 +L = +� +iξ,iξ′ +2∈Sfill,jξ′,jξ2∈Semp +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Hνf (q)ξξ′,ξ′ +2ξ2 +� +(S480) +where we introduce a conventional parameterization of Hνf (q)ξξ′,ξ′ +2ξ2 +Hνf (q)ξξ′,ξ′ +2ξ2 = +� +����� +∆1 +2 + ( +1 +4m0 + +1 +4m1 )|q|2 − ∆1 +2 + ( +1 +4m0 − +1 +4m1 )|q|2 +V +√ +2q+ +− V +√ +2q− +∆1 +2 + ( +1 +4m0 − +1 +4m1 )|q|2 +∆1 +2 + ( +1 +4m0 − +1 +4m1 )|q|2 +− V +√ +2q+ +V +√ +2q− +V +√ +2q− +− V +√ +2q− +∆2 + |q|2 +2m2 +q2 +− +2m3 +− V +√ +2q+ +V +√ +2q+ +q2 ++ +2m3 +∆2 + |q|2 +2m2 +� +����� +ξξ′,ξ′ +2ξ2 +(S481) +where the row and column indices are (++, −−, +−, −+), q± = qx ± iqy. The parameters are +∆1 = 2C0,0, +∆2 = C0,1 +m0 = +1 +2(C2,0 + C2,2), +m1 = +1 +2(C2,0 − C2,2), +m2 = +1 +2C2,1 +, +m3 = +1 +2C2,3 +V = − +√ +2C1,1 +(S482) + +99 +Parameter +∆1 +∆2 a2 +M/m0 a2 +M/m1 a2 +M/m2 a2 +M/m3 V +Value at νf = 0 (meV) +6.3 +6.7 +6.0 +0.9 +0.7 +1.0 +2.5 +Value at νf = −2 (meV) 22.8 16.4 +11.8 +-3.1 +0.9 +3.3 +7.3 +Supplementary Table S1. Numeircal values of parameters in Eq. S485 at νf = 0, −2. +We then take the following new basis +uj,i(q, iω) = (u(j+,i+)(q, ω) + u(j−,i−)(q, iω))/ +√ +2 +U T +j,i(q, iω) = ((u(j+,i+)(q, iω) − u(j−,i−))(q, iω)/ +√ +2, u(j+,i−)(q, iω), u(j−,i+)(q, iω)) +(S483) +The Lagrangian (Eq. S480) can then be written as +L = LGoldstone + Lgapped, +LGoldstone = +� +jξ∈Sfill +iξ∈Semp +u† +j,i(q, iω) +� +iω − q2/2m0 +� +uj,i(q, iω), +Lgapped = +� +jξ∈Sfill +iξ∈Semp +U † +j,i(q, iω)[iω ˆI − H(q)]Uj,i(q, iω), +(S484) +where +H(q) = +� +� +� +q+q− +2m1 + ∆1 +V q+ +−V q− +V q− +q+q− +2m2 + ∆2 +q2 +− +2m3 +−V q+ +q2 ++ +2m3 +q+q− +2m2 + ∆2 +� +� +� +(S485) +and ˆI is a 4 × 4 identity matrix. After transforming from Matsubara frequency to real frequency (iω → ω), we reach the same +formula given in the main text (Eq.[11] and Eq.[12]). +We also provide the dispersion of gapped modes. By diagonalizing H(q) in Eq. S485, we find the dispersion of three gapped +modes are +Egap,1 +q += ∆2 + ( 1 +2m2 ++ +1 +2m3 +)|q|2 +Egap,2 +q += ∆1 + ∆2 +2 ++ |q|2 +4 +� 1 +m1 ++ 1 +m2 +− 1 +m3 +� ++ +� +2V 2|q|2 + +�∆1 − ∆2 +2 ++ |q|2 +4 +� 1 +m1 +− 1 +m2 ++ 1 +m3 +��2 +Egap,3 +q +− ∆1 + ∆2 +2 ++ |q|2 +4 +� 1 +m1 ++ 1 +m2 +− 1 +m3 +� +− +� +2V 2|q|2 + +�∆1 − ∆2 +2 ++ |q|2 +4 +� 1 +m1 +− 1 +m2 ++ 1 +m3 +��2 +(S486) +We provide the numerical values of parameters (derived from Eq. S465 and Eq. S482) at νf = 0, −2 in Tab. S1. In Fig. S5, +we also compare the dispersion from the effective model (Eq. S485) with the dispersion from directly evaluating Eq. S399. We +note that the effective model S485 correctly predicts the long wavelength (small k) behaviors but failed to capture the large +momentum behaviors such as the roton modes. To recover the roton mode, we need to keep high-order contribution (q3 term). +M. +νf = −1 +We now consider νf = −1. We first discuss the symmetry of νf = −1 ground state. It turns out, the remaining discrete +symmetry of the ground state (Eq. S335) at νf = −1 is C3z, because one of the valley-spin flavors at νf = −1 is filled with +only one electron. In addition, the original flat U(4) symmetry is broken to U(1) × U(1) × U(2) symmetry. +We next discuss the fluctuations at νf = −1. From Eq. S334 and Eq. S337, we find +|u(R)⟩R = +� +R +exp +� +− i +� +ij,ξξ′ +uiξ,jξ′(R)ψf,ξ,† +R,i ψf,ξ′ +R,j +� +|ψ0⟩ ≈ +� +1 − i +� +ij,ξξ′ +uiξ,jξ′(R)ψξ,† +f,R,iψf,ξ′ +R,j +� +|ψ0⟩ +(S487) + +100 +(a) +(b) +FIG. S5. Excitation spectrum from the effective model in Eq. S485 (brown) and from numerical evaluation of Eq. S399 (blue). +Therefore at the leading-order, uiξ,jξ′(R) (iξ ̸= jξ′) describe the procedure of moving one f-electron from jξ′ flavor at site +R to iξ flavor at the same site. This allows us to classify the u fields into four groups. Taking the ground state in Eq. S335 at +νf = −1, four groups of uiξ,jξ′(R) are (1) full-empty sector with i = 3, 4, j = 1 or i = 1, j = 3, 4; (2) half-empty sector with +i = 2, j = 3, 4 or i = 3, 4, j = 2 ; (3) full-half sector with i = 1, j = 2 or j = 2, i = 1; (4) half-half sector with i = 2, j = 2. +Group (1) describes the fluctuations between fully filled and empty spin-valley flavors. Group (2) describes the fluctuations +between half-filled and empty spin-valley flavors. Group (3) describes the fluctuations between fully filled and empty spin- +valley flavors. Group (4) describes the fluctuations between half-filled and half-filled spin-valley flavors. Correspondingly, we +can separate the Seff,off into four parts: +Seff,off = Seff,off,fe + Seff,off,he + Seff,off,fh + Seff,off,hh +(S488) +where Seff,off,fe, Seff,off,he, Seff,off,fh, Seff,off,hh describe the effective theory of full-empty sector, half-empty sector, +full-half sector, and half-half sector respectively. +1. +Full-empty sector +For the full-empty sector, we have action (Eq. S399) +Seff,off,fe = +1 +2πNM +� � +q +� +i∈{1},j∈{3,4} +� +ξ,ξ′ +2,ξ′,ξ2 +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2[iωδξ2,ξ′δξ′ +2,ξ − F ij +ξξ′,ξ′ +2ξ2(q)]dω +(S489) +where i = 1, j ∈ {3, 4}. i = 1 flavor is filled with two f electrons and j = 3, 4 are empty (we take the ground sate in Eq. S335). +Utilizing the U(1) × U(1) × U(2), symmetry, we have +χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i = 1, j) = χA +ξξ′,ξ′ +2ξ2(q, iω = 0, i = 1, j′) +, +j, j′ ∈ {3, 4} +N1ξ − Njξ′ = N1ξ − Nj′ξ′ +, +j, j′ ∈ {3, 4} +(S490) +Therefore F ij +ξξ′,ξ′ +2ξ′ +2 take the same value for i = 1, j ∈ {3, 4}. We thus define +Hνf =−1,fe(q)ξξ′,ξ′ +2ξ2 = F ij +ξξ′,ξ′ +2ξ′ +2(q) +, +i = 1, j ∈ {3, 4} +(S491) +The effective theory can be written as +Seff,off,fe = +1 +2πNM +� � +q +� +i∈{1},j∈{3,4} +� +ξ,ξ′ +2,ξ′,ξ2 +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Hνf =−1,fe(q)ξξ′,ξ′ +2ξ2 +� +dω +(S492) +The excitation spectrum can be derived by finding the eigenvalues of Hνf =−1,fe(q). +Here we comment that ˆHc,order has a larger symmetry group than the symmetry group of the ground state. Even though, the +ground state does not have C2zT and C2x symmetries, ˆHc,order has a special type of C2zT and C2x symmetry, which we call +(C2zT)′ and (C2x)′. For the ground state in Eq. S335, (C2zT)′ ((C2x)′) are defined as C2zT(C2x) transformations that only +act on the valley-spin flavors i = 1, 3, 4 (Note that, C2zT and C2x will not flip valley and spin indices). Then ˆHc,order satisfies + +101 +(C2zT)′ and (C2x)′ symmetries, so is the Hνf =−1,fe(q)ξξ′,ξ′ +2ξ2. We comment that (C2zT)′, (C2x)′ are not real symmetries of +the system. +Consequently, we have +gξξ′,i +aa′ (k, τ, ) = (g−ξ−ξ′,i +aa′ +(C2zTk, τ))∗ +, +a, a′ ∈ {c′, c′′} +gξξ′,i +aa′ (k, τ) = g−ξ−ξ′,i +aa′ +(C2xk, τ) +, +a, a′ ∈ {c′, c′′} +gξξ′,i +c′c′ (k, τ) = gξξ′,i +c′c′ (C3zk, τ)ei 2π +3 (ξ′−ξ) +, +gξξ′,i +c′′c′′(k, τ) = gξξ′ +c′′c′′,i(C3zk, τ) +gξξ′,i +c′c′′ (k, τ) = gξξ′,i +c′c′′ (C3zk, τ)ei 2π +3 (−ξ) +, +gξξ′,i +c′′c′ (k, τ) = gξξ′,i +c′′c′ (C3zk, τ)ei 2π +3 (ξ′) . +(S493) +where i = 1, 3, 4. Thus, using Eq. S491, Eq. S397 and Eq. S493, we find +Hνf =−1,fe(q)ξξ′,ξ′ +2ξ2 = (Hνf (C2zTq)−ξ−ξ′,−ξ′ +2−ξ2)∗ +Hνf =−1,fe(q)ξξ′,ξ2ξ′ +2 = Hνf =−1,fe(C2xq)−ξ−ξ′,−ξ′ +2−ξ2 +Hνf =−1,fe(q)ξξ′,ξ2ξ′ +2 = Hνf =−1,fe(C3zq)ξξ′,ξ′ +2ξ2ei 2π +3 (ξ′−ξ2+ξ′ +2−ξ) +(S494) +In addition from Eq. S438 (with i = 1, j = 3, ξ = ξ′ = +) +Hνf =−1,fe(q = 0)++,++ + Hνf =−1,fe(q = 0)+−,−− = 0 +(S495) +Clearly, Hνf =−1,fe(q) satisfy the same constrain as ˆHνf =0(q) (Eq. S453). Consequently, Hνf =−1,fe(q) has the same +long-wavelength behavior as shown in Eq. S464. +2. +Full-half sector +For the full-half sector, we take i = 1, j = 2 where i = 1 flavor is filled with two f electrons, and j = 2 flavor is filled with +one f electrons (Eq. S335). +Seff,off,fh = +1 +2πNM +� � +q +� +ξ,ξ′ +u(q, iω)1ξ,2−u(−q, −iω)2−,1ξ′ +� +iωδξ,ξ′ − F 12 +ξ−,ξ′−(q) +� +dω +(S496) +We thus introduce the following matrix +Hνf =−1,fh(q)ξ,ξ′ = F 12 +ξ−,ξ′−(q) +(S497) +which is a 2 × 2 matrix. The effective action can be written as +Seff,off,fh = +1 +2πNM +� � +q +� +ξ,ξ′ +u(q, iω)1ξ,2−u(−q, −iω)2−,1ξ′ +� +iωδξ,ξ′ − Hνf =−1,fh(q)ξ,ξ′ +� +dω +(S498) +Ground state at νf = −3 (Eq. S335) only has C3z symmetry. We use Eq. S440, Eq. S443 and find +gξξ′,i +c′c′ (k, τ) = gξξ′,i +c′c′ (C3zk, τ)ei 2π +3 (ξ′−ξ) +, +gξξ′,i +c′′c′′(k, τ) = gξξ′,i +c′′c′′(C3zk, τ) +gξξ′,i +c′c′′ (k, τ) = gξξ′,i +c′c′′ (C3zk, τ)ei 2π +3 (−ξ) +, +gξξ′,i +c′′c′ (k, τ) = gξξ′,i +c′′c′ (C3zk, τ)ei 2π +3 (ξ′) . +(S499) +Then combining Eq. S499 and Eq. S397, we have +χA +ξ−,ξ′−(q, iω = 0, 2, 1) = χA +ξ−,ξ′−(C3zq, iω = 0, 2, 1)ei2π/3(ξ′−ξ) +(S500) +Thus using Eq. S397, Eq. S497 and Eq. S514 +ˆHνf =−1,fh(q)ξ,ξ′ = ˆHνf =−1,fh(C3zq)ξ,ξ′ei2π/3(ξ′−ξ) +(S501) +In the long-wavelength limit, we assume Hνf =1,fh +1 +(q) takes the form of +ˆHνf =1,fh +1 +(q)ξξ′ = +� +Cfh +0,0 + Cfh +µ,1,0qµ + Cfh +µν,2,0qµqν Cfh +0,1 + Cfh +µ,1,1qµ + Cfh +µν,2,1qµqν +Cfh +0,3 + Cfh +µ,1,3qµ + Cfh +µν,2,3qµqν Cfh +0,4 + Cfh +µ,1,4qµ + Cfh +µν,2,4qµqν +� +ξξ′ +(S502) + +102 +Combining Eq. S502 and Eq. S501 we find +Cfh +0,1 = e−i4π/3Cfh +0,1, +Cfh +0,3 = ei4π/3Cfh +0,3 +Cfh +x,1,0 = −1 +2Cfh +x,1,0 + +√ +3 +2 Cfh +y,1,0, +Cfh +y,1,0 = −− +√ +3 +2 +Cfh +x,1,0 − 1 +2Cfh +y,1,0 +Cfh +x,1,4 = −1 +2Cfh +x,1,4 + +√ +3 +2 Cfh +y,1,4, +Cfh +y,1,4 = −− +√ +3 +2 +Cfh +x,1,4 − 1 +2Cfh +y,1,4 +Cfh +x,1,1 = (−1 +2Cfh +x,1,1 + +√ +3 +2 Cfh +y,1,1)e−i4π/3, +Cfh +y,1,1 = (−− +√ +3 +2 +Cfh +x,1,1 − 1 +2Cfh +y,1,1)e−i4π/3 +Cfh +x,1,3 = (−1 +2Cfh +x,1,3 + +√ +3 +2 Cfh +y,1,3)ei4π/3, +Cfh +y,1,3 = (−− +√ +3 +2 +Cfh +x,1,3 − 1 +2Cfh +y,1,3)ei4π/3 +Cfh +xx,2,0/4 = +Cfh +xx,2,0/4 + 3Cfh +yy,2,0/4 − +√ +3Cfh +xy,2,0/4 − +√ +3Cfh +yx,2,0/4 +4 +Cfh +yy,2,0/4 = +3Cfh +xx,2,0/4 + Cfh +yyv + +√ +3Cfh +xy,2,0/4 + +√ +3Cfh +yx,2,0/4 +4 +Cfh +xy,2,0/4 = +√ +3Cfh +xx,2,0/4 − +√ +3Cfh +yy,2,0/4 + Cfh +xy,2,0/4 − 3Cfh +yx,2,0/4 +4 +Cfh +yx,2,0/4 = +√ +3Cfh +xx,2,0/4 − +√ +3Cfh +yy,2,0/4 − 3Cfh +xy,2,0/4 + Cfh +yx,2,0/4 +4 +Cfh +xx,2,1 = Cfh +xx,2,1 + 3Cfh +yy,2,1 − +√ +3Cfh +xy,2,1 − +√ +3Cfh +yx,2,1 +4 +e−i4π/3 +Cfh +yy,2,1 = 3Cfh +xx,2,1 + Cfh +yy,n,0 + +√ +3Cfh +xy,2,1 + +√ +3Cfh +yx,2,1 +4 +e−i4π/3, +Cfh +xy,2,1 = +√ +3Cfh +xx,2,1 − +√ +3Cfh +yy,2,1 + Cfh +xy,2,1 − 3Cfh +yx,2,1 +4 +e−i4π/3 +Cfh +yx,2,1 = +√ +3Cfh +xx,2,1 − +√ +3Cfh +yy,2,1 − 3Cfh +xy,2,1 + Cfh +yx,2,1 +4 +e−i4π/3 +Cfh +xx,2,3 = Cfh +xx,2,3 + 3Cfh +yy,2,1 − +√ +3Cfh +xy,2,1 − +√ +3Cfh +yx,2,3 +4 +ei4π/3 +Cfh +yy,2,3 = 3Cfh +xx,2,1 + Cfh +yy,n,0 + +√ +3Cfh +xy,2,3 + +√ +3Cfh +yx,2,3 +4 +ei4π/3, +Cfh +xy,2,3 = +√ +3Cfh +xx,2,3 − +√ +3Cfh +yy,2,3 + Cfh +xy,2,3 − 3Cfh +yx,2,3 +4 +ei4π/3, +Cfh +yx,2,3 = +√ +3Cfh +xx,2,3 − +√ +3Cfh +yy,2,3 − 3Cfh +xy,2,3 + Cfh +yx,2,3 +4 +ei4π/3 +(S503) +We also utilize the fact that Hνf =1,fh +1 +(q) is a Hermitian matrix (from Eq. S497 and Eq. S400). Then we find the non-vanishing +components are +Cfh +0,0, +Cfh +xx/yy,2,0, +Cfh +0,4, +Cfh +xx/yy,2,4, +Cµ,1,3, +Cfh +µ,1,1, +Cfh +µ,1,3 +(S504) +with +Cfh +0,0 ∈ R, +Cfh +0,4 ∈ R, +Cfh +µµ,2,0 ∈ R, +Cfh +µµ,2,4 ∈ R +Cfh +xx,2,0 = Cfh +yy,2,4, +Cfh +xx,2,0 = Cfh +yy,2,4 +Cfh +y,1,1 = −iCfh +x,1,1 = −Cfh +y,1,3 = −iCfh +x,1,3 +(S505) +Thus we have +ˆHνf =1,fh(q)ξξ′ = +� +Cfh +0,0 + Cfh +xx,2,0|q|2 +Cfh +x,1,1(qx − iqy) +Cfh +x,1,1(qx + iqy) +Cfh +0,4 + Cfh +xx,2,4|q|2 +� +ξξ′ +(S506) + +103 +We next use Eq. S438 (we take i = 1, j = 2, ξ2 = +, ξ′ +2 = −) and Eq. S492, and find +Hνf =1,fh(q = 0)−− = 0 ⇒ Cfh +0,4 = 0 +(S507) +Then we have +Hνf =1,fh(q)ξξ′ = +� +Cfh +0,0 + Cfh +xx,2,0|q|2 Cfh +x,1,1(qx − iqy) +Cfh +x,1,1(qx + iqy) +Cfh +xx,2,4|q|2 +� +ξξ′ +(S508) +where we observe Cfh +0,4 = 0 produces a Goldstone mode. The eigenvalues of Eq. S508 are +E(q) = Cfh +0,0 + Cfh +xx,2,0|q|2 + Cfh +xx,2,4|q|2 +2 +± +��Cfh +0,0 + Cfh +xx,2,0|q|2 − Cfh +xx,2,4|q|2 +2 +�2 ++ |Cfh +x,1,1|2|q|2 +(S509) +3. +Half-empty sector +For the half-empty sector, we take i = 2, j = 3, 4, where i = 2 flavor is filled with two f electrons, and j = 3, 4 flavors are +empty (Eq. S335). Then +Seff,off,he = +1 +2πNM +� � +q +� +j∈{3,4} +� +ξ,ξ2 +u(q, iω)2+,jξu(−q, −iω)jξ2,2+ +� +iωδξ2,ξ − F 2j ++ξ,+ξ2(q) +� +dω, +j = 3, 4 +(S510) +The U(2) symmetry that act on the valley-spin flavors i = 3, 4 indicates +− N2+ − Njξ +2 += −N2+ − Nj′ξ +2 +, +j, j′ ∈ {3, 4} +1 +4 +� +χA ++ξ,+ξ2(q, iω = 0, i = 2, j) + χA +ξ2+,ξ+(−q, iω = 0, j, i = 2) +� += 1 +4 +� +χA ++ξ,+ξ2(q, iω = 0, i = 2, j′) + χA +ξ2+,ξ+(−q, iω = 0, j′, i = 2) +� +, +j, j′ ∈ {3, 4} +(S511) +Therefore, F 23 ++ξ,+ξ2(q) = F 24 ++ξ,+ξ2(q). We can introduce the following matrix +Hνf =−1,he(q)ξ,ξ2 = F 2j ++ξ,+ξ2(q) +, +j ∈ {3, 4} +(S512) +which is a 2 × 2 matrix and does not depend on j. The effective action can be written as +Seff,off,he = +1 +2πNM +� � +q +� +j∈{3,4} +� +ξ,ξ2 +u(q, iω)2+,jξu(−q, −iω)jξ2,2+[iωδξ,ξ2 − Hνf =−1,he(q)ξ,ξ2]dω +(S513) +Wround state at νf = −3 (Eq. S335) has C3z symmetry. We combine Eq. S499 and Eq. S397, and find (j = 3, 4) +χA ++ξ,+ξ2(q, iω = 0, j, 2) = χA ++ξ,+ξ2(C3zq, iω = 0, j, 2)ei2π/3(ξ′−ξ) +(S514) +Thus using Eq. S397, Eq. S497 and Eq. S514 +Hνf =−1,he(q)ξ,ξ2 = Hνf =−1,he(C3zq)ξ,ξ2ei2π/3(ξ−ξ2) +(S515) +We next study the long-wavelength limit, we assume Hνf =1,fh +1 +(q) takes the form of +Hνf =1,he(q)ξξ2 = +� +Cfe +0,0 + Cfe +µ,1,0qµ + Cfh +µν,2,0qµqν Cfe +0,1 + Cfe +µ,1,1qµ + Cfe +µν,2,1qµqν +Cfh +0,3 + Cfe +µ,1,3qµ + Cfe +µν,2,3qµqν Cfh +0,4 + Cfe +µ,1,4qµ + Cfe +µν,2,4qµqν +� +ξξ2 +(S516) + +104 +with row and column indices {+, −}. Combining Eq. S516 and Eq. S515 we find +Che +0,1 = e−i4π/3Che +0,1, +Che +0,3 = ei4π/3Che +0,3 +Che +x,1,0 = −1 +2Che +x,1,0 + +√ +3 +2 Che +y,1,0, +Che +y,1,0 = −− +√ +3 +2 +Che +x,1,0 − 1 +2Che +y,1,0 +Che +x,1,4 = −1 +2Che +x,1,4 + +√ +3 +2 Che +y,1,4, +Che +y,1,4 = −− +√ +3 +2 +Che +x,1,4 − 1 +2Che +y,1,4 +Che +x,1,1 = (−1 +2Che +x,1,1 + +√ +3 +2 Che +y,1,1)ei4π/3, +Che +y,1,1 = (−− +√ +3 +2 +Che +x,1,1 − 1 +2Che +y,1,1)ei4π/3 +Che +x,1,3 = (−1 +2Che +x,1,3 + +√ +3 +2 Che +y,1,3)e−i4π/3, +Che +y,1,3 = (−− +√ +3 +2 +Che +x,1,3 − 1 +2Che +y,1,3)e−i4π/3 +Che +xx,2,0/4 = +Che +xx,2,0/4 + 3Che +yy,2,0/4 − +√ +3Che +xy,2,0/4 − +√ +3Che +yx,2,0/4 +4 +Che +yy,2,0/4 = +3Che +xx,2,0/4 + Che +yyv + +√ +3Che +xy,2,0/4 + +√ +3Che +yx,2,0/4 +4 +Che +xy,2,0/4 = +√ +3Che +xx,2,0/4 − +√ +3Che +yy,2,0/4 + Che +xy,2,0/4 − 3Che +yx,2,0/4 +4 +Che +yx,2,0/4 = +√ +3Che +xx,2,0/4 − +√ +3Che +yy,2,0/4 − 3Che +xy,2,0/4 + Che +yx,2,0/4 +4 +Che +xx,2,1 = Che +xx,2,1 + 3Che +yy,2,1 − +√ +3Che +xy,2,1 − +√ +3Che +yx,2,1 +4 +ei4π/3 +Che +yy,2,1 = 3Che +xx,2,1 + Che +yy,n,0 + +√ +3Che +xy,2,1 + +√ +3Che +yx,2,1 +4 +ei4π/3, +Che +xy,2,1 = +√ +3Che +xx,2,1 − +√ +3Che +yy,2,1 + Che +xy,2,1 − 3Che +yx,2,1 +4 +ei4π/3 +Che +yx,2,1 = +√ +3Che +xx,2,1 − +√ +3Che +yy,2,1 − 3Che +xy,2,1 + Che +yx,2,1 +4 +e−i4π/3 +Che +xx,2,3 = Che +xx,2,3 + 3Che +yy,2,1 − +√ +3Che +xy,2,1 − +√ +3Che +yx,2,3 +4 +e−i4π/3 +Che +yy,2,3 = 3Che +xx,2,1 + Che +yy,n,0 + +√ +3Che +xy,2,3 + +√ +3Che +yx,2,3 +4 +e−i4π/3, +Che +xy,2,3 = +√ +3Che +xx,2,3 − +√ +3Che +yy,2,3 + Che +xy,2,3 − 3Che +yx,2,3 +4 +e−i4π/3, +Che +yx,2,3 = +√ +3Che +xx,2,3 − +√ +3Che +yy,2,3 − 3Che +xy,2,3 + Che +yx,2,3 +4 +e−i4π/3 +(S517) +We also utilize the fact that Hνf =1,he(q) is a Hermitian matrix (from Eq. S515 and Eq. S400). Then we find the non-vanishing +components are +Che +0,0, +Che +xx/yy,2,0, +Che +0,4, +Che +xx/yy,2,4, +Cµ,1,3, +Che +µ,1,1, +Che +µ,1,3 +(S518) +with +Che +0,0 ∈ R, +Che +0,4 ∈ R, +Che +µµ,2,0 ∈ R, +Che +µµ,2,4 ∈ R +Che +xx,2,0 = Che +yy,2,4, +Che +xx,2,0 = Che +yy,2,4 +Che +y,1,1 = iChe +x,1,1 = −Che +y,1,3 = iChe +x,1,3 +(S519) +Thus we have +ˆHνf =1,he(q)ξξ′ = +�Che +0,0 + Che +xx,2,0|q|2 +Che +x,1,1(qx + iqy) +Che +x,1,1(qx − iqy) +Che +0,4 + Che +xx,2,4|q|2 +� +ξξ′ +(S520) + +105 +We next use Eq. S438 (we let i = 2, j = 3, ξ2 = +, ξ′ +2 = +) and Eq. S513, and find +Hνf =1,fh(q = 0)++ = 0 ⇒ Cfe +0,0 = 0 +(S521) +Then +Hνf =1,he(q)ξξ′ = +� +Che +xx,2,0|q|2 +Che +x,1,1(qx + iqy) +Che +x,1,1(qx − iqy) Che +0,4 + Che +xx,2,4|q|2 +� +ξξ′ +(S522) +where we notice Cfe +0,0 = 0 = 0 produces a Goldstone mode for each q. The eigenvalues of Eq. S522 are +E(q) = Che +0,4 + Che +xx,2,4|q|2 + Che +xx,2,0|q|2 +2 +± +��Che +0,4 + Che +xx,2,4|q|2 − Che +xx,2,0|q|2 +2 +�2 ++ |Che +x,1,1|2|q|2 +(S523) +4. +Half-half sector +For the half-half sector, we take i = 2, j = 2, where i = j = 2 flavor is filled with one f electron (Eq. S335). We then have +(Eq. S399) +Seff,off,hh = +1 +2πNM +� � +q +u(q, iω)2+,2−u(−q, −iω)2−,2+[iω − F 22 +−+,−+(q)]dω +(S524) +We introduce Hνf =−1,hh(q) +Hνf =−1,hh(q) = F 22 +−+,−+(q) +(S525) +and have +Seff,off,hh = +1 +2πNM +� � +q +u(q, iω)2+,2−u(−q, −iω)2−,2+[iω − Hνf =−1,hh(q)]dω +(S526) +From Eq. S400, F 22 +−+,−+(q) and also Hνf =−1,hh(q) is real. +We now discuss the effect of C3z symmetry. From Eq. S499, Eq. S397 and Eq. S398, we have +F 22 +−+,−+(q) = F 22 +−+,−+(C3zq) +Hνf =−1,hh(q) = Hνf =−1,hh(C3zq) +(S527) +We perform a small q expansions of Hνf =−1,hh(q) +Hνf =−1,hh(q) = Chh +0 ++ +� +µ +Chh +µ,1qµ + +� +µν +Chh +µν,2qµqν +(S528) +From Eq. S527, we find +Chh +x,1 = −1 +2Chh +x,1 + +√ +3 +2 Chh +y,1, +Chh +y,1 = −− +√ +3 +2 +Chh +x,1 − 1 +2Chh +y,1 +Chh +xx,2 = Chh +xx,2 + 3Chh +yy,2 − +√ +3Chh +xy,2 − +√ +3Chh +yx,2 +4 +, +Chh +yy,2 = 3Chh +xx,2 + Chh +yy,2 + +√ +3Chh +xy,2 + +√ +3Chh +yx,2 +4 +Chh +xy,2 = +√ +3Chh +xx,2 − +√ +3Chh +yy,2 + Chh +xy,2 − 3Chh +yx,2 +4 +, +Chh +yx,2 = +√ +3Chh +xx,2 − +√ +3Chh +yy,2 − 3Chh +xy,2 + Chh +yx,2 +4 +(S529) +The non-zero components are +Chh +0 , Chh +xx,2, Chh +yy,2 , +(S530) +and we also have +Chh +0 +∈ R, +Chh +xx,2 ∈ R +, Chh +yy,2 ∈ R +Chh +xx,2 = Chh +yy,2 . +(S531) +Then we have +Hνf =−1,hh(q) = Chh +0 ++ Chh +xx,2|q|2 +(S532) + +106 +5. +Effective theory in the long-wavelength limit +Here, we summarize the effective theory in the long-wavelength limit at νf = −1. From Eq. S492, Eq. S498, Eq. S513 and +Eq. S526 +Seff,off = Seff,off,fe + Seff,off,he + Seff,off,fh + Seff,off,hh +Seff,off,fe = +1 +2πNM +� � +q +� +i∈{1},j∈{3,4} +� +ξ,ξ′ +2,ξ′,ξ2 +u(q, iω)iξ,jξ′u(−q, −iω)jξ2,iξ′ +2 +� +iωδξ2,ξ′δξ′ +2,ξ − Hνf =−1,fe(q)ξξ′,ξ′ +2ξ2 +� +dω +Seff,off,fh = +1 +2πNM +� � +q +� +ξ,ξ′ +u(q, iω)1ξ,2−u(−q, −iω)2−,1ξ′ +� +iωδξ,ξ′ − Hνf =−1,fh(q)ξ,ξ′ +� +dω +Seff,off,he = +1 +2πNM +� � +q +� +j∈{3,4} +� +ξ,ξ2 +u(q, iω)2+,jξu(−q, −iω)jξ2,2+ +� +iωδξ,ξ2 − Hνf =−1,he(q)ξ,ξ2 +� +dω +Seff,off,hh = +1 +2πNM +� � +q +u(q, iω)2+,2−u(−q, −iω)2−,2+ +� +iω − Hνf =−1,hh(q) +� +dω +(S533) +where (from Eq. S508, Eq. S522 and Eq. S532) +Hνf =−1,fe = +� +���� +Cfe +0,0 + Cfe +2,0|q|2 +−Cfe +0,0 + Cfe +2,2|q|2 +Cfe +1,1(−qx − iqy) +Cfe +1,1(qx − iqy) +−Cfe +0,0 + Cfe +2,2|q|2 +Cfe +0,0 + Cfe +2,0|q|2 +Cfe +1,1(qx + iqy) +Cfe +1,1(−qx + iqy) +Cfe +1,1(−qx + iqy) +Cfe +1,1(qx − iqy) +Cfe +0,1 + Cfe +2,1|q|2 +Cfe +2,3(q2 +x − q2 +y − 2iqxqy) +Cfe +1,1(qx + iqy) +Cfe +1,1(−qx − iqy) +Cfe +2,3(q2 +x − q2 +y + 2iqxqy) +Cfe +0,1 + Cfe +2,1|q|2 +� +���� +Hνf =1,fh(q)ξξ′ = +� +Cfh +0,0 + Cfh +xx,2,0|q|2 Cfh +x,1,1(qx − iqy) +Cfh +x,1,1(qx + iqy) +Cfh +xx,2,4|q|2 +� +ξξ′ +Hνf =1,he(q)ξξ′ = +� +Che +xx,2,0|q|2 +Che +x,1,1(qx + iqy) +Che +x,1,1(qx − iqy) Che +0,4 + Che +xx,2,4|q|2 +� +ξξ′ +Hνf =−1,hh(q) = Chh +0 ++ Chh +xx,2|q|2 +(S534) +(We note that Hνf =−1,fe takes the same structure as Hνf =0,−2(q) in Eq.S464). We provide the numerical values of the +parameters in Tab. S2. In Fig. S6 (a), we also compare the dispersion from effective model S533 with the dispersion from +directly evaluating Eq. S399. We note that the effective model (Eq. S533) correctly predicts the long wavelength (small k) +behaviors. In addition, we observe a soft mode with a tiny gap in the half-half sector, as shown in Fig. S6 (b). Here we also +provide the expression of the gap ∆hh(= Chh +0 ) in the half-half sector. Using Eq. S532 and Eq. S526 +∆ = Chh +0 += Hνf =−1,hh(q = 0) = F 22 +−+,−+(q = 0) +(S535) +Using the expression of F 22 +−+,−+(q) (Eq. S398), we have +∆ = N2− − N2+ +2 +− 1 +4 +� +χA +−+,−+(q = 0, iω = 0, 2, 2) + χA ++−,+−(q = 0, iω = 0, 2, 2) +� +(S536) +where Niξ, χA +ξξ′,ξ′ +2ξ2(q, iω, i, j) are defined in Eq. S536. A direct numerical evaluation of Eq. S536 gives ∆ ≈ 0.1meV, which +indicates a small gap at ΓM. +6. +Number of Goldstone modes +We discuss the number of Goldstone modes at νf = −1. The symmetry of the ground state at νf = −1 is U(1)×U(1)×U(2) +(Eq. S335), which has rank 1 + 1 + 4 = 6. The symmetry of the original Hamiltonian at M = 0 has flat U(4) symmetry with +rank 16. We thus have (16 − 6)/2 = 5 Goldstone modes [86]. We find two of the Goldstone modes are given by Seff,off,fe +(full-empty sector), one of the Goldstone modes is given by Seff,off,fh (full-half sector) and the last two Goldstone modes are +given by Seff,off,he(half-empty sector). + +107 +(a) +(b) +FIG. S6. +(a) Excitation spectrum from the effective model in Eq. S533 (brown) and from numerical evaluation of Eq. S399 (blue, orange, +red, green). Blue, orange, red and green denote the full-empty sector, half-empty sector, full-half sector, and half-half sector respectively. (b) +Illustration of soft mode in the half-half sector (green). +Parameter +Cfe +0,0 +Cfe +2,0a2 +M +Cfe +2,1a2 +M +Cfe +0,1 +Cfe +2,2a2 +M Cfe +2,3a2 +M Cfe +1,1aM +Value (meV) +4.0 +1.8 +0.4 +7.9 +1.5 +0.6 +-2.2 +Parameter +Cfh +0,0 +Cfh +2,0a2 +M +Cfh +2,4a2 +M Cfh +1,1aM +Values (meV) 4.5 +0.2 +2.0 +-2.3 +Parameter +Che +0,4 +Che +2,0a2 +M +Che +2,4a2 +M Che +1,1aM +Value (meV) +4.5 +1.8 +0.2 +2.1 +Parameter +Chh +0 +Chh +xx,2a2 +M +Value (meV) +0.1 +0.5 +Supplementary Table S2. Parameters of the effective model in Eq. S533 (and also Eq. S534). +S9. +SINGLE-PARTICLE GREEN’S FUNCTION +A. +Single-particle Green’s function at M = 0 +In this section, we evaluate the single-particle propagator of conduction c electrons in the non-ordered state at νc = 0 and +M = 0, which has been used in Sec. S3 and Sec. S5. The Hamiltonian takes the form of +ˆH′ +c = ˆHc + ˆHMF +V ++ ˆHW − µ +� +k,αηs +c† +k,αηsck,αηs +(S537) +where µ is the chemical potential. We treat ˆHMF +V +is the mean-field Hamiltonian of ˆHV (Eq. S9) and has the form of +ˆHMF +V += V (0) +2Ω0 +NMν2 +c + V (0) +Ω0 +νc +� +k,aηs +(c† +k,aηsck,aηs − 1/2) +(S538) +The filling of f-electron is fixed to be νf at each site. Then ˆHW (Eq. S7 becomes +ˆHW = W +� +k,aηs +νfc† +k,aηsck,aηs +(S539) +Then ˆH′ +c only contains the one-body term +ˆH′ +c = +� +k,a,a′,η,s +H(c,η) +a,a′ (k)c† +k,aηsc† +k,a′ηs + (V (0)νc +Ω0 ++ W − µ) +� +k +c† +k,aηsck,aηs + const +(S540) + +108 +We then consider νc = 0 (νf = νt = 0, −1, −2). This can be realized by setting µ = +V (0)νc +Ω0 ++ W = −W. Then the +single-particle Hamiltonian becomes +ˆH′ +c = +� +k,a,a′,η,s +H(c,η) +a,a′ (k)c† +k,aηsc† +k,a′ηs + const +H(c,η)(k) = +� +02×2 +v⋆(ηkxσ0 + ikyσz) +v⋆(ηkxσ0−ikyσz) +02×2 +� +(S541) +which is equivalent to the non-interacting Hamiltonian of conduction electrons (with an additional constant term). We consider +M = 0 limit, and calculate the single-particle Green’s function. The Green’s functions in the imaginary-time τ are defined as +Gaa′,η(k, τ) = −⟨Tτck,aηs(τ)c† +k,a′ηs(0)⟩ +(S542) +The corresponding Green’s function in the Matsubara frequency domain is +(iωn − H(c,η)(k))−1 = +1 +−ω2 − |v⋆|2k2 +� +�� +iωn +0 +v⋆(ηkx + iky) +0 +0 +iωn +0 +v⋆(ηkx − iky) +v⋆(ηkx − iky) +0 +iωn +0 +0 +v⋆(ηkx + iky) +0 +iωn +� +�� +where k = +� +k2x + k2y, ωn = (2n + 1)π/β and β = 1/T the inverse temperature. where +G11,η(k, τ) = −⟨Tτck,1ηs(τ)c† +k,1ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ +G22,η(k, τ) = −⟨Tτck,2ηs(τ)c† +k,2ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ +G33,η(k, τ) = −⟨Tτck,3ηs(τ)c† +k,3ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ +G44,η(k, τ) = −⟨Tτck,4ηs(τ)c† +k,4ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ +G13,η(k, τ) = −⟨Tτck,1ηs(τ)c† +k3ηs(0)⟩ = 1 +β +� +iωn +v⋆(ηkx + iky) +−ω2n − |v⋆|2k2 e−iωnτ +G24,η(k, τ) = −⟨Tτck,2ηs(τ)c† +k4ηs(0)⟩ = 1 +β +� +iωn +v⋆(ηkx − iky) +−ω2n − |v⋆|2k2 e−iωnτ +G31,η(k, τ) = −⟨Tτck,3ηs(τ)c† +k1ηs(0)⟩ = 1 +β +� +iωn +v⋆(ηkx − iky) +−ω2n − |v⋆|2k2 e−iωnτ +G42,η(k, τ) = −⟨Tτck,4ηs(τ)c† +k2ηs(0)⟩ = 1 +β +� +iωn +v⋆(ηkx + iky) +−ω2n − |v⋆|2k2 e−iωnτ +(S543) +We let +g0(τ, k) = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ +g2,η(τ, k) = 1 +β +� +iωn +v⋆(ηkx + iky) +−ω2n − |v⋆|2k2 e−iωnτ . +Then, we have +G11,η(k, τ) = G22,η(k, τ) = G33,η(k, τ) = G44,η(k, τ) = g0(k, τ) +G13,η(τ, k) = g2,η(τ, k) +, +G24,η(τ, k) = g∗ +2,η(−τ, k) +, +G31,η(τ, k) = g∗ +2,η(−τ, k) +, +G42,η(τ, k) = g2,η(τ, k) +(S544) +and the corresponding Fourier transformation +g0(R, τ) = 1 +N +� +k +g1(k, τ)eik·R +, +g2(R, τ) = 1 +N +� +k +g2(k, τ)eik·R +We next give the analytical expression of g0(R, τ), g1(R, τ). + +109 +1. +g0(R, τ) +We calculate g0(R, τ) in the thermodynamic limit. +g0(R, τ) = +1 +AMBZ +� +g1(k, τ)eik·Rd2k += +1 +AMBZ +� 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτeik·Rd2k +where we have replace � +k with +1 +AMBZ +� +d2k where AMBZ is the area of moire Brillouin zone. We introduce the following +expansion +eik·R = J0(kr) + 2 +� +m=1 +imJm(kr) cos(m(θ − θR)) +(S545) +where k = |k|, r = |R|, θ = arctan(ky/kx), θR = arctan(Ry/Rx) and Jm(x) is the Bessel function [139]. We then have +g0(R, τ) = +1 +AMBZ +� 2π +0 +� Λc +0 +1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 [J0(kr) + 2 +� +m=1 +imJm(kr)]e−iωnτkdkdθ += +2π +AMBZ +� Λc +0 +1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 J0(kr)e−iωnτkdkdθ +(S546) +where Λc is the momentum cutoff. +We then perform Matsubara summation. We use the following identity +1 +β +� +iωn +1 +iωn − ϵe−iωnτ = −(1 − f(ϵ))e−ϵτ +(S547) +where β > τ > 0 and f(x) is the Fermi-Driac function. This can be derived by replacing the summation with contour integral +1 +β +� +iωn +1 +iωn − ϵe−iωnτ = +� +C +dz +2πi +e−zτ +z − ϵ(1 − f(z)) +(S548) +where C = [0+ − i∞, 0+ + i∞] ∪ [0 − +i∞, 0 − −i∞]. We comment that depending on the sign of τ, we could have either +1 − f(z) prefactor or f(z) prefactor [138]. For β > τ > 0 as we considered here, because e−zτ goes to zero at Re[z] > 0, we +need to choose 1 − f(z) prefactor, such that the integrand also goes to zero at Re[z] < 0. There is a pole on the real axis at +z = ϵ. We use the residue theorem and find +1 +β +� +iωn +1 +iωn − ϵe−iωnτ = 2πi(−1) +� 1 +2πi +e−zτ +z − ϵ(1 − f(ϵ)) +� +z=ϵ += 2πi 1 +2πi(−1)e−ϵτ(1 − f(ϵ)) = −e−ϵτ(1 − f(ϵ)) +(S549) +We then perform summation over Matsubara frequency of Eq. S546. Combining Eq. S546 and Eq. S549, at β > τ > 0, we +find +g0(R, τ) = +2π +AMBZ +� Λc +0 +1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 J0(kr)e−iωnτkdkdθ += +2π +AMBZ +� Λc +0 +1 +β +� +iωn +1 +2 +� +1 +iωn − |v⋆|k + +1 +iωn + |v⋆|k +� +e−iωnτJ0(kr)kdk += +2π +AMBZ +� Λc +0 +1 +2(1 − f(|v⋆|k)) +� +− e−|v⋆|kτ − e−|v⋆|k(β−τ) +� +J0(kr)kdk . +(S550) +Finally, we then take the zero-temperature limit and set Λc = ∞ to find the analytical expression: +g1(R, τ) = +2π +AMBZ +� ∞ +0 +1 +2(−e−|v⋆|kτ)J0(kr)kdk = +−π +AMBZ +|v⋆|τ +((v⋆τ)2 + r2)3/2 + +110 +For −β < −τ < 0, we note that +1 +β +� +iωn +1 +iωn − ϵe−iωn(−τ) = (−1) 1 +β +� +iωn +1 +iωn − ϵe−iωn(β−τ) +(S551) +Using Eq. S547, we have +1 +β +� +iωn +1 +iωn − ϵe−iωn(−τ) = (−1) 1 +β +� +iωn +1 +iωn − ϵe−iωn(β−τ) = (1 − f(ϵ))e−(β−τ)ϵ +(S552) +Using Eq S552, g0(R, −τ) at negative imaginary time −β < −τ < 0 can be evaluated as +g0(R, −τ) = +2π +AMBZ +� Λc +0 +1 +β +� +iωn +1 +2 +� +1 +iωn − |v⋆|k + +1 +iωn + |v⋆|k +� +eiωnτJ0(kr)kdk += +2π +AMBZ +� Λc +0 +1 +2(1 − f(|v⋆|k)) +� +e−|v⋆|k(β−τ) + e−|v⋆|kτ +� +J0(kr)kdk . +Similarly, we let Λc = ∞ and β = ∞. Then +g0(R, −τ) = +2π +AMBZ +� ∞ +0 +1 +2(e−|v⋆|kτ)J0(kr)kdk = +π +AMBZ +|v⋆|τ +((v⋆τ)2 + r2)3/2 +In summary, +g0(R, τ) = −sgn(τ) +π +AMBZ +|v⋆τ| +((v⋆τ)2 + r2)3/2 +(S553) +where we observe g0(R, τ) ∝ sgn(τ). We note that for the particle-hole symmetric system (realized at νc = 0), we have +g0(R, β − τ) = g0(R, τ). Then g0(R, −τ) = −g(R, β − τ) = −g0(R, τ). +2. +g2,η(τ, R) +We calculate g2,η(τ, R) in this section. For β > τ > 0 +g2,η(τ, R) = +1 +AMBZ +� +g2,η(τ, k)eik·Rd2k += +1 +AMBZβ +� � +iωn +v⋆(ηkx + iky) +−ω2n − |v⋆|2k2 e−iωnτeik·Rd2k += +1 +AMBZβ +� � +iωn +v⋆(η cos(θ) + i sin(θ)) +−ω2n − |v⋆|2k2 +e−iωnτ +� +J0(kr) + 2 +� +m=1 +imJm(kr) cos(m(θ − θR)) +� +k2dkdθ += +1 +AMBZβ +� � +iωn +v⋆2πi(η cos(θR) + i sin(θR)) +−ω2n − |v⋆|2k2 +e−iωnτJ1(kr)k2dk += +1 +AMBZβ +� � +iωn +|v⋆|2πi(η cos(θR) + i sin(θR)) +2|v⋆|k +[ +1 +iωn − |v⋆k| − +1 +iωn + |v⋆k|]e−iωnτJ1(kr)k2dk += +1 +AMBZ +� |v⋆|2πi(η cos(θR) + i sin(θR)) +2|v⋆|k +(1 − f(|v⋆|k))(e−|v⋆|kτ − e−|v⋆|k(β−τ)J1(kr)k2dk +We let β → ∞ and set the momentum cutoff to infinity. +g2,η(τ, R) =iπ(η cos(θR) + i sin(θR) +AMBZ +� ∞ +0 +e−|v⋆|kτJ1(kr)kdk = iπ(η cos(θR) + i sin(θR)r +AMBZ((v⋆τ)2 + r2)3/2 + +111 +For negative time (−β < −τ < 0) +g2,η(−τ, R) = −g2,η(β − τ, R) += − +1 +AMBZ +� |v⋆|2πi(η cos(θR) + i sin(θR)) +2|v⋆|k +(1 − f(|v⋆|k))(e−|v⋆|k(β−τ) − e−|v⋆|k(τ)J1(kr)k2dk +We let β → ∞ and set the momentum cutoff to infinity. +g2,η(−τ, R) = −iπ(η cos(θR) + i sin(θR) +AMBZ +� ∞ +0 +e−|v⋆|kτJ1(kr)kdk = −iπ(η cos(θR) + i sin(θR)r +AMBZ((v⋆τ)2 + r2)3/2 +In summary, +g2,η(τ, R) = sgn(τ)iπ(η cos(θR) + i sin(θR)r +AMBZ((v⋆τ)2 + r2)3/2 +(S554) +and also g2,−(τ, R) = [g2,+(τ, R)]∗ . +B. +Single-particle Green’s function at M ̸= 0 +In this section, we calculate the single-particle Green’s function at M ̸= 0, νc = 0, which has been used in Sec. S3. The +Hamiltonian is given in Eq. S541 (but with a non-zero M) and is also listed below. +ˆH′ +c = +� +k,η,a,a′,s +H(c,η) +aa′ (k)c† +k,aηsck,a′ηs + const +H(c,η)(k) = +� +02×2 +v⋆(ηkxσ0 + ikyσz) +v⋆(ηkxσ0 + ikyσz) +Mσx +� +(S555) +The corresponding Green’s function in the Matsubara frequency ωn is +(iωn − H(c,η)(k))−1 += +1 +−ω2 − |v⋆|2k2 +� +�� +iωn +0 +v⋆(ηkx + iky) +0 +0 +iωn +0 +v⋆(ηkx − iky) +v⋆(ηkx − iky) +0 +iωn +0 +0 +v⋆(ηkx + iky) +0 +iωn +� +�� ++ +M +(−ω2n − |v⋆|2k2)2 +� +�� +0 +|v⋆|2(ηkx + iky)2 +0 +iωv⋆(ηkx + iky) +|v⋆|2(ηkx − iky)2 +0 +iωv⋆(ηkx − iky) +0 +0 +iωv⋆(ηkx + iky) +0 +−ω2 +iωv⋆(ηkx − iky) +0 +−ω2 +0 +� +�� + O(M 2) +(S556) +where k = +� +k2x + k2y and we expand in powers of M. We consider the following single-particle Green’s functions in this case +G33,η(k, τ) = −⟨Tτck,3ηs(τ)ck,3ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ + O(M 2) +G44,η(k, τ) = −⟨Tτck,4ηs(τ)ck,4ηs(0)⟩ = 1 +β +� +iωn +iωn +−ω2n − |v⋆|2k2 e−iωnτ + O(M 2) +G34,η(k, τ) = −⟨Tτck,3ηs(τ)†ck,4ηs⟩ = 1 +β +� +iωn +−Mω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ + O(M 2) +G43,η(k, τ) = −⟨Tτck,4ηs(τ)†ck,3ηs⟩ = 1 +β +� +iωn +−Mω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ + O(M 2) +We let +g1(k, τ) = 1 +β +� +iωn +−Mω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ . +(S557) + +112 +Then, we have +G33,η(k, τ) = G44,η(k, τ) = g0(k, τ) +G34,η(k, τ) = G43,η(k, τ) = g1(k, τ) +and the corresponding Fourier transformation +g1(R, τ) = +1 +NM +� +k +g1(k, τ)eik·R +We next give the analytical expression of g1(R, τ) at zero temperature β → ∞ and thermodynamic limit. In the thermody- +namic limit, we can replace the momentum summation with the momentum integral +1 +NM +� +k +→ +1 +AMBZ +� +|k|<Λc +where AMBZ is the area of first moir´e Brillouin zone and Λc is the momentum cutoff. In addition, during the calculation, we +take Λc = ∞ to obtain the analytical expression. +g1(R, τ) = +1 +AMBZ +� +g1(k, τ)eik·Rd2k += +M +AMBZ +� 1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτeik·Rd2k +Using Eq. S545, we have +g1(R, τ) = +M +AMBZ +� 2π +0 +� Λc +0 +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 [J0(kr) + 2 +� +m=1 +imJm(kr)]e−iωnτkdkdθ += 2πM +AMBZ +� Λc +0 +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 J0(kr)e−iωnτkdk +(S558) +Now, we evaluate the following Matsubara summation via contour integral: +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ +We fist consider β > τ > 0 case. We change Matsubara summation to a contour integral +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ = +� +C +dz +2πi(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ +where the contour C = [0+ + i∞, 0+ − i∞] ∪ [0− − i∞, 0− + i∞]. Since, the poles only appear on the real axis (Im[z] = 0), +We distort the contour to C′ = [∞ + i0+, −∞ + i0+] ∪ [−∞ − i0+, ∞ − i0+]. This gives +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ = +� +C′ +dz +2πi(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ +We aim to evaluate the integral via residual theorem. The residues around two poles at z = ±|v⋆k| are +Res +� +(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ, |v⋆k| +� += e−|v⋆k|τ +4|v⋆k| +� +(1 − |v⋆k|τ)f(−|v⋆k|)2 + [1 + |v⋆k|(β − τ)]f(−|v⋆k|)(1 − f(−|v⋆k|)) +� +Res +� +(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ, −|v⋆k| +� += −e|v⋆k|τ +4|v⋆k| +� +(1 + |v⋆k|τ)f(|v⋆k|)2 + [1 − |v⋆k|(β − τ)]f(−|v⋆k|)(1 − f(−|v⋆k|)) +� + +113 +Using the residue theorem, we have +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωnτ = Res +� +(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ, |v⋆k| +� ++ Res +� +(f(z) − 1) +z2 +(z2 − |v⋆|2k2)2 e−zτ, −|v⋆k| +� +=−|v⋆k|τ cosh(|v⋆k|(β − τ)) + |v⋆k|(β − τ) cosh(|v⋆k|τ) + sinh(|v⋆k|(β − τ)) − sinh(|v⋆k|τ) +4|v⋆k|(1 + cosh(|v⋆k|β) +(S559) +where β > τ > 0. For the negative time, we can use the fact that +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωn(−τ) = − 1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωn(β−τ) +(S560) +Then we can replace −τ with β − τ and add a minus sign to Eq. S559. This leads to +1 +β +� +iωn +−ω2 +n +(−ω2n − |v⋆|2k2)2 e−iωn(−τ) += − −|v⋆k|τ cosh(|v⋆k|τ) + |v⋆k|τ cosh(|v⋆k|(β − τ)) + sinh(|v⋆k|τ) − sinh(|v⋆k|(β − τ)) +4|v⋆k|(1 + cosh(|v⋆k|β) +(S561) +Combining Eq. S558 and Eq. S559 we have (τ > 0) +g1(R, τ) += 2πM +AMBZ +� Λc +0 +J0(kr)−|v⋆k|τ cosh(|v⋆k|(β − τ)) + |v⋆k|(β − τ) cosh(|v⋆k|τ) + sinh(|v⋆k|(β − τ)) − sinh(|v⋆k|τ) +4|v⋆k|(1 + cosh(|v⋆k|β) +kdk +(S562) +At zero temperature and Λc = ∞, it becomes +g1(R, τ) = 2πM +AMBZ +� ∞ +0 +J0(kr)−e−|v⋆k|τ(−1 + |v⋆k|τ) +4|v⋆k| +kdk += − +πM +2AMBZ|v⋆| +r2 +� +|v⋆τ|2 + r2 +�3/2 +For the negative time, combining Eq. S558 and Eq. S559, we have (τ > 0) +g1(R, −τ) = 2πM +AMBZ +� Λc +0 +J0(kr)(−1)−|v⋆k|τ cosh(|v⋆k|τ) + |v⋆k|τ cosh(|v⋆k|(β − τ)) + sinh(|v⋆k|τ) − sinh(|v⋆k|(β − τ)) +4|v⋆k|(1 + cosh(|v⋆k|β) +kdk +At zero temperature and Λc = ∞, it becomes +g1(R, −τ) = 2πM +AMBZ +� ∞ +0 +J0(kr)−e−|v⋆k|τ(−1 + |v⋆k|τ) +4|v⋆k| +kdk += − +πM +2AMBZ|v⋆| +r2 +� +|v⋆τ|2 + r2 +�3/2 +In summary, we have +g1(R, τ) = − +πM +2AMBZ|v⋆| +r2 +� +|v⋆τ|2 + r2 +�3/2 +(S563) +Here, unlike g0(R, τ), g1(R, τ) is not proportional to sgn(τ). This is because g1(R, τ) = −⟨Tτck,aηs(τ)ck,aηs(0)⟩ which is +the correlation functions of two fermionic operators with same k, αηs indices. However, g2(R, τ) = −⟨Tτck,3ηs(τ)ck,4ηs(0)⟩ +is the correlation functions of two fermionic operators with different k, αηs indices. Thus, the particle-hole symmetry will not +enforce e g2(R, τ) ∝ sgn(τ). + +114 +S10. +FOUR-FERMIOIN CORRELATION FUNCTION I +We now calculate the following correlation function in Eq. S104 +χc(R, τ, ξ, ξ2) = − +� +k,q +� +a1,a2=3,4 +� +ηs +1 +2 +1 +2N 2 +M +δξ,(−1)a−1ηδξ2,(−1)a2−1ηGa2a,η(k + q, −τ)Gaa2,η(k, τ)iq·R +For ξ = ξ2, +χc(R, τ, ξ, ξ) = − +� +k,q +� +a1,a2=3,4 +� +ηs +1 +2 +1 +2N 2 +M +δξ,(−1)a−1ηδξ,(−1)a2−1ηGa2a,η(k + q, −τ)Gaa2,η(k, τ)iq·R += − +� +k +1 +N 2 +M +g0(k + q, −τ)g0(k, τ)eiq·R + o(M 2) += − g0(R, −τ)g0(−R, τ) + o(M 2) = +π2 +A2 +MBZ +|v⋆τ|2 +(|v⋆τ|2 + r2)3 + O(M 3) +(S564) +where we use the analytical expression of Green’s function at β = ∞, Λc = ∞ as shown in Eq. S553. +For ξ = −ξ2, +χc(R, τ, ξ, −ξ) = − +� +k,q +� +a1,a2=3,4 +� +ηs +1 +2 +1 +2N 2 +M +δξ,(−1)a−1ηδ−ξ,(−1)a2−1ηGa2a,η(k + q, −τ)Gaa2,η(k, τ)iq·R += − +� +k +1 +N 2 +M +g1(k + q, −τ)g1(k, τ)eiq·R + O(M 3) += − g1(R, −τ)g1(−R, τ) + o(M 2) = − +π2M 2 +4A2 +MBZ|v⋆|2 +r4 +� +|v⋆τ|2 + r2 +�3 + O(M 3) +(S565) +where we use the analytical expression of Green’s function at β = ∞, Λc = ∞ as shown in Eq. S563. +S11. +FOUR-FERMIOIN CORRELATION FUNCTION II +In this section, we evaluate the four-fermion correlations at M = 0, ν = νf = 0, −1, −2, νc = 0, that has been used in +Sec. S5. Unlike Sec. S10, we consider M = 0 limit and work in the ψ basis. We consider the following correlators +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ +2ξ2) +µ2ν2 +(r′ +2, r2, 0) :⟩0,con += − +� +n1,n2,n3,n4 +[T µν]n1,n2[T µ2ν2]n3,n4⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n3 (0)⟩0⟨ψc′,ξ2 +r2,n4(τ)ψc′,ξ,† +r1n1 (0)⟩0 +⟨: ˆΣ(c′′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ2ξ′ +2) +µ2ν2 +(r′ +2, r2, 0) :⟩0,con += − +� +n1,n2,n3,n4 +[T µν]n1,n2[T µ2ν2]n3,n4⟨ψc′′,ξ′ +r′ +1,n2(τ)ψc′′,ξ′ +2,† +r′ +2,n3 (0)⟩0⟨ψc′′,ξ2 +r2,n4(τ)ψc′′,ξ,† +r1n1 (0)⟩0 +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ +2ξ2) +µ2ν2 +(r′ +2, r2, 0) :⟩0,con += − +� +n1,n2,n3,n4 +[T µν]n1,n2[T µ2ν2]n3,n4⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′′,ξ′ +2,† +r′ +2,n3 (0)⟩0⟨ψc′′,ξ2 +r2,n4(τ)ψc′,ξ,† +r1n1 (0)⟩0 +(S566) +Since S0 has flat U(4) symmetry, only components with µν = µ2ν2 can be non-zero. In addition, at M = 0, the single- +particle Green’s function −⟨Tτck,αηs(τ)ck′,α′η′s′(0)⟩ equals to zero if (−1)α+1η ̸= (−1)α′+1η′. This is because the single- +particle Hamiltonian of conduction electrons does not have a hybridization term between two c electrons with opposite ξ indices. +Therefore, only components with ξ = ξ2, ξ′ = ξ′ +2 is non-zero, and we only need to calculate +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con +⟨: ˆΣ(c′′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con + +115 +where ⟨AB⟩0,con = ⟨AB⟩0 − ⟨A⟩0⟨B⟩0 +We next prove all µν components are equivalent. For the components of µν ̸= 00, they are connected by flat U(4) rotation +and are equivalent. To show it, we consider a SU(4) ⊂ U(4) rotation g that will rotates ˆΣ(c′,ξξ′) +µν +(µν ̸= 00). +gˆΣ(c′,ξξ′) +µν +g−1 = +� +µ2ν2 +Rµν,µ2ν2(g)ˆΣ(c′,ξξ′) +µ2ν2 +(S567) +where Rµν,µ2ν2(g) is the representation matrix of SU(4) rotation corresponding to the adjoint representation of SU(4) (Note +that {ˆΣ(c′,ξξ′) +µν +}µν̸=00 form an adjoint representation of SU(4) group (subgroup of U(4)). Due to the SU(4) symmetry, we have +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨g : ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) : g−1⟩0,con += +� +µ2ν2,µ3ν3 +Rµν,µ2ν2(g)Rµν,µ3ν3(g)⟨: ˆΣ(c′,ξξ′) +µ2ν2 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µ3ν3 +(r′ +2, r2, 0) :⟩0,con += +� +µ2ν2 +Rµν,µ2ν2(g)Rµν,µ3ν3(g)⟨: ˆΣ(c′,ξξ′) +µ2ν2 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µ2ν2 +(r′ +2, r2, 0) :⟩0,con +(S568) +Eq. S568 needs to be satisfied for any SU(4) rotation g, which requires all the µν ̸= 00 components are equivalent (here we +pick 0z components as a representation) +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con, +µν ̸= 00 +(S569) +Similarly, for other correlators, we also have +⟨: ˆΣ(c′′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con, +µν ̸= 00 +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con, +µν ̸= 00 +(S570) +We next prove ⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con. +⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con takes the form of +⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con += − +� +n1,n2,n3,n4 +[T 0z]n1,n2[T 0z]n3,n4⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n3 (0)⟩0⟨ψc′,ξ2 +r2,n4(τ)ψc′,ξ,† +r1,n1(0)⟩0 += − 1 +4 +� +n1,n2 +sn1sn3⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n2 (0)⟩0⟨ψc′,ξ2 +r2,n1(τ)ψc′,ξ,† +r1n1 (0)⟩0δn1,n2δn3,n4δn2,n3δn1,n4 += − 1 +4 +� +n1,n2 +⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n2 (0)⟩0⟨ψc′,ξ2 +r2,n1(τ)ψc′,ξ,† +r1n1 (0)⟩0 +where sn ∈ {+1, −1} is the spin index of ψc′,ξ′ +r,n electrons. For µν = 00 component, we have +⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con += − 1 +4 +� +n1,n2,n3,n4 +[T 00]n1,n2[T 00]n3,n4⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n3 (0)⟩0⟨ψc′,ξ2 +r2,n4(τ)ψc′,ξ,† +r1,n1(0)⟩0 += − 1 +4 +� +n1,n2 +⟨ψc′,ξ′ +r′ +1,n2(τ)ψc′,ξ′ +2,† +r′ +2,n2 (0)⟩0⟨ψc′,ξ2 +r2,n1(τ)ψc′,ξ,† +r1n1 (0)⟩0δn1,n2δn3,n4δn2,n3δn1,n4 +=⟨: ˆΣ(c′,ξξ′) +0z +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +0z +(r′ +2, r2, 0) :⟩0,con . +Therefore all µν components are equivalent. For the same reason, this is true for all three correlators, and we have +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con +⟨: ˆΣ(c′′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = ⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con +(S571) + +116 +Using Wick’s theorem and single-particle Green’s function, the first correlator in Eq. S566 becomes +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con +=⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con = 1 +4 +� +a,b +⟨: ψc′,ξ,† +r1,a (τ)ψc′,ξ′ +r′ +1,a (τ) :: ψc′,ξ′,† +r′ +2,b +(0)ψc′,ξ +r2,b(0) :⟩0,con += − 1 +4 +� +a,b +⟨ψc′,ξ +r2,b(0)ψc′,ξ,† +r1,a (τ)⟩⟨ψc′,ξ′ +r′ +1,a (τ)ψc′,ξ′,† +r′ +2,b +(0)⟩ = −g0(r2 − r1, −τ)g0(r′ +1 − r′ +2, τ) . +(S572) +The second correlator in Eq. S566 is +⟨: ˆΣ(c′′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con +=⟨: ˆΣ(c′′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con = 1 +4 +� +a,b +⟨: ψc′′,ξ,† +r1,a (τ)ψc′′,ξ′ +r′ +1,a (τ) :: ψc′′,ξ′,† +r′ +2,b +(0)ψc′′,ξ +r2,b (0) :⟩0,con += − 1 +4 +� +a,b +⟨ψc′′,ξ +r2,b (0)ψc′′,ξ,† +r1,a (τ)⟩⟨ψc′′,ξ′ +r′ +1,a (τ)ψc′′,ξ′,† +r′ +2,b +(0)⟩ = −g0(r2 − r1, −τ)g0(r′ +1 − r′ +2, τ) . +(S573) +The third correlator in Eq. S566 is +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con +=⟨: ˆΣ(c′,ξξ′) +00 +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +00 +(r′ +2, r2, 0) :⟩0,con = 1 +4 +� +a,b +⟨: ψc′,ξ,† +r1,a (τ)ψc′,ξ′ +r′ +1,a (τ) :: ψc′′,ξ′,† +r′ +2,b +(0)ψc′′,ξ +r2,b (0) :⟩0,con += − 1 +4 +� +a,b +⟨ψc′′,ξ +r2,b (0)ψc′,ξ,† +r1,a (τ)⟩⟨ψc′,ξ′ +r′ +1,a (τ)ψc′′,ξ′,† +r′ +2,b +(0)⟩ = −1 +4 +� +a +⟨ψc′′,ξ +r2,a(0)ψc′,ξ,† +r1,a (τ)⟩⟨ψc′,ξ′ +r′ +1,a (τ)ψc′′,ξ′,† +r′ +2,a +(0)⟩ . +(S574) +For Eq. S574, we consider each ξ, ξ′ combination. For ξ = ξ′ = +1: +⟨: ˆΣ(c′,+1+1) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,+1+1) +µν +(r′ +2, r2, 0) :⟩0,con = −1 +4 +� +a +⟨ψc′′,+1 +r2,a (0)ψc′,+1,† +r1,a +(τ)⟩⟨ψc′,+1 +r′ +1,a (τ)ψc′′,+1,† +r′ +2,a +(0)⟩ += − 1 +2 +� +G31,+(r2 − r1, −τ)G13,+(r′ +1 − r′ +2, τ) + G42,−(r2 − r1, −τ)G24,−(r′ +1 − r′ +2, τ) +� += − 1 +2[g∗ +2,+(r1 − r2, τ)g2,+(r′ +1 − r′ +2, τ) + g2,−(r2 − r1, −τ)g∗ +2,−(r′ +2 − r′ +1, −τ)] . +(S575) +For ξ = ξ′ = −1: +⟨: ˆΣ(c′,−1−1) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,−1−1) +µν +(r′ +2, r2, 0) :⟩0,con = −1 +4 +� +a +⟨ψc′′,−1 +r2,a (0)ψc′,−1,† +r1,a +(τ)⟩⟨ψc′,−1 +r′ +1,a (τ)ψc′′,−1,† +r′ +2,a +(0)⟩ += − 1 +2 +� +G31,−(r2 − r1, −τ)G13,−(r′ +1 − r′ +2, τ) + G42,+(r2 − r1, −τ)G24,+(r′ +1 − r′ +2, τ) +� += − 1 +2[g∗ +2,−(r1 − r2, τ)g2,−(r′ +1 − r′ +2, τ) + g2,+(r2 − r1, −τ)g∗ +2,+(r′ +2 − r′ +1, −τ)] . +(S576) +For ξ = 1, ξ′ = −1, +⟨: ˆΣ(c′,+1−1) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,−1+1) +µν +(r′ +2, r2, 0) :⟩0,con = −1 +4 +� +a +⟨ψc′′,+1 +r2,a (0)ψc′,−1,† +r1,a +(τ)⟩⟨ψc′,−1 +r′ +1,a (τ)ψc′′,+1,† +r′ +2,a +(0)⟩ += − 1 +2 +� +G31,+(r2 − r1, −τ)G13,−(r′ +1 − r′ +2, τ) + G42,−(r2 − r1, −τ)G24,+(r′ +1 − r′ +2, τ) +� += − 1 +2[g∗ +2,+(r1 − r2, τ)g2,−(r′ +1 − r′ +2, τ) + g2,−(r2 − r1, −τ)g∗ +2,+(r′ +2 − r′ +1, −τ)] . +(S577) + +117 +For ξ = −1, ξ′ = 1, +⟨: ˆΣ(c′,−1+1) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,+1−1) +µν +(r′ +2, r2, 0) :⟩0,con = −1 +4 +� +a +⟨ψc′′,−1 +r2,a (0)ψc′,+1,† +r1,a +(τ)⟩⟨ψc′,+1 +r′ +1,a (τ)ψc′′,−1,† +r′ +2,a +(0)⟩ += − 1 +2 +� +G31,−(r2 − r1, −τ)G13,+(r′ +1 − r′ +2, τ) + G42,+(r2 − r1, −τ)G24,−(r′ +1 − r′ +2, τ) +� += − 1 +2[g∗ +2,−(r1 − r2, τ)g2,+(r′ +1 − r′ +2, τ) + g2,+(r2 − r1, −τ)g∗ +2,−(r′ +2 − r′ +1, −τ)] . +(S578) +In summary, combining Eq. S575, Eq. S576, Eq. S577 and Eq. S578 +⟨: ˆΣ(c′,ξξ′) +µν +(r1, r′ +1, τ) :: ˆΣ(c′′,ξ′ξ) +µν +(r′ +2, r2, 0) :⟩0,con = − 1 +2[g∗ +2,ξ(r1 − r2, τ)g2,ξ′(r′ +1 − r′ +2, τ) + g2,−ξ(r2 − r1, −τ)g∗ +2,−ξ′(r′ +2 − r′ +1, −τ)] += − g∗ +2,ξ(r1 − r2, τ)g2,ξ′(r′ +1 − r′ +2, τ) . +(S579) +S12. +FOURIER TRANSFORMATIONS +In this section, we provide the Fourier transformation of the following functions that have been used in Sec. S5, Eq. S234. +� +R +1 +|R|3 e−ik· R, +� +R +1 +|R|5 e−ik· R, +� +R +Rx +|R|5 e−ik· R, +� +R +Ry +|R|5 e−ik· R, +� +R +R2 +x − R2 +y − 2iξRxRy +|R|7 +e−ik· R +(S580) +We first consider the Fourier transformation of |R|−3 +� +R +1 +|R|3 e−ik· R +We replace the summation with the integral, � +R → AMBZ +4π2 +� +d2R and have +AMBZ +4π2 +� +dθRrdr 1 +r3 +� +J0(kr) + 2 +� +m=1 +(−i)mJm(kr) cos(m(θk − θR)) +� +=AMBZ +2π +� +1 +r2 J0(kr)dr +We then regularize the integral by replacing r with +� +r2 + a2 +M where aM is the moir´e lattice constant. Then we find +� +R +1 +|R|3 e−ik· R ≈ AMBZ +2π +� +1 +(r2 + a2 +M)J0(kr)dr = AMBZ +4aM +(I0(kaM) − L0(kaM)) +where In(x) is the modified Bessel function of the first kind [139], Ln(x) is the modified Struve function [139]. It is also useful +to evaluate its behavior at small k +AMBZ +4aM +(I0(kaM) − L0(kaM)) = AMBZ +4aM +(1 − 2aM +π +k + a2 +Mk2 +4 +) + o(k2) +We next consider 1/|R|5. Following the same strategy +� +R +1 +|R|5 e−ik· R ≈AMBZ +4π2 +� +dθRdr +1 +(r2 + a2 +M)2 +� +J0(kr) + 2 +� +m=1 +(−i)mJm(kr) cos(m(θk − θR)) +� +=AMBZ +2π +� +1 +(r2 + a2 +M)2 J0(kr)dr += AMBZ +24πka4 +M +� +3π(−2 + k2a2 +M)L1(kaM) + qaM +� +4qaM + 3πI0(qaM) − 3πqaMI1(qaM) − 3πL2(qaM) +�� +≈AMBZ +8a3 +M +(1 − 1 +4(kaM)2) + o(k2) + +118 +In the final line, we evaluated the behavior at small k. +We next consider the Fourier transformation of Rx/r5. Following the same strategy +� +R +Rx +|R|5 e−ik· R ≈AMBZ +4π2 +� +dθRdr +cos(θR) +(r2 + a2 +M)3/2 +� +J0(kr) + 2 +� +m=1 +(−i)mJm(kr) cos(m(θk − θR)) +� +=−i cos(θk)AMBZ +2π +� +dr +1 +(r2 + a2 +M)3/2 J1(kr) +=−iAMBZkx +2πk2a3 +M +� +1 − e−kaM (1 + kaM) +� +≈−iAMBZkx +2πaM +(1 +2 − kaM +3 +) + o(k2) +In the final line, we evaluated the behavior at small k. +We next consider the Fourier transformation of Ry/r5. Following the same strategy +� +R +Ry +|R|5 e−ik· R ≈AMBZ +4π2 +� +dθRdr +sin(θR) +(r2 + a2 +M)3/2 +� +J0(kr) + 2 +� +m=1 +(−i)mJm(kr) cos(m(θk − θR)) +� +=−i sin(θk)AMBZ +2π +� +dr +1 +(r2 + a2 +M)3/2 J1(kr) +=−iAMBZky +2πk2a3 +M +� +1 − e−kaM (1 + kaM) +� +≈−iAMBZky +2πaM +(1 +2 − kaM +3 +) + o(k2) +In the final line, we evaluated the behavior at small k. +We next consider the Fourier transformation of +R2 +x−R2 +y−2iξRxRy +|R|7 +. +� +R +R2 +x − R2 +y − 2iξRxRy +|R|7 +e−ik· R +≈AMBZ +4π2 +� +dθRdrcos(2θR) − iξ sin(2θR) +(r2 + a2 +M)2 +� +J0(kr) + 2 +� +m=1 +(−i)mJm(kr) cos(m(θk − θR)) +� +=−2πAMBZ +4π2 +� +drcos(2θk) + iξ sin(2θk) +(r2 + a2 +M)2 +J2(kr) += − AMBZk +2π +(k2 +x − k2 +y + 2iξkxky) +� +− 1 +30 + +π +4k3a3 +M +� +I2(kaM) + kaMI3(kaM) +� +− +π +4k3a3 +M +� +L2(kaM) + kaML3(kaM) +�� +≈ − AMBZ +64aM +(k2 +x − k2 +y + 2iξkxky) +In the final line, we evaluated the behavior at small k. In(x) is the modified Bessel function of the first kind [139], Ln(x) is the +modified Struve function [139]. + +119 +In summary, +� +R +1 +|R|3 e−ik· R =AMBZ +4aM +(I0(kaM) − L0(kaM)) ≈ AMBZ +4aM +(1 − 2aM +π +k + a2 +Mk2 +4 +) +� +R +1 +|R|5 e−ik· R = AMBZ +24πka4 +M +� +3π(−2 + k2a2 +M)L1(kaM) + qaM +� +4qaM + 3πI0(qaM) − 3πqaMI1(qaM) +− 3πL2(qaM) +�� +≈ AMBZ +8a3 +M +(1 − 1 +4(kaM)2) +� +R +Rx +|R|5 e−ik· R =−iAMBZkx +2πk2a3 +M +� +1 − e−kaM (1 + kaM) +� +≈ −iAMBZkx +2πaM +(1 +2 − kaM +3 +) +� +R +Ry +|R|5 e−ik· R =−iAMBZky +2πk2a3 +M +� +1 − e−kaM (1 + kaM) +� +≈ −iAMBZky +2πaM +(1 +2 − kaM +3 +) +� +R +R2 +x − R2 +y − 2iξRxRy +|R|7 +e−ik· R = − AMBZk +2π +(k2 +x − k2 +y + 2iξkxky) +� +− 1 +30 + +π +4k3a3 +M +� +I2(kaM) + kaMI3(kaM) +� +− +π +4k3a3 +M +� +L2(kaM) + kaML3(kaM) +�� +≈ −AMBZ +64aM +(k2 +x − k2 +y − 2iξkxky) +(S581) +S13. +SINGLE PARTICLE GREEN’S FUNCTION IN THE ORDERED STATE +In this section, we discuss the single-particle Green’s function of the conduction electrons in the ordered phase that has been +used in Sec. S8. The Hamiltonian of c electrons is defined in Eq. S359 and is also written here +ˆHc,order += +� +k,i +� +ψξ=+,c′,† +k,i +ψξ=+,c′′,† +k,i +ψξ=−,c′,† +k,i +ψξ=−,c′′,† +k,i +� +� +���� +E+,i +0,k + E+,i +3,k +v⋆(kx + iky) vi +k(kx − iky) +0 +v⋆(kx − iky) E+,i +0,k − E+,i +3,k +0 +0 +vi +k(kx + iky) +0 +E−,i +0,k + E−,i +3,k +v⋆(kx − iky) +0 +0 +v⋆(kx + iky) E−,i +0,k − E−,i +3,k +� +���� +� +����� +ψξ,c′ +k,i +ψξ,c′′ +k,i +ψξ=−,c′ +k,i +ψξ=−,c′′ +k,i +� +����� +(S582) +The single-particle Green’s function can be obtained by +gξξ′,i +aa′ (k, iω) = +� +� +� +� +� +� +� +� +� +iωI4 − +� +���� +E+,i +0,k + E+,i +3,k +v⋆(kx + iky) vi +k(kx − iky) +0 +v⋆(kx − iky) E+,i +0,k − E+,i +3,k +0 +0 +vi +k(kx + iky) +0 +E−,i +0,k + E−,i +3,k +v⋆(kx − iky) +0 +0 +v⋆(kx + iky) E−,i +0,k − E−,i +3,k +� +���� +� +� +� +� +� +� +� +� +� +ξa,ξ′a′ +(S583) +where a, a′ ∈ {c′, c′′}. By diagonalizing the matrix, we have +gξξ,i +c′c′ (k, iω) = f1(|k|, iω, ξ, i) +gξξ,i +c′′c′′(k, iω) = f2(|k|, iω, ξ, i) +gξξ,i +c′c′′(k, iω) = v⋆(kx + iξky)f3(|k|, iω, ξ, i) +gξξ,i +c′′c′(k, iω) = v⋆(kx − iξky)f3(|k|, iω, ξ, i) +gξ−ξ,i +c′c′ +(k, iω) = v′ +⋆(kx − iξky)f4(|k|, iω, i) +gξ−ξ,i +c′′c′′ (k, iω) = v′ +⋆v2 +⋆(kx − iξky)3f5(|k|, iω, i) +gξ−ξ,i +c′c′′ (k, iω) = v′ +⋆v⋆(kx − iξky)2f6(|k|, iω, ξ, i) +gξ−ξ,i +c′′c′ (k, iω) = v′ +⋆v⋆(kx + iξky)2f6(|k|, iω, −ξ, i) +(S584) + +120 +where we introduce the following functions +F|k|,i = +�� +iω − E+,i +0,k +�2 +− v2 +⋆|k|2 − +� +E+,i +3,k +�2 +− v2 +⋆|k|2 +��� +iω − E−,i +0,k +�2 +− v2 +⋆|k|2 − +� +E−,i +3,k +�2 +− v2 +⋆|k|2 +� +f1(|k|, iω, ξ, i) = +1 +F|k|,i +(iω + Eξ,i +3,k − Eξ,i +0,k) +� +(iω − E−ξ,i +0,k )2 − (E−ξ,i +3,k )2 − v2 +⋆|k|2 +� +f2(|k|, iω, ξ, i) = +1 +F|k|,i +� +(iω − Eξ,i +3,k − Eξ,i +0,k) +� +(iω − E−ξ,i +0,k )2 − (E−ξ,i +3,k )2 − v2 +⋆|k|2 +� ++ (iω + E−ξ,i +3,k − E−ξ,i +0,k )(v′ +⋆)2|k|2 +� +f3(|k|, iω, ξ, i) = +1 +F|k|,i +� +(iω − E−ξ,i +0,k )2 − (E−ξ,i +3,k )2 − v2 +⋆|k|2 +� +f4(|k|, iω, i) = +1 +F|k|,i +(iω + E+,i +3,k − E+,i +0,k)(iω + E−,i +3,k − E−,i +0,k) +f5(|k|, iω, i) = +1 +F|k|,i +f6(|k|, iω, ξ, i) = +1 +F|k|,i +(iω + Eξ,i +3,k − Eξ,i +0,k) +(S585) +where +F|k|,i = +�� +iω − E+,i +0,k +�2 +− v2 +⋆|k|2 − +� +E+,i +3,k +�2 +− v2 +⋆|k|2 +��� +iω − E−,i +0,k +�2 +− v2 +⋆|k|2 − +� +E−,i +3,k +�2 +− v2 +⋆|k|2 +� +− (v′ +⋆)2|k|2(iω − E+,i +0,k + E+,i +3,k)(iω − E−,i +0,k + E−,i +3,k) . +(S586) +We also mention that Eξ,i +3,k, Eξ,i +0,k, F|k|,i, f1(|k|, iω, ξ, i), f2(|k|, iω, ξ, i), f3(|k|, iω, ξ, i), f4(|k|, iω, i), f5(|k|, iω, i), f6(|k|, iω, ξ, i) +are functions of |k| instead of k. + diff --git a/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/load_file.txt b/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a053f3a2d624b46afc49edf7f0ab24e290fba5c --- /dev/null +++ b/AtE3T4oBgHgl3EQfsgvt/content/tmp_files/load_file.txt @@ -0,0 +1,10874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf,len=10873 +page_content='Kondo Lattice Model of Magic-Angle Twisted-Bilayer Graphene: Hund’s Rule, Local-Moment Fluctuations, and Low-Energy Effective Theory Haoyu Hu,1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig,2, 1, 3, ∗ and Alexei M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Tsvelik4 1Donostia International Physics Center, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 3IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 4Division of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA We apply a generalized Schrieffer–Wolff transformation to the extended Anderson-like topological heavy fermion (THF) model for the magic-angle (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='05◦) twisted bilayer graphene (MATBLG) (Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 129, 047601 (2022)), to obtain its Kondo Lattice limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In this limit localized f-electrons on a triangular lattice interact with topological conduction c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' By solving the exact limit of the THF model, we show that the integer fillings ν = 0, ±1, ±2 are controlled by the heavy f-electrons, while ν = ±3 is at the border of a phase transition between two f-electron fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For ν = 0, ±1, ±2, we then calculate the RKKY interactions between the f-moments in the full model and analytically prove the SU(4) Hund’s rule for the ground state which maintains that two f-electrons fill the same valley-spin flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Our (ferromagnetic interactions in the) spin model dramatically differ from the usual Heisenberg antiferromagnetic interactions expected at strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We show the ground state in some limits can be found exactly by employing a positive semidefinite ”bond- operators” method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We then compute the excitation spectrum of the f-moments in the ordered ground state, prove the stability of the ground state favored by RKKY interactions, and discuss the properties of the Goldstone modes, the (reason for the accidental) degeneracy of (some of) the excitation modes, and the physics of their phase stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We develop a low-energy effective theory for the f-moments and obtain analytic expressions for the dispersion of the collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We discuss the relevance of our results to the spin-entropy experiments in twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Introduction— The discovery of the correlated insulating phase [1] and superconductivity [2] in the MATBLG [3] has driven considerable theoretical [4–20] and experimental ef- forts [21–44] to understand its topology [45–61] and corre- lation physics [45, 60–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Theoretically, correlated insula- tors [77–88], ferromagnetic order [85, 89–92], superconduc- tivity [81, 93–108], and other exotic quantum phases [109– 116] have been identified and systematically studied which all point to rich physics [60] of the MATBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The recent experiments [28, 64, 117, 118] have provided evidence for fluctuating local moments and Hubbard-like physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Mean- while the theoretical understanding is challenging since the stable topology [52, 57] of the flat bands obstructs the sym- metric real-space description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A real-space extended Hubbard model [68, 70, 119–121] can still be constructed, but a certain symmetry (C2zT or P) becomes non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' To address this problem the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [122] have introduced an exact mapping of MATBG to a THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' THF is a version of the extended Anderson lattice model describing localized f- electrons interacting with topological conduction c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The f-electrons have zero kinetic energy and strong Hubbard interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' they admit a description in terms of localized Wannier orbitals centered at the AA-stacking region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The topological flat bands can be recovered from the hybridization between the f- and the c-electrons [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In this letter, we map the THF model to a Kondo lattice model using a generalized Schrieffer-Wolff (SW) transforma- tion which takes into account the density-density interaction term between f- and c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In this limit, the dynamics ∗ bernevig@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='edu of the localized orbitals becomes the one of the f-moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' By solving exactly a particular limit of the THF model, we show that the integer fillings ν = 0, ±1, ±2 are controlled by the f-electrons, while the situation is drastically different for ν = ±3 which sits at the phase transition between two f-electrons fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In the Kondo lattice model of MATBLG, the local moments formed by the fully-localized f-electrons interact with topological conduction electron bands via Kondo superexchange and direct ferromagnetic exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These two types of exchange interactions induce an RKKY in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = 0, −1, −2, the RKKY interaction dominates the physics and stabilizes the ferromagnetic ground states that obey the Hund’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This result provides an analytic deriva- tion of the Hund’s rule found numerically in the Hartree-Fock calculations of THF model [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We then proceed to inves- tigate fluctuations of the f-moments in the symmetry broken ground states by developing the low-energy effective theory and calculating the excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Schrieffer–Wolff transformation and Kondo lattice model— The single-particle Hamiltonian of the THF model contains the kinetic term ˆHc describing the topological conduction c- electron bands (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I), and the hybridization be- tween the f- and the c-electrons ˆHfc [122, 123]: ˆHfc = � |k|<Λc,R i,ξ,ξ′ �eik·R− |k|2λ2 2 ˜H(fc) ξξ′ (k) √NM ψf,ξ,† R,i ψc′,ξ′ k,i + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' � ˜H(fc)(k) = � γ v′ ⋆(kx − iky) v′ ⋆(kx + iky) γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (1) where ψc′,ξ,† k,i creates Γ3 “conduction” c-electron with mo- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='04669v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='str-el] 11 Jan 2023 2 mentum k, valley-spin flavor i ∈ {1, 2, 3, 4} (with (1, 2, 3, 4) corresponding to (+ ↑, − ↑, + ↓, − ↓)), and ”orbital” index ξ = (−1)a+1η (with a = 1, 2 are original orbital indices and η = ± are valley indices defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [122]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,ξ,† R,i creates f-electron at moir´e unit cell R with valley-spin flavor i, and orbital index ξ = (−1)a+1η, where a = 1, 2, η = ± [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Λc is the momentum cutoff, NM is the total number of moir´e unit cells and λ is the damping factor [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In the hybridiza- tion matrix ˜H(fc)(k), we only keep the first two terms (γ and v′ ⋆) in the expansion in powers of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The flat band limit is re- alized by setting M = 0, where M is taken as a parameter of ˆHc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The interaction Hamiltonian of the THF model is ˆHI = ˆHU + ˆHJ + ˆHV + ˆHW , where ˆHU, ˆHJ describe respec- tively the on-site Hubbard interaction of the f-electrons (U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='95meV), and the ferromagnetic exchange between the f- and the c-electrons (J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='38meV), ˆHV , ˆHW describe re- spectively the repulsion between the c-electrons (∼ 48meV) and the repulsion between the f- and the c-electrons (W = 47meV) [122, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The full Hamiltonian is ˆHc + ˆHfc + ˆHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The model possesses the U(4)×U(4) symmetry in the chiral- flat limit (M = 0, v′ ⋆ = 0), a flat U(4) symmetry in the nonchiral-flat limit (M = 0, v′ ⋆ ̸= 0), a chiral U(4) sym- metry in the chiral-nonflat limit (M ̸= 0, v′ ⋆ = 0) and a U(2) × U(2) symmetry in the nonchiral-nonflat limit (M ̸= 0, v′ ⋆ ̸= 0) [4, 85, 119, 122, 123, 125, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At sufficiently strong on-site Coulomb interaction U, the f- electrons are fully localized and give rise to local f-moments which are defined as ˆΣ(f,ξξ′) µν (R) = � i,j 1 2T µν ij ψf,ξ,† R,i ψf,ξ′ R,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (2) {T µν ij } with µ, ν ∈ {0, x, y, z} are given in SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 are the generators of the U(8) group (8 = 2(orbital) × 2(valley) × 2(spin)) where the U(1) charge component can be gauged away for the fully-localized f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We first analyze the zero-hybridization limit of the model (γ = 0, v′ ⋆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' v′ ⋆ is small [122] while γ changes rapidly and goes through zero at the value of w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='9 close to the actual MATBLG value w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hence this limit can be thought as a meaningful approximation close to the MATBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The zero-hybridization model is exactly solv- able at zero Coulomb repulsion between c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Here, we treat ˆHV in the mean-field approximation ( ˆHMF V ) and drop the ˆHJ which is relatively weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We then solve the model under the assumption that each site is filled with νf + 4 f- electrons with an integer νf (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We use νc to denote the filling of the c-electrons and use ν = νf + νc to denote the total fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1 (a), we plot νf and νc of the ground state as a function of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = 0, −1, −2, −3, the ground state has νf = ν and νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν = −3 is close to the transition point between νf = −3 state and νf = −2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, at ν = −3, our assumption of uniform charge distri- bution may be violated [115], and a Kondo model description fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This is consistent with the special place that ν = −3 has in the TBG physics [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' + ↑ + ↓ − ↑ − ↓ ξ = + 1 ξ = − 1 ξ = + 1 ξ = − 1 ξ = + 1 ξ = − 1 ν = 0 ν = − 1 ν = − 2 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a) Filling of f electrons(νf) and c-electrons(νc) as a func- tion of total filling(ν) in the zero hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (b), (c), (d) Illustrations of ground states at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The red dot means the filling of one f-electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = 0, −1, −2, we fix the filling of the f-electrons (ac- cording to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1 (a)) and perform a generalized SW transfor- mation (SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' IV), which leads to the following Kondo lattice Hamiltonian: ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ + ˆHMF V + ˆHW (3) where ˆHK and ˆHcc are the Kondo interaction and one-body scattering term generated by the SW transformation respec- tively [123, 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The Kondo interaction ˆHK takes the form of ˆHK = � R � |k|<Λc,|k+q|<Λc � µνξξ′ e−iq·Re− |k|2+|k+q|2 2 λ2 NM � γ2 Dνc,νf : ˆΣ(f,ξξ′) µν (R) :: ˆΣ(c′,ξ′ξ) µν (k, q) : + (4) �v′ ⋆γ(kx − iξky) Dνc,νf : ˆΣ(f,ξξ′) µν (R) :: ˆΣ(c′,ξ′−ξ) µν (k, q) : +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' �� where the colon symbols represent the normal ordering and Σ(c′,ξ′ξ) µν (k, q) (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I) is the U(8) moment of c-electrons defined in the manner similar to Σ(f,ξ′ξ) µν (R) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The parameter Dνc,νf at ν = νf = 0, −1, −2 is defined as 1 Dνc=0,νf = − 1 (U − W)νf − U 2 + 1 (U − W)νf + U 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (5) The distinct feature of this expression is the presence W absent in the standard Kondo Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In addition, we perform a k-expansion in the square bracket of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 4, and keep only the zeroth and the linear order terms in k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' as was done in the expression for the hybridization matrix ˜H(fc,η)(k) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S195).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The zeroth order Kondo coupling has strength γ2/Dνc=0,νf = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3meV, 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3meV, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='6meV at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The RKKY interactions and the Hund’s rule— By integrat- ing out the c-electrons in the Kondo Hamiltonian(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3), one 3 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Excitation spectrum at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Blue, orange, red and green denote the fluctuations in the full-empty sector, half-empty sector, full-half sector and half-half sector respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' induces an RKKY interaction between the f-moments [128– 130] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We restrict ourselves to deriving the RKKY interac- tions [123] in the leading order in ˆHK, ˆHJ at integer fillings ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In the chiral-flat limit (v′ ⋆ = 0, M = 0), the RKKY interactions can be described by the following U(4) × U(4) symmetric Hamiltonian ˆHv′ ⋆=0,M=0 RKKY = � R,R′ µν,ξξ′ [JRKKY 0 (R′) + JRKKY 1 (R′)δξ,ξ′] : Σ(f,ξξ′) µν (R) :: Σ(f,ξ′ξ) µν (R + R′) : , (6) where both JRKKY 0 (R′)(≤ 0) and JRKKY 1 (R′)(≤ 0) can be analytically obtained and are ferromagnetic (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using bond operators Aξ,ξ′ R,R′ = � i ψf,ξ,† R,i ψf,ξ′ R′,i, we show that ˆHv′ ⋆=0,M=0 RKKY is a Positive Semidefinite Hamil- tonian [4, 68], where the exact ground state can be ob- tained [4, 68, 85, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The grounds states are ferromagnetic with the form of (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VI) � R � ν+ f +2 � n=1 ψf,+,† R,in νf +4 � m=ν+ f +3 ψf,−,† R,im � |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (7) where ν+ f denotes the filling of the f-electrons with index ξ = 1, and ν+ f = 0 at ν = 0, ν+ f = −1, 0 at ν = −1, and ν+ f = −1 at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' {in} are chosen arbitrarily and |0⟩ is the vacuum with ψf,ξ R,i|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We note that our ground states form a subset of the ground states in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [85], due to the additional non-zero kinetic energy generated by f-c hybridizations in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We then consider the nonchiral-flat limit (v′ ⋆ ̸= 0, M = 0), where the RKKY interactions are flat-U(4) symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' They lift the ground-state degeneracy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We obtain the ground states by treating v′ ⋆-induced RKKY interactions as perturbations (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VI) , and the true ground state is selected by the following RKKY interactions � R,R′ µν,ξ JRKKY 2 (R′) : ˆΣ(f,ξξ) µν (R) :: ˆΣ(f,−ξ−ξ) µν (R + R′) : (8) where JRKKY 2 (R)(≤ 0) is analytically obtained and ferro- magnetic (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' JRKKY 2 (R2 − R) tends to align two f-moments : ˆΣ(f,++) µν (R) : and : ˆΣ(f,−−) µν (R2) : and stabilize the ground states obeying the Hund’s rule (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The corresponding ground states are consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [68, 85] (nonchiral-flat limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Moreover, we also derive the RKKY interactions in the zero-hybridization limit (γ = 0, v′ ⋆ = 0) with non-zero J(̸= 0) (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' III), where the corresponding ground states are consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [85] (chiral-nonflat limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fluctuations of the f-moments— We check the stability of the ferromagnetic ground state derived from RKKY Hamil- tonian by studying small fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We restrict ourselves to the flat-U(4) nonchiral-flat limit M = 0, v′ ⋆ ̸= 0 at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' To describe the fluctuations we introduce for each site a 8 × 8 traceless Hermitian matrix uiξ,jξ′(R), where i, j ∈ {1, 2, 3, 4} are valley-spin flavors and ξ, ξ′ ∈ {+, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The f-moments in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 can then be written as (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆΣ(f,ξξ′) µν (R) = � ij T µν ij [eiu(R)Λe−iu(R)]iξ,jξ′ (9) where Λ is an 8 × 8 matrix defined as Λiξ,jξ′ = ⟨ψ0| : ψf,ξ,† R,i ψf,ξ′ R,j : |ψ0⟩/2 and |ψ0⟩ is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A non- zero uiξ,jξ′(R) will generate fluctuations by rotating the f- moments from their ground-state expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The ferromagnetic order in the ground state opens a gap in the single-particle spectrum of the c-electrons which allows us to safely integrate them out [123] and to develop an effective theory for small fluctuations by expanding the action to the second order in uiξ,jξ′(R, τ), where τ is the imaginary time (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The Lagrangian of the effective theory is provided in SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VIII, where we find the diagonal components uiξ,iξ(R, τ) only contribute a total derivative and we will focus on the off-diagonal components: uiξ,jξ′(R, τ) with iξ ̸= jξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We then introduce two sets Sfill and Semp to character- ize the ground state, where Sfill and Semp denote the sets of iξ indices that are filled with one and zero number of the f- electrons at each site, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A fluctuation generated by uiξ,jξ′(R) (iξ ̸= jξ′) is described by the procedure of mov- ing one f-electron from jξ′ flavor at site R to iξ flavor at 4 the same site (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This procedure can only be valid when iξ ∈ Semp, jξ′ ∈ Sfill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Consequently, only uiξ,jξ′(R, τ) with jξ′ ∈ Sfill, iξ ∈ Semp and also its com- plex conjugate ujξ′,iξ(R, τ) that describes the reverse proce- dure appear in our effective Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We diagonalize the Lagrangian and plot the excitation spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We first analyze the spectrum at ν = 0, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The La- grangian density in the long-wavelength limit and in mo- mentum (k) and frequency (ω) spaces has the form of L = LGoldstone + Lgapped: LGoldstone = � jξ∈Sfill iξ∈Semp u† j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ω) � ω − k2 2m0 � uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lgapped = � jξ∈Sfill iξ∈Semp U † j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ω)[ω ˆI − ˆH(k)]Uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H(k) = � � � � k+k− 2m1 + ∆1 V k+ −V k− V k− k+k− 2m2 + ∆2 k2 − 2m3 −V k+ k2 + 2m3 k+k− 2m2 + ∆2 � � � � (10) where uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i = (u(j+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i+) + u(j−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i−))/ √ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' and U T j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i = ((u(j+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i+) − u(j−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i−))/ √ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' u(j+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' u(j−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='i+)) and k± = kx ± iky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' k = |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' m0, m1, m2, V, ∆1, ∆2, V are analyti- cally defined constants that depend on the parameters of the original THF Hamiltonian (SM [123], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The Gold- stone modes with quadratic dispersion decouple from the rest (gapped modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Their stiffness is 1/m0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='9meV · a2 M (at ν = 0), 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2meV · a2 M (at ν = −2), where aM is the moir´e lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The gaps at k = 0 are ∆1 and ∆2, with ∆2 correspond- ing to two-fold degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This feature is repro- duced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The exceptional case when all three modes are degenerate (∆1 = ∆2) is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 (a) and the condition for the degeneracy is α = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='37 + � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='14 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='23(v′⋆)2/(γλ)2]γ2/(JDνcνf ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the re- alistic values of parameters, we find α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='07 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' By directly evaluating the Lagrangian, we also observe ”roton” minima in the gapped modes (most obviously at ν = −2) at |k| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3|bM1| with bM1 the moir´e reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The roton mode has small anisotropy with the minimum lo- cated along the ΓM-MM line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We next discuss the number of the Goldstone modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Each combination of (i, j) that satisfies i+, i− ∈ Semp, j+, j− ∈ Sfill produce a Goldstone mode [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This leads to four Goldstone modes at ν = 0 and three Goldstone modes at ν = −2 [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Furthermore, all excitation modes depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 are four-fold degenerate at ν = 0 and three-fold degen- erate at ν = −2, due to the remaining U(2) × U(2) symme- tries at ν = 0 and the remaining U(1) × U(3) symmetries at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We now analyze the spectrum at ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Unlike ν = 0, −2 where each valley-spin flavor is filled with either two or zero f- electrons, at ν = −1, there is one valley-spin flavor (de- noted by i = 1) filled with two f-electrons and one valley-spin flavor (denoted by i = 2) filled with one f-electron and two empty valley-spin flavors (denoted by i = 3, 4) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This allows us to classify uiξ,jξ′(R, τ) at ν = −1 into four sectors: (1) full-empty sector with i = 3, 4, j = 1 or i = 1, j = 3, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (2) half-empty sector with i = 2, j = 3, 4 or i = 3, 4, j = 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (3) full-half sector with i = 1, j = 2 or j = 2, i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (4) half-half sector with i = 2, j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2, we label the excitation in different sectors with differ- ent colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Due to the remaining U(1) × U(1) × U(2) sym- metry of the ν = −1 ground state , each mode is two-fold de- generate in the full-empty and half-empty sectors and is non- degenerate in the full-half and half-half sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We find 2 de- generate Goldstone modes with stiffness 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='7meV · a2 M in the full-empty sector, 2 degenerate Goldstone modes with stiff- ness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3meV · a2 M in the half-empty sector, and 1 Goldstone mode with stiffness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8meV · a2 M in the full-half sector [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Several remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Firstly, some of the gapped modes at ν = 0, −1, −2 are relatively flat as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Secondly, at ν = −1, one of the flat modes (green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 (b)) with eigenfunction u2−,2+(k) has a tiny gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='12meV at ΓM point and a very narrow bandwidth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='5meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thirdly, the dispersion along MM to KM is also relatively flat which is related to the approximate C∞ symmetry of the excitation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Summary and discussions— We have constructed and stud- ied a Kondo lattice model for MATBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Its distinct feature is the Dirac character of the c-electron spectrum: at integer fillings, there is no Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hence the Kondo screening is irrelevant and the low energy physics is dominated by the RKKY interactions, which is also responsible for the Hund’s rule and ferromagnetism of the ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have devel- oped an effective theory describing small fluctuations of the local moments and found their excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have also discussed the properties of the Goldstone and gapped modes and their degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We believe that our work pro- vides insight into the correlated ground states and the local- moment fluctuations of the MATBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We also comment on the connection with previous works [86, 131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Some features, including the soft modes and accidental degeneracy of gapped modes at ΓM, also appear in the projected Coulomb model [86], but the roton modes do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We note that the projected Coulomb model [86] has ignored the effect of remote bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Roton modes have also been seen in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [131], where the remote bands have been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' However, our model gives a larger bandwidth of the gapped modes than Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We point out that, we take a different set of parameter values (including dielectric con- stant, and gate distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Besides, in our model, all Goldstone modes have quadratic dispersions, in contrast to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [131] which found a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The difference in the results has its origin in the different symmetry properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the flat limit of the model, and the quadratic dispersion comes from the broken flat U(4) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [131] takes non-flat bands, and the linear Goldstone modes come from the broken U(1) valley symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We conclude the paper with a discussion of the relevance of our results to the recent entropy experiments [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ex- perimentally, the entropy in TBG near ν = −1 has been found to be of the order of Boltzmann’s constant and is suppressed by the applications of magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This large entropy can 5 be explained by the presence of soft mode at ν = −1, which will be suppressed by the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Finally, we point out that the fluctuations of the f-moments could potentially gen- erate attractive interactions between c-electrons and drive the system to the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, our work also es- tablishes a platform for understanding the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Note added— After finishing this work, we have learned that related, but not identical, results had recently been ob- tained by the S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Das Sarma’s [132], P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Coleman’s [133] and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Song’s groups [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Acknowledgements— We thank Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Song for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We also thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Coleman and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Das Sarma for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B.’s work was primarily supported by the DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' DE-SC0016239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 101020833).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Further support was provided by the Gordon and Betty Moore Foundation through the EPiQS Initiative, Grant GBMF11070 and the European Research Council (ERC) un- der the European Union’s Horizon 2020 research and inno- vation program (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 101020833) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' was supported by the Office of Basic Energy Sciences, Ma- terial Sciences and Engineering Division, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Department of Energy (DOE) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [1] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Tomarken, Jason Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Luo, Javier D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Sanchez- Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, Ray C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ashoori, and Pablo Jarillo-Herrero, “Cor- related insulator behaviour at half-filling in magic-angle graphene superlattices,” Nature 556, 80–84 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [2] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo- Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature 556, 43–50 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [3] Rafi Bistritzer and Allan H MacDonald, “Moir´e bands in twisted double-layer graphene,” Proceedings of the National Academy of Sciences 108, 12233–12237 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, Zhi-Da Song, Nicolas Regnault, and Biao Lian, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' interacting hamilto- nian and exact symmetries,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 103, 205413 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [5] Kasra Hejazi, Chunxiao Liu, Hassan Shapourian, Xiao Chen, and Leon Balents, “Multiple topological transitions in twisted bilayer graphene near the first magic angle,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 99, 035111 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6] Bikash Padhi, Apoorv Tiwari, Titus Neupert, and Shin- sei Ryu, “Transport across twist angle domains in moir\\’e graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 2, 033458 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Dodaro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kivelson, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Schattner, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Sun, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Wang, “Phases of a phenomenological model of twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 075154 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [8] Tomohiro Soejima, Daniel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Parker, Nick Bultinck, Johannes Hauschild, and Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, “Efficient simulation of moir\\’e materials using the density matrix renormalization group,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 205111 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [9] Paul Eugenio and Ceren Dag, “DMRG study of strongly in- teracting $\\mathbb{Z} 2$ flatbands: A toy model inspired by twisted bilayer graphene,” SciPost Physics Core 3, 015 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [10] Yixuan Huang, Pavan Hosur, and Hridis K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Pal, “Quasi- flat-band physics in a two-leg ladder model and its relation to magic-angle twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 155429 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [11] Xu Zhang, Gaopei Pan, Yi Zhang, Jian Kang, and Zi Yang Meng, “Momentum Space Quantum Monte Carlo on Twisted Bilayer Graphene,” Chinese Physics Letters 38, 077305 (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='07010 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='str-el].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [12] Jian Kang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Oskar Vafek, “Cas- cades between light and heavy fermions in the normal state of magic angle twisted bilayer graphene,” arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='01145 [cond-mat] (2021), arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='01145 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [13] Johannes S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hofmann, Eslam Khalaf, Ashvin Vishwanath, Erez Berg, and Jong Yeon Lee, “Fermionic Monte Carlo Study of a Realistic Model of Twisted Bilayer Graphene,” Physical Review X 12, 011061 (2022), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='12112 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='str-el].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [14] Ya-Hui Zhang, Dan Mao, Yuan Cao, Pablo Jarillo-Herrero, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, “Nearly flat Chern bands in moir\\’e superlat- tices,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 99, 075127 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [15] Xiao-Chuan Wu, Chao-Ming Jian, and Cenke Xu, “Coupled- wire description of the correlated physics in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 99, 161405 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [16] Rafael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fernandes and J¨orn W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Venderbos, “Nematicity with a twist: Rotational symmetry breaking in a moir´e super- lattice,” Science Advances 6, eaba8834 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [17] Justin H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Wilson, Yixing Fu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Das Sarma, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Pixley, “Disorder in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 2, 023325 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [18] Jie Wang, Yunqin Zheng, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Millis, and Jennifer Cano, “Chiral approximation to twisted bilayer graphene: Ex- act intravalley inversion symmetry, nodal structure, and im- plications for higher magic angles,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 3, 023155 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, Zhi-Da Song, Nicolas Regnault, and Biao Lian, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Matrix elements, ap- proximations, perturbation theory, and a k ˙p two-band model,” Physical Review B 103, 205411 (2021), publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [20] Jiabin Yu, Ming Xie, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Sankar Das Sarma, to be published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [21] Yuan Cao, Debanjan Chowdhury, Daniel Rodan-Legrain, Oriol Rubies-Bigorda, Kenji Watanabe, Takashi Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, and Pablo Jarillo-Herrero, “Strange metal in magic-angle graphene with near planckian dissipation,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 076801 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [22] Yuan Cao, Daniel Rodan-Legrain, Jeong Min Park, Noah F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Yuan, Kenji Watanabe, Takashi Taniguchi, Rafael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fernan- des, Liang Fu, and Pablo Jarillo-Herrero, “Nematicity and competing orders in superconducting magic-angle graphene,” Science 372, 264–271 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [23] Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mo- hammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji Watan- abe, Takashi Taniguchi, Guangyu Zhang, Adrian Bachtold, Allan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Superconduc- tors, orbital magnets and correlated states in magic-angle bi- layer graphene,” Nature 574, 653–657 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [24] Matthew Yankowitz, Shaowen Chen, Hryhoriy Polshyn, Yux- uan Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, David Graf, Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 6 Young, and Cory R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Dean, “Tuning superconductivity in twisted bilayer graphene,” Science 363, 1059–1064 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [25] Petr Stepanov, Ipsita Das, Xiaobo Lu, Ali Fahimniya, Kenji Watanabe, Takashi Taniguchi, Frank H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Koppens, Johannes Lischner, Leonid Levitov, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Unty- ing the insulating and superconducting orders in magic-angle graphene,” Nature 583, 375–378 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [26] Ming Xie and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, “Weak-Field Hall Resistivity and Spin-Valley Flavor Symmetry Breaking in Magic-Angle Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 127, 196401 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [27] Yuhang Jiang, Xinyuan Lai, Kenji Watanabe, Takashi Taniguchi, Kristjan Haule, Jinhai Mao, and Eva Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei, “Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene,” Nature 573, 91–95 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [28] Dillon Wong, Kevin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Nuckolls, Myungchul Oh, Biao Lian, Yonglong Xie, Sangjun Jeon, Kenji Watanabe, Takashi Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Ali Yazdani, “Cas- cade of electronic transitions in magic-angle twisted bilayer graphene,” Nature 582, 198–202 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [29] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zondiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rodan-Legrain, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Cao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Queiroz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Oreg, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' von Op- pen, Ady Stern, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Berg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Jarillo-Herrero, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ilani, “Cas- cade of phase transitions and Dirac revivals in magic-angle graphene,” Nature 582, 203–208 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [30] Ipsita Das, Xiaobo Lu, Jonah Herzog-Arbeitman, Zhi- Da Song, Kenji Watanabe, Takashi Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic- angle bilayer graphene,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 17, 710–714 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [31] Jeong Min Park, Yuan Cao, Kenji Watanabe, Takashi Taniguchi, and Pablo Jarillo-Herrero, “Flavour Hund’s cou- pling, Chern gaps and charge diffusivity in moir´e graphene,” Nature 592, 43–48 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [32] Xiaobo Lu, Biao Lian, Gaurav Chaudhary, Benjamin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Piot, Giulio Romagnoli, Kenji Watanabe, Takashi Taniguchi, Mar- tino Poggio, Allan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Multiple flat bands and topological Hofs- tadter butterfly in twisted bilayer graphene close to the second magic angle,” PNAS 118 (2021), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2100006118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [33] Maximilian Otteneder, Stefan Hubmann, Xiaobo Lu, Dmitry A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kozlov, Leonid E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Golub, Kenji Watanabe, Takashi Taniguchi, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Sergey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ganichev, “Terahertz photogalvanics in twisted bilayer graphene close to the second magic angle,” Nano Letters 20, 7152–7158 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [34] Simone Lisi, Xiaobo Lu, Tjerk Benschop, Tobias A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' de Jong, Petr Stepanov, Jose R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Duran, Florian Margot, Ir`ene Cuc- chi, Edoardo Cappelli, Andrew Hunter, Anna Tamai, Vik- tor Kandyba, Alessio Giampietri, Alexei Barinov, Johannes Jobst, Vincent Stalman, Maarten Leeuwenhoek, Kenji Watan- abe, Takashi Taniguchi, Louk Rademaker, Sense Jan van der Molen, Milan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Allan, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Felix Baumberger, “Observation of flat bands in twisted bilayer graphene,” Nature Physics 17, 189–193 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [35] Tjerk Benschop, Tobias A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' de Jong, Petr Stepanov, Xiaobo Lu, Vincent Stalman, Sense Jan van der Molen, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Milan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Allan, “Measuring local moir´e lattice heterogene- ity of twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3, 013153 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [36] Paul Seifert, Xiaobo Lu, Petr Stepanov, Jos´e Ram´on Dur´an Retamal, John N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Moore, Kin-Chung Fong, Alessan- dro Principi, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Magic-angle bi- layer graphene nanocalorimeters: Toward broadband, energy- resolving single photon detection,” Nano Letters 20, 3459– 3464 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [37] Niels C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hesp, Iacopo Torre, Daniel Rodan-Legrain, Pietro Novelli, Yuan Cao, Stephen Carr, Shiang Fang, Petr Stepanov, David Barcons-Ruiz, Hanan Herzig Sheinfux, Kenji Watan- abe, Takashi Taniguchi, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, Efthimios Kaxi- ras, Pablo Jarillo-Herrero, Marco Polini, and Frank H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Koppens, “Observation of interband collective excitations in twisted bilayer graphene,” Nature Physics 17, 1162–1168 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [38] Jaime Diez-Merida, Andr´es D´ıez-Carl´on, SY Yang, Y-M Xie, X-J Gao, Kenji Watanabe, Takashi Taniguchi, Xiaobo Lu, Kam Tuen Law, and Dmitri K Efetov, “Magnetic josephson junctions and superconducting diodes in magic angle twisted bilayer graphene,” arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='01067 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [39] Petr Stepanov, Ming Xie, Takashi Taniguchi, Kenji Watanabe, Xiaobo Lu, Allan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Competing zero-field chern insulators in superconducting twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 127, 197701 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hubmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Soul, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Di Battista, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hild, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ganichev, “Nonlin- ear intensity dependence of photogalvanics and photoconduc- tance induced by terahertz laser radiation in twisted bilayer graphene close to magic angle,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 6, 024003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [41] Alexandre Jaoui, Ipsita Das, Giorgio Di Battista, Jaime D´ıez- M´erida, Xiaobo Lu, Kenji Watanabe, Takashi Taniguchi, Hiroaki Ishizuka, Leonid Levitov, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Quantum critical behaviour in magic-angle twisted bilayer graphene,” Nature Physics 18, 633–638 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [42] Ipsita Das, Cheng Shen, Alexandre Jaoui, Jonah Herzog- Arbeitman, Aaron Chew, Chang-Woo Cho, Kenji Watanabe, Takashi Taniguchi, Benjamin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Piot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Observation of reentrant correlated in- sulators and interaction-driven fermi-surface reconstructions at one magnetic flux quantum per moir´e unit cell in magic- angle twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 128, 217701 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [43] Sameer Grover, Matan Bocarsly, Aviram Uri, Petr Stepanov, Giorgio Di Battista, Indranil Roy, Jiewen Xiao, Alexan- der Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Meltzer, Yuri Myasoedov, Keshav Pareek, Kenji Watan- abe, Takashi Taniguchi, Binghai Yan, Ady Stern, Erez Berg, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Eli Zeldov, “Chern mosaic and berry-curvature magnetism in magic-angle graphene,” Nature Physics 18, 885–892 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [44] Giorgio Di Battista, Paul Seifert, Kenji Watanabe, Takashi Taniguchi, Kin Chung Fong, Alessandro Principi, and Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, “Revealing the thermal properties of super- conducting magic-angle twisted bilayer graphene,” Nano Let- ters 22, 6465–6470 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [45] Kevin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Nuckolls, Myungchul Oh, Dillon Wong, Biao Lian, Kenji Watanabe, Takashi Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Ali Yazdani, “Strongly correlated Chern insulators in magic- angle twisted bilayer graphene,” Nature 588, 610–615 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [46] Youngjoon Choi, Hyunjin Kim, Yang Peng, Alex Thom- son, Cyprian Lewandowski, Robert Polski, Yiran Zhang, Harpreet Singh Arora, Kenji Watanabe, Takashi Taniguchi, Ja- son Alicea, and Stevan Nadj-Perge, “Correlation-driven topo- logical phases in magic-angle twisted bilayer graphene,” Na- ture 589, 536–541 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [47] Yu Saito, Jingyuan Ge, Louk Rademaker, Kenji Watanabe, Takashi Taniguchi, Dmitry A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Abanin, and Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' , 1–4 7 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [48] Shuang Wu, Zhenyuan Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, and Eva Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei, “Chern insulators, van Hove singulari- ties and topological flat bands in magic-angle twisted bilayer graphene,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 20, 488–494 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [49] Liujun Zou, Hoi Chun Po, Ashvin Vishwanath, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, “Band structure of twisted bilayer graphene: Emergent sym- metries, commensurate approximants, and Wannier obstruc- tions,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 085435 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [50] Dmitry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efimkin and Allan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, “Helical net- work model for twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 035404 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [51] Jian Kang and Oskar Vafek, “Symmetry, Maximally Localized Wannier States, and a Low-Energy Model for Twisted Bilayer Graphene Narrow Bands,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 8, 031088 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [52] Zhida Song, Zhijun Wang, Wujun Shi, Gang Li, Chen Fang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “All magic angles in twisted bi- layer graphene are topological,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 123, 036401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [53] Hoi Chun Po, Liujun Zou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, and Ashvin Vishwanath, “Faithful tight-binding models and fragile topology of magic- angle bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 99, 195455 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [54] Bikash Padhi, Chandan Setty, and Philip W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phillips, “Doped Twisted Bilayer Graphene near Magic Angles: Proximity to Wigner Crystallization, Not Mott Insulation,” Nano Letters 18, 6175–6180 (2018), arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='01101 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='str-el].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [55] Kasra Hejazi, Xiao Chen, and Leon Balents, “Hybrid Wannier Chern bands in magic angle twisted bilayer graphene and the quantized anomalous Hall effect,” Physical Review Research 3, 013242 (2021), publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [56] Jian Kang and Oskar Vafek, “Non-Abelian Dirac node braid- ing and near-degeneracy of correlated phases at odd integer filling in magic-angle twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 035161 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [57] Zhi-Da Song, Biao Lian, Nicolas Regnault, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' An- drei Bernevig, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Stable sym- metry anomaly,” Physical Review B 103, 205412 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='11872 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [58] Biao Lian, Fang Xie, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Landau level of fragile topology,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 041402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [59] Kasra Hejazi, Chunxiao Liu, and Leon Balents, “Landau lev- els in twisted bilayer graphene and semiclassical orbits,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 100, 035115 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [60] Leon Balents, Cory R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Dean, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “Superconductivity and strong correlations in moir´e flat bands,” Nature Physics 16, 725–733 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [61] Eva Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, Pablo Jarillo-Herrero, Al- lan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, Kin Fai Mak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, Emanuel Tutuc, Ali Yazdani, and Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “The marvels of moir´e materials,” Nature Reviews Materials 6, 201–206 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [62] Hryhoriy Polshyn, Matthew Yankowitz, Shaowen Chen, Yux- uan Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, Cory R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Dean, and Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “Large linear-in-temperature resistivity in twisted bilayer graphene,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 15, 1011–1016 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [63] Xiaoxue Liu, Zhi Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, Oskar Vafek, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Li, “Tuning electron correlation in magic- angle twisted bilayer graphene using Coulomb screening,” Science 371, 1261–1265 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [64] Yonglong Xie, Biao Lian, Berthold J¨ack, Xiaomeng Liu, Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' An- drei Bernevig, and Ali Yazdani, “Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene,” Nature 572, 101–105 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [65] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex Thom- son, Harpreet Arora, Robert Polski, Yiran Zhang, Hechen Ren, Jason Alicea, Gil Refael, Felix von Oppen, Kenji Watanabe, Takashi Taniguchi, and Stevan Nadj-Perge, “Electronic corre- lations in twisted bilayer graphene near the magic angle,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 15, 1174–1180 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [66] Alexander Kerelsky, Leo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' McGilly, Dante M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kennes, Lede Xian, Matthew Yankowitz, Shaowen Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Taniguchi, James Hone, Cory Dean, Angel Rubio, and Abhay N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Pasupathy, “Maximized electron interactions at the magic angle in twisted bilayer graphene,” Nature 572, 95–100 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [67] Hoi Chun Po, Liujun Zou, Ashvin Vishwanath, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, “Origin of mott insulating behavior and superconductivity in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 8, 031089 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [68] Jian Kang and Oskar Vafek, “Strong Coupling Phases of Par- tially Filled Twisted Bilayer Graphene Narrow Bands,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 122, 246401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [69] Fang Xie, Aditya Cowsik, Zhi-Da Song, Biao Lian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, and Nicolas Regnault, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' An exact diagonalization study at nonzero integer filling,” Physical Review B 103, 205416 (2021), publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [70] J¨orn W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Venderbos and Rafael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fernandes, “Correlations and electronic order in a two-orbital honeycomb lattice model for twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 245103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [71] Yi-Zhuang You and Ashvin Vishwanath, “Superconductivity from valley fluctuations and approximate SO(4) symmetry in a weak coupling theory of twisted bilayer graphene,” npj Quan- tum Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 4, 1–12 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [72] Nick Bultinck, Shubhayu Chatterjee, and Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, “Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 166601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [73] Alex Thomson, Shubhayu Chatterjee, Subir Sachdev, and Mathias S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Scheurer, “Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 075109 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [74] Yuan Da Liao, Zi Yang Meng, and Xiao Yan Xu, “Valence Bond Orders at Charge Neutrality in a Possible Two-Orbital Extended Hubbard Model for Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 123, 157601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [75] Yuan Da Liao, Jian Kang, Clara N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Breiø, Xiao Yan Xu, Han- Qing Wu, Brian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andersen, Rafael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fernandes, and Zi Yang Meng, “Correlation-Induced Insulating Topological Phases at Charge Neutrality in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 11, 011014 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [76] Noah F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Yuan and Liang Fu, “Model for the metal-insulator transition in graphene superlattices and beyond,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 045103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [77] Ming Xie and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, “Nature of the Correlated Insulator States in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 097601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [78] Masayuki Ochi, Mikito Koshino, and Kazuhiko Kuroki, “Pos- sible correlated insulating states in magic-angle twisted bi- layer graphene under strongly competing interactions,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 081102 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [79] Jianpeng Liu, Junwei Liu, and Xi Dai, “Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 99, 155415 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [80] Kangjun Seo, Valeri N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kotov, and Bruno Uchoa, “Ferromag- netic Mott state in Twisted Graphene Bilayers at the Magic Angle,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 122, 246402 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [81] Maine Christos, Subir Sachdev, and Mathias S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Scheurer, “Superconductivity, correlated insulators, and 8 Wess–Zumino–Witten terms in twisted bilayer graphene,” PNAS 117, 29543–29554 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [82] Jianpeng Liu and Xi Dai, “Theories for the correlated insulat- ing states and quantum anomalous Hall effect phenomena in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 103, 035427 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [83] Tommaso Cea and Francisco Guinea, “Band structure and in- sulating states driven by Coulomb interaction in twisted bi- layer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 045107 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [84] Yi Zhang, Kun Jiang, Ziqiang Wang, and Fuchun Zhang, “Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 035136 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [85] Biao Lian, Zhi-Da Song, Nicolas Regnault, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efe- tov, Ali Yazdani, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' exact insulator ground states and phase diagram,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 103, 205414 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [86] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, Biao Lian, Aditya Cowsik, Fang Xie, Nicolas Regnault, and Zhi-Da Song, “Twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' exact analytic many-body excitations in coulomb hamiltonians: Charge gap, goldstone modes, and absence of cooper pairing,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 103, 205415 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [87] Huaiming Guo, Xingchuan Zhu, Shiping Feng, and Richard T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Scalettar, “Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 97, 235453 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [88] Jonah Herzog-Arbeitman, Aaron Chew, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Reentrant correlated insulators in twisted bilayer graphene at 25 t (2π flux),” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 129, 076401 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [89] Jed H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Pixley and Eva Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei, “Ferromagnetism in magic- angle graphene,” Science 365, 543–543 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [90] Fengcheng Wu and Sankar Das Sarma, “Collective Excitations of Quantum Anomalous Hall Ferromagnets in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 046403 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [91] C´ecile Repellin, Zhihuan Dong, Ya-Hui Zhang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, “Ferromagnetism in Narrow Bands of Moir\\’e Su- perlattices,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 187601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [92] Jianpeng Liu, Zhen Ma, Jinhua Gao, and Xi Dai, “Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 9, 031021 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [93] Biao Lian, Zhijun Wang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Twisted Bilayer Graphene: A Phonon-Driven Superconductor,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 122, 257002 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [94] Fengcheng Wu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MacDonald, and Ivar Martin, “The- ory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 121, 257001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [95] Cenke Xu and Leon Balents, “Topological Superconductiv- ity in Twisted Multilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 121, 087001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [96] Francisco Guinea and Niels R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Walet, “Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers,” PNAS 115, 13174–13179 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [97] Hiroki Isobe, Noah F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Yuan, and Liang Fu, “Unconven- tional Superconductivity and Density Waves in Twisted Bi- layer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 8, 041041 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [98] Cheng-Cheng Liu, Li-Da Zhang, Wei-Qiang Chen, and Fan Yang, “Chiral Spin Density Wave and $d+id$ Superconduc- tivity in the Magic-Angle-Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 121, 217001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [99] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Gonz´alez and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Stauber, “Kohn-Luttinger Superconductiv- ity in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 122, 026801 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [100] Eslam Khalaf, Shubhayu Chatterjee, Nick Bultinck, Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, and Ashvin Vishwanath, “Charged skyrmions and topological origin of superconductivity in magic-angle graphene,” Science Advances 7, eabf5299 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [101] Cyprian Lewandowski, Debanjan Chowdhury, and Jonathan Ruhman, “Pairing in magic-angle twisted bilayer graphene: Role of phonon and plasmon umklapp,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 103, 235401 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [102] Shang Liu, Eslam Khalaf, Jong Yeon Lee, and Ashvin Vishwanath, “Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 3, 013033 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [103] Dante M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kennes, Johannes Lischner, and Christoph Kar- rasch, “Strong correlations and $d+\\mathit{id}$ supercon- ductivity in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 241407 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [104] Tongyun Huang, Lufeng Zhang, and Tianxing Ma, “Antifer- romagnetically ordered Mott insulator and d+id superconduc- tivity in twisted bilayer graphene: A quantum Monte Carlo study,” Science Bulletin 64, 310–314 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [105] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Julku, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Peltonen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Liang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Heikkil¨a, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' T¨orm¨a, “Superfluid weight and Berezinskii-Kosterlitz- Thouless transition temperature of twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 101, 060505 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [106] Fang Xie, Zhida Song, Biao Lian, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Topology-Bounded Superfluid Weight in Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 167002 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [107] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' K¨onig, Piers Coleman, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Tsvelik, “Spin mag- netometry as a probe of stripe superconductivity in twisted bi- layer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 102, 104514 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [108] Dmitry V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Chichinadze, Laura Classen, and Andrey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Chubukov, “Nematic superconductivity in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 101, 224513 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [109] Ahmed Abouelkomsan, Zhao Liu, and Emil J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Bergholtz, “Particle-Hole Duality, Emergent Fermi Liquids, and Frac- tional Chern Insulators in Moir\\’e Flatbands,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 124, 106803 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [110] Yves H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kwan, Glenn Wagner, Tomohiro Soejima, Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, Steven H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Simon, Siddharth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Parameswaran, and Nick Bultinck, “Kekul\\’e spiral order at all nonzero integer fillings in twisted bilayer graphene,” arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='05857 [cond- mat] (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='05857 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [111] C´ecile Repellin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Senthil, “Chern bands of twisted bi- layer graphene: Fractional Chern insulators and spin phase transition,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 2, 023238 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [112] Patrick J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ledwith, Grigory Tarnopolsky, Eslam Khalaf, and Ashvin Vishwanath, “Fractional Chern insulator states in twisted bilayer graphene: An analytical approach,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Research 2, 023237 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [113] Leon Balents, Cory R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Dean, Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Efetov, and Andrea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “Superconductivity and strong correlations in moir´e flat bands,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 16, 725–733 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [114] Peter Cha, Aavishkar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Patel, and Eun-Ah Kim, “Strange metals from melting correlated insulators in twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 127, 266601 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [115] Fang Xie, Jian Kang, B Andrei Bernevig, Oskar Vafek, and Nicolas Regnault, “Phase diagram of twisted bilayer graphene at filling factor ν = −3,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='14322 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [116] Yarden Sheffer and Ady Stern, “Chiral magic-angle twisted bilayer graphene in a magnetic field: Landau level correspon- dence, exact wave functions, and fractional chern insulators,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 104, L121405 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [117] Yu Saito, Fangyuan Yang, Jingyuan Ge, Xiaoxue Liu, Takashi 9 Taniguchi, Kenji Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Li, Erez Berg, and An- drea F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Young, “Isospin pomeranchuk effect in twisted bilayer graphene,” Nature 592, 220–224 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [118] Asaf Rozen, Jeong Min Park, Uri Zondiner, Yuan Cao, Daniel Rodan-Legrain, Takashi Taniguchi, Kenji Watanabe, Yuval Oreg, Ady Stern, Erez Berg, Pablo Jarillo-Herrero, and Shahal Ilani, “Entropic evidence for a Pomeranchuk effect in magic- angle graphene,” Nature 592, 214–219 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [119] Oskar Vafek and Jian Kang, “Lattice model for the coulomb interacting chiral limit of magic-angle twisted bilayer graphene: Symmetries, obstructions, and excitations,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 104, 075143 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [120] Mikito Koshino, Noah F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Yuan, Takashi Koretsune, Masayuki Ochi, Kazuhiko Kuroki, and Liang Fu, “Maxi- mally Localized Wannier Orbitals and the Extended Hubbard Model for Twisted Bilayer Graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' X 8, 031087 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [121] Xiao Yan Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Law, and Patrick A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lee, “Kekul\\’e va- lence bond order in an extended Hubbard model on the hon- eycomb lattice with possible applications to twisted bilayer graphene,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B 98, 121406 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [122] Zhi-Da Song and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “Magic-angle twisted bilayer graphene as a topological heavy fermion problem,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 129, 047601 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [123] Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [124] Dumitru C˘alug˘aru, Maksim Borovkov, Zhida Song, Liam Lau, Piers Coleman, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Andrei Bernevig, “TBG as Topological Heavy Fermion: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Analytical approximations of the model parameters,” In preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [125] Nick Bultinck, Eslam Khalaf, Shang Liu, Shubhayu Chatter- jee, Ashvin Vishwanath, and Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, “Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling,” Physical Review X 10, 031034 (2020), publisher: American Physical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [126] Oskar Vafek and Jian Kang, “Renormalization Group Study of Hidden Symmetry in Twisted Bilayer Graphene with Coulomb Interactions,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 125, 257602 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [127] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Schrieffer and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Wolff, “Relation between the an- derson and kondo hamiltonians,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 149, 491–492 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [128] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ruderman and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Kittel, “Indirect exchange coupling of nuclear magnetic moments by conduction electrons,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 96, 99–102 (1954).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [129] Tadao Kasuya, “A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model,” Progress of Theoretical Physics 16, 45–57 (1956), https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='com/ptp/article- pdf/16/1/45/5266722/16-1-45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [130] Kei Yosida, “Magnetic properties of cu-mn alloys,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 106, 893–898 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [131] Eslam Khalaf, Nick Bultinck, Ashvin Vishwanath, and Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zaletel, “Soft modes in magic angle twisted bilayer graphene,” arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='14827 [cond-mat] (2020), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='14827 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [132] Yang-Zhi Chou and Sankar Das Sarma, “Kondo lattice model in magic-angle twisted bilayer graphene,” arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='15682 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [133] Liam L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lau and Piers Coleman, to be published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [134] Geng-Dong Zhou and Zhi-Da Song, to be published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [135] Elliott H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lieb, “Flux phase of the half-filled band,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 73, 2158–2161 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [136] Alexei Kitaev, “Anyons in an exactly solved model and be- yond,” Annals of Physics 321, 2–111 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [137] Alexander Weisse, Gerhard Wellein, Andreas Alvermann, and Holger Fehske, “The kernel polynomial method,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 78, 275–306 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [138] Piers Coleman, Introduction to many-body physics (Cam- bridge University Press, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [139] Milton Abramowitz, Irene A Stegun, and Robert H Romer, “Handbook of mathematical functions with formulas, graphs, and mathematical tables,” (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [140] Alexander Altland and Ben D Simons, Condensed matter field theory (Cambridge university press, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1 Supplementary Materials CONTENTS References 5 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Model and notation 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' U(4) moments 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψ basis and U(8) moments 5 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zero-hybridziation limit 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Symmetries of the zero-hybridization Limit 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zero hybridization limit without ˆHJ 9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Solution of the one f-electron problem above (or below) uniform integer filling 11 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Charge density waves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 14 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zero-hybridization Model with ˆHJ 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Coherent states of the f-moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 15 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Trial wavefunction 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Perturbation theory 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Z0 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Z1/Z0 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Z2/Z0 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Free energy: F 24 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Path integral formula 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ground state of the zero hybridization model at M = 0, νf = 0, −1, −2 31 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ground state of the zero hybridization model at M ̸= 0, νf = 0, −1, −2 33 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Schrieffer-Wolff transformation 35 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Hamiltonian 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆH0 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆH1 38 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Procedure of Schrieffer–Wolff transformation 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Expression of S 39 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effective Hamiltonian from SW transformation 41 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effect of ˆHJ and PH ˆHfcPH 46 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effective Kondo model 46 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' RKKY interactions at M = 0 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −⟨(SK + SJ)Scc⟩0,con 51 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −⟨SKSJ⟩0,con 53 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' − 1 2⟨S2 K⟩0,con 54 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' − 1 2⟨S2 J⟩0,con 55 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' RKKY interactions 56 S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ground state of f-moments 58 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ground state at v′ ⋆ = 0, M = 0 58 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Ground state at v′ ⋆ ̸= 0, M = 0 61 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparisons of the ground states at different limits 64 S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fluctuation spectrum of f-moments based on RKKY interactions 64 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' RKKY Hamiltonian 65 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Excitation spectrum from RKKY interaction at νf = 0, −2 66 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Excitation spectrum from RKKY interaction at νf = −1 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Full-empty sector 73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Full-half sector 73 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Half-empty sector 74 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Half-half sector 75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Number of Goldstone modes 75 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Discussion 75 S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effective theory of f-moments 75 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Single-particle Green’s function of conduction electrons 81 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ⟨Sint⟩0 82 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ⟨Sint,2⟩0 83 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' − 1 2⟨S′ KS′ K⟩0,con 84 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' − 1 2⟨S′ JS′ J⟩0,con 85 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −⟨S′ JS′ K⟩0,con 85 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effective action 86 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Flat U(4) symmetry and Noether’s theorem 89 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Symmetry 93 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf = 0 94 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf = −2 97 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Lagrangian at νf = 0, −2 in the long-wavelength limit 98 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf = −1 99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Full-empty sector 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Full-half sector 101 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Half-empty sector 103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Half-half sector 105 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Effective theory in the long-wavelength limit 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Number of Goldstone modes 106 S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Single-particle Green’s function 107 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Single-particle Green’s function at M = 0 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' g0(R, τ) 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' g2,η(τ, R) 110 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Single-particle Green’s function at M ̸= 0 111 S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Four-fermioin correlation function I 114 S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Four-fermioin correlation function II 114 S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Fourier transformations 117 S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Single particle Green’s function in the ordered state 119 3 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' MODEL AND NOTATION The topological heavy-fermion model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [122] takes the following Hamiltonian ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV (S1) The single-particle Hamiltonian of conduction c-electrons has the form of ˆHc = � η,s,a,a′,|k|<Λc H(c,η) a,a′ (k)c† kaηscka′ηs , H(c,η)(k) = � 02×2 v⋆(ηkxσ0 + ikyσz) v⋆(ηkxσ0 − ikyσz) Mσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' � (S2) where σ0,x,y,z are identity and Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ckaηs represents the annihilation operator of the a(= 1, 2, 3, 4)-th conduction band basis of the valley η(= ±) and spin s(=↑, ↓) at the moir´e momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ΓM point (k = 0) of the moir´e Brillouin zone, ck1ηs, ck2ηs form a Γ3 irreducible representation (of P6′2′2 group), ck3ηs, ck4ηs form a Γ1 ⊕ Γ2 reducible (into Γ1 and Γ2 - as they are written, the ck3ηs, ck4ηs are just the σx linear combinations of Γ1 ± Γ2 ) representation (of P6′2′2 group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Λc is the momentum cutoff for the c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' N is the total number of moir´e unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Parameter values are v⋆ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='303eV · ˚A, M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='697meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The hybridization between f and c electrons has the form of ˆHfc = 1 √NM � |k|<Λc R � αaηs � eik·R− |k|2λ2 2 H(fc,η) αa (k)f † Rαηsckaηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' � , (S3) where fRαηs represents the annihilation operators of the f electrons with orbital index α(= 1, 2), valley index η(= ±) and spin s(=↑, ↓) at the moir´e unit cell R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' NM is the number of moir´e unit cells and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3376aM is the damping factor, where aM is the moir´e lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The hybridization matrix H(fc,η) has the form of H(fc,η)(k) = �γσ0 + v′ ⋆(ηkxσx + kyσy), 02×2 � (S4) which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Parameter values are γ = −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='75meV, v′ ⋆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='622eV · ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆHU (U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='89meV) describes the on-site interactions of f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆHU = U 2 � R : nf R :: nf R :, (S5) where nf R = � αηs f † RαηsfRαηs is the f-electrons density and the colon symbols represent the normal ordered operator with respect to the normal state: : f † Rα1η1s1fRα2η2s2 := f † Rα1η1s1fRα2η2s2 − 1 2δα1η1s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The ferromagnetic exchange interaction between f and c electrons ˆHJ is defined as HJ = − J 2NM � Rs1s2 � αα′ηη′ � |k1|,|k2|<Λc ei(k1−k2)·R(ηη′ + (−1)α+α′) : f † Rαηs1fRα′η′s2 :: c† k2,α′+2,η′s2ck1,α+2,ηs1 : (S6) where J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='38meV and : c† k2,α′+2,η′s2ck1,α+2,ηs1 := c† k2,α′+2,η′s2ck1,α+2,ηs1 − 1 2δk1,k2δα,α′δη,η′δs1,s2 The repulsion between f and c electrons ˆHW has the form of ˆHW = � η,s,η′,s′,a,α � |k|<Λc,|k+q|<Λc Wae−iq·R : f † R,aηsfR,aηs :: c† k+q,aη′s′ck,aη′s′ : (S7) where W1 = W2, W3 = W4 [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We further require W1 = W2 = W3 = W4 = W = 47meV (the difference is about 10-15% ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The Coulomb interaction between c electrons has the form of ˆHV = 1 2Ω0NM � η1s1a1 � η2s2a2 � |k1|,|k2|<Λc � q |k1+q|,|k2+q|<Λc V (q) : c† k1a1η1s1ck1+qa1η1s1 :: c† k2+qa2η2s2ck2a2η2s2 : (S8) 4 where Ω0 is the area of the moir´e unit cell and V (q = 0)/Ω0 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='33meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='We will always take a mean-field treatment of ˆHV , which gives ˆHMF V = −V (0) 2Ω0 NMν2 c + V (0) Ω0 νc � |k|<Λc,aηs (c† k,aηsck,aηs − 1/2) (S9) where νc is the filling of c-electrons and |ψ0⟩ νc = ⟨ψ0| ˆνc|ψ0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S10) |ψ0⟩ is the ground state and the density operator ˆνc of c-electrons is defined ˆνc = 1 NM � |k|<Λc,aηs c† k,aηsck,aηs − 1/2 (S11) and |ψ0⟩ is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Clearly, at the mean-field level ˆHMF V is equivalent to a chemical potential shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In addition, the energy loss from ˆHMF V is ⟨ψ0| ˆHMF V |ψ0⟩ = V (0) 2Ω0 NMν2 c (S12) The model has a U(4) × U(4) symmetry at chiral-flat limit M = 0, v′ ⋆ = 0, a flat U(4) symmetry at flat limit M = 0 and a chiral U(4) symmetry at v′ ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In addition, the particle-hole conjugation transformation will map the ground state of the model at ν (total filling of f and c electrons) to the ground state at −ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus we only consider ν ≤ 0 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' U(4) moments To observe the symmetry of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' we introduce the following flat U(4) moments (flat U(4) symmetry at M = 0) ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) µν (R) =δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='(−1)α−1η 2 Aµν αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′f † RαηsfRα′η′s′ ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) µν (q) =δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='(−1)a−1η 2NM � |k|<Λc Aµν aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a′η′s′c† k+qaηscka′η′s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' a′ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2) ˆΣ(c′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) µν (q) =δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='(−1)a−1η 2NM � |k|<Λc Bµν aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a′η′s′c† k+qaηscka′η′s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' a′ = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 4) (S13) where repeated indices should be summed over and Aµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Bµν (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' z) are eight-by-eight matrices Aµν ={σ0τ0ςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' σyτxςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' σyτyςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' σ0τzςν} Bµν ={σ0τ0ςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −σyτxςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −σyτyςν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' σ0τzςν} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S14) with σ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' τ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='z being the Pauli or identity matrices for the orbital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' valley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' and spin degrees of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The ±1 valued index ξ, equal to (−1)α−1η or (−1)a−1η in the generators the f and c electrons respectively, labels different fundamental representations of the flat-U(4) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The global flat-U(4) rotations are generated by ˆΣµν = � ξ=±1 � ˆΣ(f,ξ) µν + ˆΣ(c′,ξ) µν + ˆΣ(c′′,ξ) µν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S15) The chiral U(4) moments (chiral limit with v′ ⋆ = 0 and different other parameters [122]) can be defined in a similar manner ˆΘ(f,ξ) µν (R) =δξ,(−1)α−1η 2 Θµν,f αηs,α′η′s′f † RαηsfRα′η′s′ ˆΘ(c′,ξ) µν (q) =δξ,(−1)a−1η 2NM � |k|<Λc Θµν,c′ aηs,a′η′s′c† k+qaηscka′η′s′, (a, a′ = 1, 2) ˆΘ(c′′,ξ) µν (q) =δξ,(−1)a−1η 2NM � |k|<Λc Θµν,c′′ aηs,a′η′s′c† k+qaηscka′η′s′, (a, a′ = 3, 4) (S16) 5 where µ, ν = 0, x, y, z, and the repeated indices should be summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In addition, Θ(0ν,f) = σ0τ0ςν, Θ(0ν,c′) = σ0τ0ςν Θ(0ν,c′′) = σ0τ0ςν , Θ(xν,f) = σxτxςν, Θ(xν,c′) = σxτxςν Θ(xν,c′′) = −σxτxςν , Θ(yν,f) = σxτyςν, Θ(yν,c′) = σxτyςν Θ(yν,c′′) = −σxτyςν , Θ(zν,f) = σ0τzςν, Θ(zν,c′) = σ0τzςν Θ(zν,c′′) = σ0τzςν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The global chiral-U(4) rotations are generated by ˆΘµν = � ξ=±1 � ˆΘ(f,ξ) µν + ˆΘ(c′,ξ) µν + ˆΘ(c′′,ξ) µν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The relations between chiral and flat U(4) moments are ˆΣf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (R) = ˆΘf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (R) ˆΣf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (R) = ˆΘf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (R) ˆΣf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ xν (R) = ξ ˆΘ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) yν (R) ˆΣf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ yν (R) = −ξ ˆΘ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) xν (R) ˆΣc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (R) = ˆΘc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (q) ˆΣc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (q) = ˆΘc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (q) ˆΣc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ xν (q) = ξ ˆΘ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) yν (q) ˆΣc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ yν (q) = −ξ ˆΘ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) xν (q) ˆΣc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (q) = ˆΘc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ 0ν (q) ˆΣc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (q) = ˆΘc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ zν (q) ˆΣc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ xν (q) = ξ ˆΘ(c′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) yν (q) ˆΣc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ yν (q) = −ξ ˆΘ(c′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ) xν (q) (S17) We note that even though two U(4) moments are related via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S17, they actually characterize two different U(4) symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψ basis and U(8) moments In this section, we introduce more general U(8) moments, where 8=2(orbital)× 2(valley) ×2(spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We first consider the following convenient basis of electron operators ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='n (n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ξ = ±): (ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4) = (fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2−↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2−↓) (ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4) = (fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1−↑fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1−↓) (ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4 ) = (ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2−↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2−↓) (ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4 ) = (ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1−↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1−↓) (ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='+ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4 ) = (ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4−↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' −ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4−↓) (ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='− k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4 ) = (ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4+↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3−↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='4+↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='3−↓) (S18) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' the index ξ = η(−1)α+1 can be understood as the index of Chern basis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The single-particle Hamiltonian ˆHc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S2) and ˆHfc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S2) in the ψ basis takes the form of ˆHc = � |k|<Λc,n,ξ (Ψc,ξ k )†v⋆ � 04×4 (kx + iξky)I4×4 (kx − iξky)I4×4 04×4 � Ψc,ξ k + � |k|<Λc,n,ξ (−1)n+1Mψc′′,ξ,† k,n ψc′′,−ξ k,n ˆHfc = 1 √NM � |k|<Λc,R,n � eik·R− |k|2λ2 2 ˜H(fc) ξξ′ (k)ψf,ξ,† R,n ψc′,ξ′ k,n + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' � , ˜H(fc)(k) = � γ v′ ⋆k− v′ ⋆k+ γ � (S19) where Ψc,ξ k = � ψc′,ξ k,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=',4 ψc′′,ξ k,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=',4 �T and k± = kx ± iky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We now define the U(8) moments with ψ basis (µ, ν = 0, x, y, z): ˆΣ(f,ξξ′) µν (R) = 1 2 � mn ψf,ξ,† R,n [T µν]nmψf,ξ′ R,m ˆΣ(c′,ξξ′) µν (k, q) = 1 2 � mn ψc′,ξ,† k+q,n[T µν]nmψc′,ξ′ k,m , ˆΣ(c′′,ξξ′) µν (k, q) = 1 2 � mn ψc′′,ξ,† k+q,n[T µν]nmψc′′,ξ′ k,m {T µν} = {ς′ νρ0, ς′ νρy, −ς′ νρx, ς′ νρz} (S20) 6 and we let ρx,y,z,0 be the Pauli matrices and the identity matrix defined in the subspace of (1+, 2−) for ξ = +1 and (2+, 1−) for ξ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ς′ x,y,z,0 are the Pauli matrices and identity matrix acting in the spin subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The previous flat U(4) moments in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S16 can be written with new electron basis as ˆΣ(f,ξ) µν (R) = 1 2 � mn ψf,ξ,† R,n [T µν]nmψf,ξ R,m = ˆΣ(f,ξξ) µν (R) ˆΣ(c′,ξ) µν (q) = 1 2NM � mn,k ψc′,ξ,† k+q,n[T µν]nmψc′,ξ k,m = 1 NM � k ˆΣ(c′,ξξ) µν (k, q) ˆΣ(c′′,ξ) µν (q) = 1 2NM � mn ψc′′,ξ,† k+q,n[T µν]nmψc′′,ξ k,m = 1 NM � k ˆΣ(c′′,ξξ) µν (k, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S21) The advantage of using the ψ basis is that the flat U(4) moments of f, c′, c′′ electrons have the same matrix structure T µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' It is also useful to consider the following Fourier transformation of electron operators and f-moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ψc′,ξ r,n = 1 √NM � k eik·rψc′,ξ k,n ψc′′,ξ r,n = 1 √NM � k eik·rψc′′,ξ k,n ˆΣ(c′,ξξ′) µν (r, r′) = 1 NM � k,q ˆΣ(c′,ξξ′) µν (k, q)eik·r′−i(k+q)r = 1 2 � mn ψc′,ξ,† r,n [T µν]nmψc′,ξ′ r′,m ˆΣ(c′′,ξξ′) µν (r, r′) = 1 NM � k,q ˆΣ(c′′,ξξ′) µν (k, q)eik·r′−i(k+q)r = 1 2 � mn ψc′′,ξ,† r,n [T µν]nmψc′′,ξ′ r′,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S22) Since the momentum of c-electron has a finite cutoff, this definition is for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' However, if we only consider the c-electrons in the first moir´e Brillouin zone, the completeness of c-electron Fourier transformation is guaranteed and a well- defined inverse Fourier transformation also exists for c electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For what follows, unless specifically mentioned, we only consider c-electrons in the first moir´e Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' we utilize the following relation � µν [T µν]ab[T µν]cd = 4δa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='dδb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c (S23) and find � µν ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2ξ2) µν (R2) = � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='bψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ2 R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a � µν ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′ 1)ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2ξ2) µν (r′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r2) = � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ r′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† r′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ2 r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a � µν ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ˆΣ(c′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2ξ2) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='R ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a ψf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='bψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='b ψc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ2 r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a (S24) 7 We change from ψ basis to f and c basis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S24 becomes(which will be used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S4) � µν � ξ,ξ′ ˆΣ(f,ξξ′) µν (R)ˆΣ(f,ξ′ξ) µν (R2) = � αα′ηη′ss′ f † R,αηsfR,α′η′s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='f † R2,α′η′s′fR2,αηs � µν � ξ,ξ′ ˆΣ(f,ξξ′) µν (R)ˆΣ(c′,ξ′ξ) µν (r, r′) = � αα′ηη′ss′ f † R,αηsfR,α′η′s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='c† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′cr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηs � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′ξ) µν (R2) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′f † R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′fR2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsη′[σx]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2 � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′ξ) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′c† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′cr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsη′[σx]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2 � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′−ξ) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′c† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′cr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′ 2ηsη′η[σx]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2[σx]α′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ξ′ ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′ξ) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′c† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′cr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηs[iσy]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2 � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ (−ξ)ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ξ′ ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′ξ) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′c† r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′ 2ηs[iσy]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2[iσy]α′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α � µν � ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξ′ ˆΣ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ξξ′) µν (R)ξ′ ˆΣ(c′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='−ξ′ξ) µν (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' r′) = � αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′ f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′η′s′c† r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2η′s′cr′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α′ 2ηs[iσy]α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α2[ησx]α′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='α (S25) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ZERO-HYBRIDZIATION LIMIT The f-c hybridization parameter γ vanishes at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='9 which is close to the actual value w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Vanishing γ corresponds to the gap closing between the remote and the flat bands in the continuum single particle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We solve the model exactly at γ = 0 and then treat it as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In this limit, the f and c electrons are coupled only through the interacting terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We now neglect ˆHV except for a mean-field treatment- for the same reason as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [122], that it will cause only a velocity renormalization of the linear conduction fermion, and keep only the remaining terms ˆHU, ˆHW , ˆHJ, ˆHMF V where ˆHMF V denotes the ˆHV with mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This now becomes a polynomially solvable Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Some remarks: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This is more complicated than the zero hybridiza- tion limit of the Anderson lattice model in which the Hilbert space factorizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This is because in both ˆHW (we have work in the limit W1 = W2 = W3 = W4), and ˆHJ terms, the f occupation number will influence the electron Hamiltonian, but in a one-body fashion, since the c operators are quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The strategy is then to solve the Hamiltonian in this limit and then add the f-c hybridization perturbatively via Schrieffer–Wolff transformation (SW) transformation [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' To solve the Hamiltonian we first need to find the on-site configurations of the f fermions (which are good quantum numbers) which then, after adding ˆHU + ˆHW + ˆHJ and the c-electron kinetic term Hc = � ηs � aa′ � |k|<Λc(H(c,η) a,a′ (k) − µδaa′)c† kaηscka′ηs − µNf (S26) give the lowest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This is a minimization problem similar in spirit to the Lieb theorems [135] with flux π (used in the Kitaev model [136]) where it is shown that the flux π configuration of a noninteracting fermion model in a background plaquette flux with each plaquette having possibly different flux is minimal at flux π per all plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Our problem, at least in the first try, might be amenable to Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Symmetries of the zero-hybridization Limit We can rewrite the ˆHU + ˆHW + ˆHJ ˆHU = U 2 � R : nfR :: nfR, ˆHW = 1 N � R : nfR : � |k|,|k′|<Λc � η2s2a Wae−i(k−k′)·R : c† kaη2s2ck′aη2s2 : , HJ = − J 2N � Rs1s2 � αα′ηη′ � |k1|,|k2|<Λc ei(k1−k2)·R(ηη′ + (−1)α+α′) : f † Rαηs1fRα′η′s2 :: c† k2,α′+2,η′s2ck1,α+2,ηs1 : (S27) Where we have defined the occupation number of the electron on-site R nfR = � αηs f † RαηsfRαηs ∈ 0, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 8 (S28) 8 which is a symmetry of the system even when W1 ̸= W3: [ ˆHU, nfR] = 0, [ ˆHW , nfR] = 0, [ ˆHJ, nfR] = 0, [Hc, nfR] = 0 (S29) The system has a large U(1)NM symmetry where NM is the number of moir´e unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' An eigenstate of the system is first indexed by nfR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Then this couples through the ˆHW and (with a smaller coefficient) through ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' There are now several questions If first we neglect ˆHJ we have a large symmetry, U(8)NM , whose generators are ˆΣ(ξξ′) µν (R) = ˆΣ(f,ξξ′) µν (R)+ˆΣ(c′,ξξ′) µν (R)+ ˆΣ(c′′,ξξ′) µν (R) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [ ˆHU, ˆΣ(ξξ′) µν (R)] = 0 , [ ˆHW , ˆΣ(ξξ′) µν (R)] = 0 , [Hc, ˆΣ(f,ξξ′) µν (R)] = 0 (S30) which gives huge number of degenerate states, all with the same occupation number nfR on-site distributed around the different α, η, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' If we take W1 = W3 = W then we have that ˆHW = W 1 N � R : nfR : � |k|,|k′|<Λc � η2s2a e−i(k−k′)·R � ηsa : c† kaηsck′aηs : = W 1 N � R : nfR : � |k|,|k′|<Λc � η2s2 e−i(k−k′)·R � n1,n2(� a U η,∗ k,an1U η,∗ k′,an2)γ† k,n1ηsγk′,n2ηs (S31) where we have introduced the eigenvectors U η k,an and eigenvalues ϵη k,n of ˆH(c,η)(k) � a′ Hc aa′(k)U η k,a′n = ϵη k,nUk,an , and the operator in the band basis γk,nηs = � a U ∗ k,anck,aηs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ˆHW is now a one-body term for c or γ, which depends on the distribution of nf R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Through Monte Carlo sampling, we can find the ground-state exactly and check whether the ground state involve uniform nf R or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' If furthermore, we assume uniform nf R = nf then the summation over R gives us a δk,k′ and we obtain an nf-dependent chemical potential of the electrons: ˆHW = Wnf � |k|,|k′|<Λc � nηs γ† knηsγknηs (S32) We can now find easily the admixture of f, c fermions at any filling N = Nf + Nc with Nf = NMnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This allows us to see, as a function of the total filling, analytically, the distribution between the f, c electrons in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We can then add the ˆHJ under the stronger assumption that f † Rαηs1fRα′η′s2 = Aαηs1,α′η′s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S33) We can then find the Aαηs1,α′η′s2 which will minimize the state energy and break the U(8) on-site symmetry to U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We can assume nf R not constant, and adopt charge density wave (CDW) or other types of translational symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We leave it for future study We can then add the f †c term through Schrieffer-Wollf (SW) transformation , which will be discussed in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 9 4 4 nu=0 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�,v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3 5 nu=-1 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�,v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 2 6 nu=-2 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�,v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 1 7 nu=-3 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 +6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�,v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H(MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 52 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' HF band structures of correlated insulator phases at ⌫ = �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (a), (b), (c) are the one-shot HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (d), (e), (f) are the self-consistent HF band structures of the VP, K-IVC, and IVC phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The color represents the composition of the energy bands, where yellow corresponds to the local orbitals and blue corresponds to the conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We have chosen w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Other parameters of the single-particle and interaction hamiltonians are given in Tables S4 and S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S7, one-shot and self-consistent HF bands of all the di↵erent states at ⌫ = �1 have a feature in common: There is a set of flat bands above the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Such flat bands are also observed in one of our previous studies [6] but have not been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (See the particle excitation spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 11 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') We now can explain the origin of the flatness through our topological heavy fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We consider the k·p expansion of the one-shot mean-field Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' For simplicity, here we mainly focus on the VP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Following the same procedure we have done to obtain the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), we obtain H (MF) VP (k) ⇡ 0 @ �W1�0⌧0&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z + iky�z⌧0)&0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�0⌧z � iky�z⌧0)&0 �W3�0⌧0&0 + M�x⌧0&0 � J 2 �0⌧z&0 0 ��0⌧0&0 + v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (kx�x⌧z + ky�y⌧0)&0 0 �(U1 + 6U2)�0⌧0&0 � U1(Of � 1 2�0⌧0&0) 1 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S332) The density matrix Of is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Comparing it to the one-shot mean field Hamiltonian at ⌫ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S305)), there are three additional terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', �W1�0⌧0&0, �W3�0⌧0&0, and �(U1 +6U2)�0⌧0&0 in the three diag- onal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' These three terms come from the Hartree channel terms ⌫fW1, ⌫fW3, ⌫f(U1 + 6U2) of HW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S284)) and HU (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S279)) and only shift the energies of the three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Without the energy shift and the couplings (�, v0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=') between c- and f-electrobs, the c-bands (the first two blocks) would have a quadratic touching at zero energy and the f-levels (the third block) have energies ±U1/2, as illustrated by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Using the parameters obtained at w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='8 (Table S6), the average energy shift of the c-electron bands (the first two blocks) is �(W1 + W3)/2 ⇡ �47meV, and the energy shift of the f-electron bands is �(U1 + 6U2) ⇡ �72meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Thus, the relative energy shift of f-electrons with respect to c-electrons is given by �E ⇡ (�72 + 47)meV ⇡ �25meV and is approximately �U1/2 ⇡ 29meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' That means, if we turn o↵ the hybridizations, the upper branch of the f-electron levels will be shifted to the quadratic touching point of the c-electrons, as shown by the red bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 3(b) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Turning on the hybridizations will gap out the quadratic touching point, then the upper branch of f-electron bands form an isolated set of flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zero hybridization γ = 0 model at different fillings of the Heavy fermions (the state assumed as the parent state carries all the filling in the heavy fermions, unlike the exact state at γ = 0 which contains both c and f fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The Fermi level is a horizontal dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The solid lines are zero hybridization scenarios while the green dashed lines are the level splittings occurring after small increase of γ from nonzero, but still smaller than the realistic value of γ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='6meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = 0 both minima of the bands are made by c electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = −1+ϵ the band is very flat while at ν = −1 − ϵ the minimum of the band is formed by the c electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This last situation changes at ν = −2, −3, at least in the absence of hybridization: both band edges are made up of heavy fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At small hybridization (green dashed) the band edges at ν = −2 ± ϵ and ν = −3 ± ϵ are still made up by heavy fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Once γ reaches its large 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='6meV value, we find that at ν = −2 − ϵ the band has changed character from heavy fermion to light fermion due to large heavy-light mixing (see violet dashed line, large curvature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' At ν = −3 − ϵ however, the portion of the heavy band as a ratio of the Brillouin zone is larger and hence upon hybridization, the portion of the mixing is smaller than at ν = −2 − ϵ, and the band has more f-character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The CDW [115] obtained at ν = −3 most likely has f-f CDW correlations between the ν = −3 + ϵ band edge (which is around K, M points and hence made up of heavy fermions) and the ν = −3 − ϵ band edge (around Γ but still made up of heavy fermions) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Zero hybridization limit without ˆHJ We first solve the zero hybridization model at J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The total Hamiltonian of the zero-hybridization model reads ˆHc + ˆHU + ˆHW + ˆHMF V (S34) We consider the solution with uniform charge distribution of f-electrons nf R = νf + 4, with integer νf ∈ [−4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=', 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' We note that, in the zero-hybridization model, the filling of f-electron of each site is a good quantum number and takes integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' The trial wavefunction of the ground state we proposed is |νf, ν⟩ = [ � R |νf⟩R]|Ψ[ν − νf, νf]⟩c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (S35) where νf denotes the filling of f electrons, ν denotes the total filling of f and c electrons, and νc = ν − νf is the filling of c electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' |νf⟩R describes a f state at site R with filling νf: � αηs : f † R,αηsfR,αηs : |νf⟩R = νf|νf⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' and we take a uniform charge distribution, so for each site, the fillings of f-electrons are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' 10 |Ψ[νc, νf]⟩s denotes a Slater determinant state corresponding to the ground state of the following one-body Hamiltonian of c electrons at filling νc = ν − νf: H′ c = ˆHc + � η,s,a � |k|<Λc Wνf : c† kaηsckaηs : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Above one-body Hamiltonian can be diagonalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' (Note that Wa is taken to be orbital independent): H′ c = � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='n (Eknη + Wνf)γ† knηsγknη where n is the band index and γ is the operator in the band basis γknηs = � a U η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='∗ kanckaηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ckaηs = � n U η kanγknηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' � a′ H(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='η) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='a′ (k)U η ka′n = EknηU η kan (S36) The energy of |νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩ state is then Eνf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ν/NM = ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHU|νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM + ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHV |νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM + ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHW |νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM + ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHc|νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHU|νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM = U 2 ν2 f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHMF V |νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM = V0 2Ω0 ν2 c = V0 2Ω0 (ν − νf)2 ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHW |νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM = Wνfνc = Wνf(ν − νf) ⟨νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν| ˆHc|νf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' ν⟩/NM = 1 NM � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='nηs (Eknη)⟨Ψ[νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf|γ† knηsγknηs|Ψ[νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf]⟩ Eνf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='ν/NM = U1 2 ν2 f + V0 2Ω0 (ν − νf)2 + Wνf(ν − νf) + 1 NM � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content='nηs (Eknη)⟨Ψ[νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf|γ† knηsγknηs|Ψ[νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' νf]⟩ For a given total filling ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' we compare the energies of different νf (νf can only be an integer) and take the one with the lowest energy as our ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' In the M = 0 limit, the analytical expression of Eνf ,ν/NM can be easily given At M = 0, c electron dispersion becomes ±v⋆|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Filling νc = ν − νf conduction electrons is equivalent to fill the 8-fold (8 = 2 × 2 × 2, 2 for spin, 2 for valley, 2 for orbital) linear-dispersive bands up to certain momentum k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' Depending on the sign of νc, we either fill electrons (νc > 0) or holes (νc < 0) to the Dirac sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE3T4oBgHgl3EQfsgvt/content/2301.04669v1.pdf'} +page_content=' This gives the following equations to determine k0 8 AMBZ � |k|