diff --git "a/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt" "b/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/FdE3T4oBgHgl3EQftAvj/content/tmp_files/load_file.txt" @@ -0,0 +1,2225 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf,len=2224 +page_content='Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Hu,1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Rai,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Crippa,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Herzog-Arbeitman,4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C˘alug˘aru,4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Wehling,2, 5 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Sangiovanni,3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Valent´ı,6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Tsvelik,7 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Bernevig4, 1, 8, ∗ 1Donostia International Physics Center, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Institute of Theoretical Physics, University of Hamburg, Notkestrasse 9, 22607 Hamburg, Germany 3Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='qmat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Universit¨at W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 97074 W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Germany 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' New Jersey 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' USA 5The Hamburg Centre for Ultrafast Imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 22761 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Germany 6Institut f¨ur Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Goethe Universit¨at Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Max-von-Laue-Strasse 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 60438 Frankfurt am Main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Germany 7Division of Condensed Matter Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' NY 11973-5000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' USA 8IKERBASQUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Basque Foundation for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Spain We use the topological heavy fermion (THF) model [1] and its Kondo Lattice (KL) formulation [2] to study the possibility of a symmetric Kondo state in twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Via a large-N approximation, we find a symmetric Kondo state in the KL model at fillings ν = 0, ±1, ±2 where a KL model can be constructed [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the symmetric Kondo state, all symmetries are preserved and the local moments are Kondo screened by the conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At the mean-field level of the THF model at ν = 0, ±1, ±2, ±3 we also find a similar symmetric state that is adiabatically connected to the symmetric Kondo state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We study the stability of the symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] and find the ordered states to have lower energy at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, moving away from integer fillings by doping holes to the light bands, our mean-field calculations find the energy difference between the ordered state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency towards Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We expect that including the Gutzwiller projection in our mean-field state will further reduce the energy of the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In order to include many-body effects beyond the mean-field approximation, we also performed dynamical mean-field theory (DMFT) calculations on the THF model in the non-ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The spin susceptibility follows a Curie behavior at ν = 0, ±1, ±2 down to ∼ 2K where the onset of screening of the local moment becomes visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This hints to very low Kondo temperatures at these fillings, in agreement with the outcome of our mean-field calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At non-integer filling ν = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5, ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='8, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 DMFT shows deviations from a 1/T-susceptibility at much higher temperatures, suggesting a more effective screening of local moments with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we study the effect of a C3z-rotational-symmetry-breaking strain via mean-field approaches and find that a symmetric phase (that only breaks C3z symmetry) can be stabilized at sufficiently large strain at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Our results suggest that a symmetric Kondo phase is strongly suppressed at integer fillings, but could be stabilized either at non-integer fillings or by applying strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Introduction— The experiments on magic-angle (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='05◦) twisted bilayer graphene (MATBLG) [4–6] have es- tablished the existence of a variety of interesting phases [7– 28], including correlated insulating phases [29–39] and super- conductivity [40–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Their discovery has been followed by considerable theoretical efforts [45–69] aimed at understand- ing their origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' An extended Hubbard model has been con- structed to analyze the interacting physics [60, 70–82], how- ever, due to the non-trivial topology of the flat bands [83– 91], certain symmetries become non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Alternatively, an approach based on a momentum space model has been con- sidered [92–100], in which correlated insulators [101–108], superconductivity [109–114], and other correlated quantum phases [115–119] have been identified and studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Besides, various numerical calculations [120–127] have also been per- formed to investigate the correlated nature of the phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, the active phase diagram including the states at non-integer fillings is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The exact map- ping between the MATBLG and topological heavy-fermion ∗ bernevig@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='edu model constructed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] could be used for develop- ments in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This mapping establishes a bridge between heavy-fermions [3, 128–131] and moir´e systems [1, 2, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The presence of localized moments in MATBG is supported by recent entropy measurements which have found a Pomeranchuk-type transition [19, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A large entropy ob- served at high-temperatures, originates from weakly interact- ing local moments whose fluctuations are quenched at low temperatures [19, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since a similar behavior is observed in heavy fermion systems [3, 128], where the fluctuating lo- cal moments are screened by conduction electrons (Kondo effect), this observation is suggestive of a Kondo state with screened local moments in MATBLG [128, 134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In this paper we use the KL model [2], to describe and study the symmetric Kondo (SK) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We focus on integer fillings ν = 0, ±1, ±2, where a KL model can be constructed [2] (a KL description fails at ν = ±3 as demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The SK phase preserves all symmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' the local moments are screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We discuss the topology and the band struc- ture of the SK state and extend the study to the THF model where we identify the symmetric state that is adiabatically connected to the SK state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In order to address integer arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='04673v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='str-el] 11 Jan 2023 2 and fractional fillings on equal footing, we perform both a mean-field and a dynamical mean-field theory (DMFT) calcu- lations of the THF defining a “periodic Anderson model” with a momentum-dependent hybridization between the correlated f- and the dispersive c-electrons in the non-ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Our mean-field calculations indicate that the energy of the symmetric state is higher than that of the ordered (symmetry- breaking) states found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] at integer filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We thus conclude that ordered states are more energetically favored at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DMFT supports this picture as we obtain a Curie behavior of the local spin susceptibility at integer fillings, down to very low temperatures ∼ 2K, hinting to a very small Kondo scale (lower than ∼ 2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Together with the mean-field results we would then expect an ordered state to be favored at low temperatures for these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Turning to the effect of doping, instead, from our mean- field analysis, we find that the energy difference between the symmetric phase and the ordered phase can be sizeably re- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Doping hence suppresses the ordering and enhances the Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This conclusion is further supported by the DMFT results at non-integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, we find clear deviations from the Curie behavior in the entire range from 10K down to ∼1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Even though it is computationally too demanding to go further down in temperature, we point out that our evidence of a clear-cut difference in the screening properties between integer and fractional fillings is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DMFT treats indeed local quantum fluctuations exactly [135] and hence takes into account the many-body processes that can potentially lead to the screening of local moments at any filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since realistic samples have intrinsic strains, we finally study the effect of a C3z-breaking strain on the symmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Our mean-field calculations show that the order is sup- pressed by the strain effect and a symmetric state can be sta- bilized at a sufficiently large strain at ν = 0, ±1, ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In summary, we conclude that a symmetric Kondo phase is absent at integer fillings of MATBLG, but could in princi- ple be stabilized either at non-integer fillings or by applying strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Topological Heavy Fermion model and the Kondo lattice model— The THF model [1] contains two types of electrons: topological conduction c-electrons (ck,aηs) and localized f- electrons (fR,αηs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The operator ck,aηs annihilates conduc- tion c-electron with momentum k, orbital a ∈ {1, 2, 3, 4}, valley η ∈ {+, −} and spin s ∈ {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At the ΓM-point for each valley and each spin projection, c-electrons in the orbital 1 and 2 transform according to the Γ3 irreducible represen- tation (of magnetic space group P6′2′2) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The remaining c-electrons (a = 3, 4) at the same valley with the same spin projection transform in the Γ1 ⊕ Γ2 reducible representation (of magnetic space group P6′2′2) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We will call them Γ3 c- electrons (a = 1, 2) and Γ1⊕Γ2 c-electrons (a = 3, 4) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' fR,αηs is the annihilation operator of the f-electron at the moir´e unit cell R with orbital α ∈ {1, 2}, valley η and spin s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Hamiltonian of the THF model [1, 136] is ˆHT HF = ˆHc + ˆHfc + ˆHU + ˆHW + ˆHV + ˆHJ (1) where ˆHc describes the kinetic term of conduction elec- trons, ˆHfc describes the hybridization between f-c elec- trons [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The interactions include an on-site Hubbard interaction of f-electrons ( ˆHU with U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='95meV), a re- pulsion between f- and c-electrons ( ˆHW with W = 48meV), a Coulomb interaction between c-electrons ( ˆHV with V (q = 0)/Ω0 = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='33meV and Ω0 the area of moir´e unit cell), and a ferromagnetic exchange coupling between f-and c-electrons ( ˆHJ with J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='38meV) [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Based on the THF model [1], a KL model of MATBLG has been constructed via a generalized Schrieffer–Wolff (SW) transformation as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The KL model is de- scribed by the following Hamiltonian ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (2) where ˆHc, ˆHJ come from the original THF model and ˆHcc, ˆHK emerge from the SW transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˆHcc is the one- body scattering term of Γ3 c-electrons with the form of ˆHcc = � |k|<Λc � a,a′∈{1,2} η,s e−|k|2λ2 : c† k,aηsck,a′ηs : � −1 Dνc,νf + −1 Dνc,νf � � γ2/2 γv′ ⋆(ηkx − iky) γv′ ⋆(ηkx + iky) γ2/2 � a,a′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (3) λ is the damping factor of the f-c hybridization in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' γ, v′ ⋆ characterize the zeroth order and linear order f-c hybridization of the THF model with v′ ⋆ characterizing a k-dependent hybridization matrix [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' D1,νc,νf and D2,νc,νf are defined as D1,νc,νf = (U − W)νf − U 2 + (−V (0) Ω0 + W)νc D2,νc,νf = (U − W)νf + U 2 + (−V (0) Ω0 + W)νc , (4) where νf, νc are the filling of f- and c-electrons deter- mined from the calculations of the THF model at the zero- hybridization limit [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We point out that in the single-orbital Kondo model, the one-body scattering term merely introduces a chemical potential shift [3, 137] of the c-electrons and is usually omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, in our model, ˆHcc cannot be ig- nored for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' First, ˆHcc is k-dependent and thus introduces additional kinetic energy to the conduction elec- trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S24, we observe the k-dependency mainly comes from the linear k term that is proportional to v′ ⋆ and can be traced back to the k-dependency of the hybridization matrix in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Secondly, even if we drop the v′ ⋆ term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S24 (v′ ⋆ = 0 corresponding to the chiral limit [1]), ˆHcc still produces an energy shift for the Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus ˆHcc leads to the energy splitting between Γ3 and Γ1 ⊕ Γ2 c- electrons and cannot be simply treated as a shift of the chem- ical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˆHK is the Kondo interaction between f- and Γ3 c-electrons whose explicit form is given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [2, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Kondo interaction consists of two parts: the zeroth order Kondo in- teraction proportional to γ2/Dνc,νf and the first order Kondo 3 interaction proportional to γv′ ⋆/Dνc,νf , where D−1 νc,νf = −D−1 1,νc,νf + D−1 2,νc,νf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The zeroth order Kondo interaction term describes the antiferromagnetic interaction between the U(8) moments of the f- and the Γ3 c-electrons and has a U(8) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The linear-order Kondo interaction only has a flat U(4) symmetry and is k-dependent [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˆHJ is the fer- romagnetic exchange interaction between Γ1 ⊕ Γ2 c- and f- electrons that already exists in the TFH model [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also note that, for both the THF model and the KL model, ground states at filling ν and −ν are connected by a charge- conjugation transformation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This can be broken by other one-body terms which we did not consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, in what follows, we only focus on ν ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field Hamiltonian of the Kondo model— We next per- form a mean-field study of the KL model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This MF sup- presses the RKKY interaction and essentially restores the hy- bridization term ˆHfc of the original periodic Anderson model, but in a renormalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' It becomes exact in the N → ∞ limit (we have N = 4 here which corresponds to the approxi- mate flat U(4) symmetry of the KL Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At the mean-field level, the Kondo interaction ˆHK can be written as (see Supplementary Materials (SM)) ˆHMF K = � R,|k|<Λc � αηs eik·R−|k|2λ2/2 √NMDνc,νf � − f † R,αηsck,aηs � γ2V ∗ 1 + γv′ ⋆V ∗ 2 V ∗ 1 (ηkx − iky) V ∗ 1 (ηkx + iky) γ2V ∗ 1 + γv′ ⋆V ∗ 2 � α,a + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � + NM � γ2|V ∗ 1 |2 + γv′ ⋆(V ∗ 1 V ∗ 2 + V ∗ 2 V ∗ 1 ) � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (5) where we have introduced the following mean fields V ∗ 1 = � R,|k|<Λc � αηs eik·R−|k|2λ2/2 √NMNM ⟨Ψ|f † R,αηsck,αηs|Ψ⟩ V ∗ 2 = � R,|k|<Λc � αaηs eik·R−|k|2λ2/2 √NMNM (ηkxσx + kyσy)αa ⟨Ψ|f † r,αηsck,αηs|Ψ⟩ (6) with |Ψ⟩ being the mean-field ground state, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' denotes the Hartree term (⟨f †f⟩, ⟨c†c⟩) whose explicit formula is in the Supplementay Materials (SM) [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Several points are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' First, as we have mentioned above, the mean field restores the hybridization of the original Anderson model, but in a renormalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' V ∗ 1 , V ∗ 2 describe the renormalized hybridization between the f- and Γ3 c-electrons driven by the Kondo interactions between two types of electrons(f and Γ3 c) [3, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Second, it is necessary to keep the Hartree con- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the canonical Kondo model, the Hartree term merely produces a chemical potential shift (in the case without symmetry breaking) and hence can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, Hartree contributions (see SM [136]) are k-dependent because of the k-dependency of the Kondo interactions, and thus contribute to the dispersion of the conduction c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Furthermore, since only Γ3 c-electrons contribute to the Kondo interaction, the Hartree term also produces an energy splitting between the Γ3 and the Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for ˆHJ, we perform a similar mean-field decoupling ˆHMF J =J � R,|k|<Λc,αηs eik·R √NM � V3δ1,η(−1)α+1f † R,αηsck,α+2ηs + V4δ−1,η(−1)α+1ηf † R,αηsck,α+2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � − JNM � |V3|2 + |V4|2 � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (7) where we have introduced the following two mean-field aver- ages that describe the f-c hybridization: V3 = � R,|k|<Λc � αη,s eik·Rδ1,η(−1)α+1 √NMNM ⟨Ψ|f † R,αηsck,α+2ηs|Ψ⟩ V4 = � R,|k|<Λc � αη,s eik·Rδ−1,η(−1)α+1 √NMNM ⟨Ψ|ηf † R,αηsck,α+2ηs|Ψ⟩ , (8) To impose the filling of the f-electrons to be νf, we intro- duce the Lagrange multiplier [134, 136, 138]: ˆHλf = � R,αηs λf � : f † R,αηsfR,αηs : −νf � (9) with λf to be determined self-consistently [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we introduce a chemical potential µc to the c-electrons ˆHµc = −µc � |k|<Λc,aηs : c† k,aηsck,aηs : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (10) In the calculation, we tune µc and λc together to fix both the total filling ν = νf + νc and the filling of f-electrons [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The final mean-field Hamiltonian of the KL model now is ˆHMF Kondo = ˆHc + ˆHcc + ˆHMF K + ˆHMF J + ˆHλf + ˆHµc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (11) We then self-consistently solve the mean-field equations (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = νf = 0, −1, −2 (where a KL model can be constructed), we identify a SK state that preserves all the symmetries and is characterized by V ∗ 1 ̸= 0, V ∗ 2 ̸= 0, V ∗ 3 = 0, V ∗ 4 = 0 [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We comment that the exchange interaction ˆHJ [1] between f- and Γ1 ⊕ Γ2 c-electrons is fer- romagnetic, and hence disfavors the singlet formation or hy- bridization (V3, V4) between f- and Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find that V3, V4 vanish (their numerical amplitudes are smaller than 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In fact, ˆHJ favors the triplet formation or pair- ing formation (f †c†), where both lead to a symmetry-breaking state at the mean-field level and are beyond our current con- sideration of SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Properties of the symmetric Kondo phase— In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1, we plot the band structure of the SK phase and compare it with the non-interacting THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find the hybridization in 4 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (a) Band structure of the non-interaction THF model at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (b), (c), (d) Band structure of the SK phase at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' the SK state defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 5 to be enhanced compared to the non-interacting limit of THF model, which is clear from the increase of the gap of the Γ3 states at the Γ point [1] from its non-interacting value 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='75meV at ν = 0, to 168meV, 190meV, 213meV at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also find that in the SK phase the bandwidths of the flat bands at ν = −1, −2 become 16meV, 53meV, which are (much) larger than the non-interacting flat-band bandwidth (= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4meV) of the THF model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, at ν = 0, the flat-band bandwidth is the same as the non-interacting flat-band band- width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This is because, at ν = 0, the one-body scattering term and the Hartree contributions from ˆHK, ˆHJ all van- ish [136], and the enhanced hybridization pushes the remote bands away from the Fermi energy and does not change much the band structures of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, unlike the non-interacting case, here we found the flat bands are mostly formed by Γ1⊕Γ2 c-electrons with orbital weights larger than 70% at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This is because the large f-c hy- bridization induced by V1, V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 5) pushes the energy of Γ3 c- and f-electrons away from the Fermi energy and reduces their orbital weights [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The flat bands in the SK phase form Γ1 ⊕ Γ2, M1 ⊕ M2 and K2K3 representations at ΓM, MM, KM respectively, and have the same topology as the flat bands in the non-interacting THF model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' More explicitly, the flat bands for each val- ley and each spin projection belong to a fragile topology [1] at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = 0, due to the additional particle-hole symmetry, flat bands have a stable topology [1, 85, 91, 136], which is characterized by the odd winding number of the Wil- son loop as shown in supplementary material [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We men- tion that the interplay between Kondo effect and the topologi- cal bands has also been studied in various other systems [139– 144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Symmetric phase in the topological heavy-fermion model— We next investigate the similar symmetric phase in the THF model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We first focus on integer fillings ν = 0, −1, −2, −3 and perform the mean-field calculations of THF as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By enforcing the mean- field Hamiltonian to preserve all the symmetries, we are able to identify a symmetric state that preserves all the symmetries at ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To observe the stability of the sym- metric phase, we compare its energy (Esym) with the energy (Eorder) of the ordered (symmetry-breaking) ground states derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The ordered ground states in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] are a Kramers inter-valley-coherent (KIVC) state at ν = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0 10 20 30 40 50 E/meV = -3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0 10 20 30 40 50 = -2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0 10 20 30 40 50 = -1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0 10 20 30 40 50 = 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Doping dependence of the ground state energy difference ∆E = Esym − Eorder near integer fillings νt = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' a KIVC+valley polarized (VP) state at ν = −1, a KIVC state at ν = −2 and a VP state at ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We point out that at ν = −3 other states with lower energy exist [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In our numerical calculations, we find ∆E = Esym − Eorder = 47meV, 40meV, 33meV, 23meV at ν = 0, −1, −2, −3 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In all integer filling cases, the symmetric states have higher energy, and the ground states cannot be the sym- metric state, which is consistent with the previous calcula- tions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1, 2, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Note that our mean-field calcula- tion does not include a Gutzwiller projection to fix the fill- ing of f-electrons at each site, and hence we expect the en- ergy of projected symmetric states will be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, as we show later, after including the effect of local correlations via the DMFT approach, we confirm that the Kondo phase, which is adiabatically connected to the symmetric phase in the mean-field calculations, is strongly suppressed at integer fillings (down to temperatures ∼ 1-2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This further supports the development of ordering at integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effects of doping— We next investigate the effects of dop- ing, first at the level of mean-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We stick to a nar- row region ν ∈ [νint−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5, νint+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5] near each integer filling νint = 0, −1, −2, −3 and compare the energies of the ordered states Eord and the symmetric states Esym in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To find the ordered state solutions, we first initialize the cal- culations with the mean-field solutions at integer filling νint and fill the mean-field bands up to current filling ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then self-consistently solve the mean-field equations and calculate the energy of the resulting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We obtain the symmetric- state solution in a similar manner but take the symmetric so- lution at νint as initialization and enforce the symmetry of the mean-field Hamiltonian during the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2 displays a plot of the difference of the ground state energies 5 ∆E = Esym −Eorder as a function of doping ∆ν = ν −νint near νint = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We observe that hole doping at ν = 0, −1, −2, −3 and electron doping at ν = 0 decreases the ∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Doping holes at ν = 0, −1, −2, −3 and doping elec- trons at ν = 0 to the ordered states is equivalent to doping the light (dispersive) bands which are mostly formed by conduc- tion c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' After doping, the conduction electrons will stay close to the Fermi energy, and then enhance the tendency towards the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, doping electrons at ν = −1, −2 to the ordered states is equivalent to doping heavy (flat) bands which mostly come from the f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Because of the flatness of the band, we find the nature of the ordered states will change with doping (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For example at ν = 2, the KIVC order is suppressed by the electron doping (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus, ∆E will be affected by both, changes of ordering and doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, a sizeable change of the order parameters is not observed for hole doping at ν = 0, −1, −2, −3 and also electron doping at ν = 0 (see SM [136]), because we are dop- ing conduction c-electrons in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also point out the complexity of ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' First, other states that break trans- lational invariant [145] could have lower energy than the VP state we currently considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Second, even for the VP state, doping electrons is equivalent to doping both heavy and light bands [1], since both light and heavy bands appear in the elec- tron doping case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In practice, as we increase ∆ν, we find that, at ν = −1, −2, ∆E will first decrease and then increase and, at ν = −3, ∆E will always increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In summary, we conclude that hole doping can suppress the long-range order and enhance the tendency towards the Kondo effect near ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Electron doping, depending on the fillings, could also enhance the tendency toward the Kondo ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, on the electron doping side, the change of or- der moments indicates the importance of the correlation effect which could be underestimated in the mean-field approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the next section, we provide a more comprehensive study of the doping effect using the DMFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dynamical mean-field theory results of the THF model— In the following, we present the dynamical mean-field the- ory resultss of the THF model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1), where we describe the local quantum many-body effects of the density-density Hub- bard term ˆHU within the f-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The ˆHW and ˆHV in- teractions involving density fluctuations on the c orbitals are accounted for at the static mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We neglect or- dered phases and perform calculations in the non-ordered one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' There, we focus in particular on lifetime effects, quasiparticle weights and exploit the ability of DMFT to take local vertex corrections to the spin-spin correlation function into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' First, DMFT finds a qualitative difference between the strong quasiparticle renormalization when the f+c manifold is occupied with an integer number of electrons and a lighter Fermi liquid at fractional fillings: this can be seen from the scattering rate Γf = −ImΣf(ω = 0) which is shown as a function of the total filling ν at T = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='6K (light blue empty circles) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The largest scattering rates are found close to ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0, progressively decreasing as one moves away from the charge neutrality point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Correspond- ingly, the spectral weight at the Fermi level (black and grey solid circles) is suppressed at these fillings, with a residual nonzero value due to the finite temperature on the one hand and the resilient f/c hybridization on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3(b) illustrates the temperature-dependent screening of the local magnetic moment on the f orbitals at different fill- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A flat T · χloc spin(ω = 0) indicates Curie behavior and a well-defined effective local moment, while deviations signal the onset of screening and a crossover towards a renormalized Pauli-like behavior, in agreement with the general expecta- tion of zero-temperature Fermi-liquid in the periodic Ander- son model [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' While at ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 and -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 the 1/T local spin susceptibility persists down to 1-2 K, the fractional fillings deviate from Curie at much higher temperatures, in line with the better Fermi-liquid nature signaled by the single- particle quantities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As in the Hartree-Fock treatment of the THF model, DMFT confirms the difference between electron doping and hole dop- ing (particle-hole asymmetry) near integer filling ν = −1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, DMFT reveals particle-hole asymmetric scattering rates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 3(a)) and also in the difference of effective local moments at ν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='8 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In summary, our DMFT calculations confirm that the Kondo phase is strongly suppressed at integer fillings ν = 0, −1, −2, increasing the propensity towards long-range order at these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, by doping the system, the develop- ment of Kondo screening (starting from ∼ 10K) is observed, which suggests that doping could enhance the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This picture is consistent with our mean-field calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effects of strain— Since twisted bilayer graphene samples exhibit intrinsic strain [147] and the ordered states are disfa- vored by strain, we investigate the effect of strain on the sym- metric state of THF model via mean-field approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We focus on ν = 0, −1, −2, −3 and introduce the following Hamilto- nian [136] that qualitatively characterizes the effect of strain ˆHstrain = α � R,ηs (f † R,1ηsfR,2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=') where α is proportional to the strain amplitude (we leave the construction of a realistic strain Hamiltonian [148–150] for a future study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A non-zero α breaks the C3z symmetry but preserves all other symmetries [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We compare the ground state energies of the symmetric states (Estrain sym ) and the or- dered states (Estrain ord ) at non-zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To obtain the sym- metric state solution, we solve the mean-field equations by requiring the mean-field Hamiltonian to satisfy all symme- tries except for the C3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We obtain the solution of the or- dered states by initializing the mean-field calculations with the ordered ground states at zero strain and then perform self- consistent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 4, we plot the difference be- tween the ground state energies of the symmetric and the or- dered states ∆Estrain = Estrain sym − Estrain order as a function of the effective strain amplitude α with 0meV < α < 20meV at ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We observe that ∆E at ν = 0, −1 van- ishes at sufficiently large strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A detailed analysis [136] of the wavefunction shows that the ordered state cannot be sta- bilized and converged to a C3z broken symmetric solution at large strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By further increasing strain, we find a symmet- ric state at ν = −2 can also be stabilized at α ∼ 45meV 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 A( = 0) a Atot Af Ac f 0 2 4 6 8 10 T [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 T loc spin( = 0) b total eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='39 total eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='34 total eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='19 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='80 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='00 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='20 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='00 2 1 0 2 0 f/c f c 0 20 40 f [meV] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DMFT solution of the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (a) Doping ν dependent low-energy spectral function at the Fermi level (A(ω = 0)) for the full system Atot, the c- (Ac) and the f-electrons (Af) at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Also shown is the scattering rate Γf as extracted from the local f-electron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (b) Effective local moment T · χloc spin(ω = 0) as a function of temperature T for different doping levels ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='0 /meV 0 2 4 6 8 10 12 14 16 Estrain/meV = 0 = 1 = 2 = 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Energy difference ∆Estrain = Estrain sym − Estrain ord between the symmetric state that only breaks C3z symmetry (Estrain sym ) and the ordered state (Estrain ord ) as a function of α - a parameter charac- terizing the strain amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that even at zero strain α = 0, a symmetric state that only breaks C3z symmetry has lower energy than the fully symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus ∆Estrain at α = 0 is smaller than the corresponding ∆E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (see SM [136]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We conclude that a symmetric phase can be stabilized by sufficiently large strain at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for ν = −3, we mention that other ordered states, that break translational symmetry and have lower energy than the VP state, exist even at zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We leave a systematical analy- sis of ν = −3 for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we comment that even at zero strain, a symmetric state that breaks C3z symmetry has lower energy than the fully symmetric state that preserves all the symmetries (including C3z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, ∆Estrain (energy difference between a symmetric state that only breaks C3z and an ordered state) at zero strain α = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 4 is smaller than the corresponding ∆E (energy difference between a fully symmetric state and an ordered state) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Discussion and summary— Our main result is that an or- dered state, instead of a SK state, will be the ground state of the system at integer filling ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Our mean- field calculations of THF model indicate ground state energy of the symmetric state is higher than the one of the ordered states at these fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Via DMFT calculations, we find the Kondo temperature to be substantially smaller than 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus, we conclude the Kondo effect is suppressed at integer filling ν = 0, −1, −2, −3, and the ground state is likely to be an ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, our mean-field calculations suggest doping can reduce the energy difference between the symmet- ric state and the ordered state enhancing the tendency towards the SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This has also been confirmed by the DMFT calculations which show a strong deviation from the Curie Weiss law at non-integer fillings ν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='8, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 al- ready around 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Furthermore, we show that a sufficiently large C3z breaking strain could also stabilize a symmetric state that only breaks the C3z symmetry at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, we conclude both doping and strain enhance the Kondo effect and could, in principle, stabilize a SK state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Our results may explain the recent entropy experiments [18, 19] which reveal a high-temperature phase with fluctuating mo- ments and a low-temperature Fermi liquid phase with unpolar- ized isospins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This could be understood as a sign of screening of the local moments and the development of the SK phase at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As far as the SK state is concerned, we have performed a systematic study of its band structure and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Via the mean-field approach, we successfully identified the SK state in the KL model, and a symmetric state, that is adiabatically connected to the SK state, in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For the SK state in the KL model, we find that the Γ3 states near the ΓM point have been pushed away, and the bandwidth of the flat bands is enlarged at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The hybridization between f-electrons and Γ3 c-electons is enhanced by the Kondo in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Consequently, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The topology of the flat bands remains the same as in the non-interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, for the sym- metric state in the THF model, the enhanced f-c hybridization does not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We mention that the mean-field solution of 7 the symmetric state in the THF model underestimates the cor- relation effect, which could be the origin of the weak f-c hy- bridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We expect introducing a Gutzwiller projector will give a more precise description of the symmetric state in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Note added— After finishing this work, we learned that re- lated, but not identical, results have also recently been ob- tained by the S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Das Sarma’s [151], P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Coleman’s [134], and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Song’s groups [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also mention that results from Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Song’s group are compatible with our DMFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Acknowledgements— B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' B.’s work was primarily sup- ported by the DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DE-SC0016239, the Si- mons Investigator Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 404513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' was supported by the European Research Council (ERC) under the Euro- pean Union’s Horizon 2020 research and innovation program (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 101020833).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' was supported by a Hertz Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' was supported by the DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DE-SC0016239 and the Simons Investigator Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 404513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' was supported by the Office of Ba- sic Energy Sciences, Material Sciences and Engineering Divi- sion, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Department of Energy (DOE) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' R, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' K, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' thank the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) for funding through QUAST FOR 5249- 449872909 (Projects P4 and P5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R.' metadata={'source': 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Andrei Bernevig, “Landau level of fragile topology,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' B 102, 041402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [88] 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' B 100, 035115 (2019).' metadata={'source': 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Quantum Systems,” Computer Physics Communications 196, 398–415 (2015), arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='01952 [cond-mat, physics:physics].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [156] Priyanka Seth, Igor Krivenko, Michel Ferrero, and Olivier Parcollet, “TRIQS/CTHYB: A Continuous-Time Quantum Monte Carlo Hybridization Expansion Solver for Quantum Impurity Problems,” Computer Physics Communications 200, 274–284 (2016), arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='00175 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [157] Markus Aichhorn, Leonid Pourovskii, Priyanka Seth, Veron- ica Vildosola, Manuel Zingl, Oleg E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Peil, Xiaoyu Deng, Jernej Mravlje, Gernot J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Kraberger, Cyril Martins, Michel Ferrero, and Olivier Parcollet, “TRIQS/DFTTools: A TRIQS application for ab initio calculations of correlated materials,” Computer Physics Communications 204, 200–208 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 13 Supplementary Materials CONTENTS References 7 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Toplogical heavy-fermion model 14 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Kondo lattice model 15 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Symmetry 16 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field solutions of the Kondo lattice model 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field decoupling of ˆHK 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock term 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Hartree term 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock and Hartree terms 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field decoupling of ˆHJ 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock term 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Hartree term 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock and Hartree terms 21 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Filling constraints and mean-field equations 21 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of the symmetric Kondo state 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Properties of the symmetric Kondo state 24 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field solutions of the topological heavy-fermion model 27 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of fully symmetric state 27 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of the symmetric state in the presence of strain 29 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effect of doping 30 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effect of strain 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ν = −1 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ν = −2 34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Discussions about ν = −3 36 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Strain 36 S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dynamical mean field theory: implementation 39 14 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' TOPLOGICAL HEAVY-FERMION MODEL The topological heavy-fermion (THF) model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] takes the following Hamiltonian ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV + ˆHµ (S12) 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � (S13) where σ0,x,y,z are identity and Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ΓM point (k = 0) of the moir´e Brillouin zone (MBZ), 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Λc is the momentum cutoff for the c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' N is the total number of moir´e unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The parameter values are v⋆ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='303eV·˚A, M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='697meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � , (S14) 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' NM is the number of moir´e unit cells and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The hybridization matrix H(fc,η) has the form of H(fc,η)(k) = �γσ0 + v′ ⋆(ηkxσx + kyσy), 02×2 � (S15) which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The parameter values are γ = −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='75meV, v′ ⋆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='622eV · ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˆHU (U = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='89meV) describes the on-site interactions of f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˆHU = U 2 � R : nf R :: nf R :, (S16) 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α2η2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.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 : (S17) where J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.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′ : (S18) where we take W1 = W2 = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='03meV and W3 = W4 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='20meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Coulomb interaction between c electrons has the form of ˆHV = 1 2Ω0N � η1s1a1 � η2s2a2 � |k1|,|k2|<Λc � q |k1+q|,|k2+q|<Λc V (q) : c† k1a1η1s1ck1+qa1η1s1 :: c† k2+qa2η2s2ck2a2η2s2 : (S19) 15 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/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='33meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We will always treat ˆHV at mean-field level (int both the THF model and the Kondo lattice (KL) model) [1] ˆHV ≈ V (0) Ω0 νc � |k|<Λc,a,η,s c† k,aηsck,aηs − V (0) 2Ω0 NMν2 c + V (0) Ω0 � |k|<Λc 8νc (S20) where νc is the filling of c electrons νc = 1 NM � |k|<Λc,a,η,s⟨Ψ| : c† k,aηsck,aηs : |Ψ⟩ with |ψ⟩ the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we introduce a chemical potential term ˆHµ = −µ � |k|<Λc,aηs c† k,aηsck,aηs − µ � R,αηs f † R,αηsfR,αηs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S21) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' KONDO LATTICE MODEL The Kondo lattice model is derived by performing a generalized Schrieffer-Wolff (SW) transformation on the topological heavy fermion model (detailed derivation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Hamiltonian has the form of ˆHKondo = ˆHc + ˆHV + ˆHW + ˆHJ + ˆHK + ˆHcc − ˆHµc (S22) where ˆHc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S13), ˆHV (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S19) and ˆHW (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S18) and ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S17) come from the original TFH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Kondo interactions and the one-body scattering term are ˆHK = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k′|<Λc � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2 NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf : f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ :: c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs : � γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a + γv′ ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a[η′k′ xσx − k′ yσy]α′a′ + γv′ ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′[ηkxσx + kyσy]αa � (S23) and ˆHcc = − � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2} e−|k|2λ2� 1 D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf + 1 D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf � � γ2/2 γv′ ⋆(ηkx − iky) γv′ ⋆(ηkx + iky) γ2/2 � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ : c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs : (S24) where D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf = (U − W)νf − U 2 + (−V0 Ω0 + W)νc ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf = (U − W)νf + U 2 + (−V0 Ω0 + W)νc Dνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf = � − 1 D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf + 1 D2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S25) We point out that, at ν = νf = νc = 0, D1,νc,νf = −D2,νc,νf and the on-body term ˆHcc(= 0) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that in the Kondo model the filling of f electron at each site is fixed to be νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then we can replace � αηs : f † R,αηsfR,αηs : with νf and ˆHW becomes ˆHW = � |k|<Λc,|k′|<Λc,aηs � Q Wνf : c† k,aη′s′ck′,aηsδk,k′+Q (S26) where Q ∈ {mbM1 + mbM2|m, n ∈ Z} and bM1, bM2 are the reciprocal lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' If we focus on the conduction electrons within the first MBZ, we can replace δk,k′+Q by δk,k′ and ˆHW = � |k|<Λc,aηs � Q Wνf : c† k,aη′s′ck,aηs (S27) which is a chemical shift of conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also set W1 = W2 = W3 = W4 = W = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='12meV in ˆHW to simplify the SW transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The realistic values of W1,2,3,4 are not identical but the difference is about 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we introduce a chemical potential µc to tune the filling of the system ˆHµc = −µc � |k|<Λc,aηs : c† k,aηsck,aηs : (S28) 16 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' SYMMETRY We now provide the symmetry transformation of electron operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For a given symmetry operation g, we let Df(g), Dc′(g), Dc′′(g) denote the representation matrix of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons: gf † R,αηsg−1 = � α′η′s′ f † gR,α′η′s′Df(g)α′η′s′,αηs gc† k,aηsg−1 = � a′∈{1,2},η′s′ c† gk,a′η′s′Dc′(g)a′η′s′,aηs, a ∈ {1, 2} gc† k,aηsg−1 = � a′∈{3,4},η′s′ c† gk,a′η′s′Dc′′(g)a′+2η′s′,a+2ηs, a ∈ {3, 4} (S29) We consider the following symmetry operations as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' T, C3z, C2x, C2zT (S30) with the following representation matrices T : Df(T) = σ0τxς0, Dc′(T) = σ0τxς0, Dc′′(T) = σ0τxς0 C3z : Df(C3z) = ei 2π 3 σzτzς0, Dc′(C3z) = ei 2π 3 σzτzς0, Dc′′(C3z) = σ0τ0ς0 C2x : Df(C2x) = σxτ0ς0, Dc′(C2x) = σxτ0ς0, Dc′′(C2x) = σxτ0ς0 C2zT : Df(C2xT) = σxτ0ς0, Dc′(C2xT) = σxτ0ς0, Dc′′(C2zT) = σxτ0ς0 (S31) where σx,y,z,0, τx,y,z,0, ςx,y,z,0 are Pauli or identity matrices of orbital, valley and spin degrees of freedom respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At M ̸= 0, v′ ⋆ ̸= 0, we also have U(1)c charge symmetry, U(1)v valley symmetry and SU(2)η spin symmetry for each valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also mention that at M = 0, we have an enlarged flat U(4) symmetry and at v′ ⋆ = 0 we have an enlarged chiral U(4) symmetry [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At M = 0, v′ ⋆ = 0, we have a U(4) × U(4) symmetry [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, we consider the case of M ̸= 0, v′ ⋆ ̸= 0, where we only have a U(1)c ×U(1)v ×SU(2)η=+ ×SU(2)η=− symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We comment that M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='698meV is relatively small and we have an approximate flat U(4) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Under U(1)c transformation gU(1)c(θc) (characterized by a real number θc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' U(1)v transformation gU(1)v(θv) (characterized by a real number θv) and SU(2)η spin transformation gSU(2)η(θµ η ) (characterized by three real numbers θµ η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' µ ∈ {x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' z} ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' we have U(1)c : Df(gU(1)c((θc)) = e−iθcσ0τ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′(gU(1)c((θc)) = e−iθcσ0τ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′′(gU(1)c((θc)) = e−iθcσ0τ0ς0 U(1)v : Df(gU(1)v((θv)) = σ0e−iθvτzς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′(gU(1)v((θv)) = σ0e−iθvτzς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′′(gU(1)v((θv)) = σ0e−iθvτzς0 SU(2)η : Df(gSU(2)η(θµ η )) = σ0e−i � µ θη µ τ0+ητz 4 ςµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′(gSU(2)η(θµ η )) = σ0e−i � µ θη µ τ0+ητz 4 ςµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dc′′(gSU(2)η(θµ η )) = σ0e−i � µ θη µ τ0+ητz 4 ςµ (S32) S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' MEAN-FIELD SOLUTIONS OF THE KONDO LATTICE MODEL The Kondo Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S22 contains two single-particle term ˆHc and ˆHcc and two interaction terms ˆHK + ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now discuss the mean-field decoupling of ˆHK, ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field decoupling of ˆHK We treat the interaction terms via mean-field decoupling ˆHK ≈ ˆHMF K = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k′|<Λc � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2 NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf � γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a + γv′ ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a[η′k′ xσx − k′ yσy]α′a′ + γv′ ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′[ηkxσx + kyσy]αa � � ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟩⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟩c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ − f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨: f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ :⟩⟨: c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs :⟩ + ⟨: f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ :⟩ : c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs : + : f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ : ⟨: c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs :⟩ � (S33) where for an operator O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ⟨O⟩ = ⟨Ψ|O|Ψ⟩ with |Ψ⟩ the mean-field ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock term We first consider the Fock term (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ), which takes the form of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k′|<Λc � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2 NMDνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf � γ2δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a + γv′ ⋆δα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a[η′k′ xσx − k′ yσy]α′a′ + γv′ ⋆δα′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′[ηkxσx + kyσy]αa � � ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟩⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟩c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ − f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ � = � R 1 Dνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='νf � γ2⟨ � |k|<Λc � αηs eik·R−|k|2λ2/2 √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟩⟨ � |k′|<Λc � α′η′s′ e−ik′·R−|k′|2λ2/2 √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − � γ2 � |k|<Λc � αηs eik·R−|k|2λ2/2 √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs⟨ � |k′|<Λc � α′η′s′ e−ik′·R−|k′|2λ2/2 √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � + γv′ ⋆⟨ � |k|<Λc � αηs eik·R−|k|2λ2/2 √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs⟩⟨ � |k′|<Λc � a′α′η′s′ e−ik′·R−|k′|2λ2/2[η′k′ xσx − k′ yσy]α′a′ √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ + γv′ ⋆⟨ � |k|<Λc � αaηs eik·R−|k|2λ2/2[ηkxσx + kyσy]αa √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs⟩⟨ � |k′|<Λc � α′η′s′ e−ik′·R−|k′|2λ2/2 √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − γv′ ⋆ � � |k|<Λc � αaηs eik·R−|k|2λ2/2[ηkxσx + kyσy]αa √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs⟨ � |k′|<Λc � α′η′s′ e−ik′·R−|k′|2λ2/2 √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ + � |k|<Λc ⟨ � αaηs eik·R−|k|2λ2/2[ηkxσx + kyσy]αa √NM f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs⟩ � |k′|<Λc � α′η′s′ e−ik′·R−|k′|2λ2/2 √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� (S34) 18 We introduce the following mean-field expectation values V1 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αηs eik·R−|k|2λ2/2 NM √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩ V2 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αaηs eik·R−|k|2λ2/2 NM √NM (ηkxσx + kyσy)αa⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs|Ψ⟩ (S35) and assume the ground state is translational invariant such that � |k|<Λc � αηs eik·R−|k|2λ2/2 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩ = 1 NM � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αηs eik·R−|k|2λ2/2 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩ = V1 � |k|<Λc � αaηs eik·R−|k|2λ2/2 √NM (ηkxσx + kyσy)αa⟨Ψ|f † r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩ = 1 NM � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αaηs eik·R−|k|2λ2/2 √NM (ηkxσx + kyσy)αa⟨Ψ|f † r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩ = V2 (S36) Then the Fock term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S34) becomes F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − γ2 Dνc,νf � R,|k|<Λc � αη,s eik·R−|k|2λ2/2 √NM � V ∗ 1 f † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � + NMγ2|V1|2 Dνc,νf − γv′ ⋆ Dνc,νf � R,|k|<Λc � α,a,η,s eik·R−|k|2λ2/2 √NM � V ∗ 1 (ηkxσx + kyσy)αaf † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � + NMγv′ ⋆V ∗ 1 V2 Dνc,νf − γv′ ⋆ Dνc,νf � R,|k|<Λc � α,η,s eik·R−|k|2λ2/2 √NM � V ∗ 2 f † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � + NMγv′ ⋆V ∗ 2 V1 Dνc,νf (S37) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Hartree term For the Hartree term (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ), we introduce the following density matrices Of, Oc′,1, Oc′,2, where Of have also been used in the mean-field calculations of the THF model as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] (however, Oc′,1, Oc′,2, V1, V2 are absent in the THF model) Of αηs,α′η′s′ = 1 NM � R ⟨Ψ| : f † R,αηsfR,α′η′s′ : |Ψ⟩ Oc′,1 aηs,a′η′s′ = 1 NM � |k|<Λc e−|k|2λ2⟨Ψ| : c† k,aηsck,a′η′s′ : |Ψ⟩, a, a′ ∈ {1, 2} Oc′,2 a′η′s′,αηs = 1 NM � |k|<Λc � a=1,2 e−|k|2λ2(ηkxσx + kyσy)αa⟨Ψ| : c† k,a′η′s′ck,aηs : |Ψ⟩, a′, α ∈ {1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S38) We then assume the ground state is translational invariance such that ⟨Ψ| : f † R,αηsfR,α′η′s′ : |Ψ⟩ = 1 NM � R ⟨Ψ| : f † R,αηsfR,α′η′s′ : |Ψ⟩ = Of αηs,α′η′s′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S39) 19 Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S39, the Hartree term can be written as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = � R, |k|<Λc,|k′|<Λc � α,α′,a,a′, η,η′,s,s′ ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2 NMDνc,νf � γ2δα′,a′δα,a + γv′ ⋆δα,a[η′k′ xσx − k′ yσy]α′a′ + γv′ ⋆δα′,a′[ηkxσx + kyσy]αa � � − ⟨: f † R,αηsfR,α′η′s′ :⟩⟨: c† k′,a′η′s′ck,aηs :⟩ + ⟨: f † R,αηsfR,α′η′s′ :⟩ : c† k′,a′η′s′ck,aηs : + : f † R,αηsfR,α′η′s′ : ⟨: c† k′,a′η′s′ck,aηs :⟩ � = � α,α′, η,η′,s,s′ NM Dνc,νf � − γ2Of αηs,αη′s′Oc′1 α′η′s′,αηs − � γv′ ⋆Of αηs,α′η′s′Oc′,2 α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + � |k|<Λc � α,α′, η,η′,s,s′ � Of αηs,α′η′s′e−|k|2λ2 : c† k,a′η′s′ck,aηs : δα,aδα′,a′ + � γv′ ⋆Of αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2 : c† k,a′η′sck,aηs : +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + � R � α,α′, η,η′,s,s′ � : f † R,αηsfR,α′η′s′ : Oc′,1 α′η′s′,αηs + � γv′ ⋆ : f † R,αηsfR′,α′η′s′ : Oc′,2 α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� (S40) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock and Hartree terms Combining Fock and Hartree (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S37 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S37) terms, we have ˆHK ≈ ˆHMF K =F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − γ2 Dνc,νf � R,|k|<Λc � αη,s eik·R−|k|2λ2/2 √NM �� V ∗ 1 f † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + NMγ2|V1|2 Dνc,νf − γv′ ⋆ Dνc,νf � R,|k|<Λc � α,a,η,s eik·R−|k|2λ2/2 √NM �� V ∗ 1 (ηkxσx + kyσy)αaf † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + NMγv′ ⋆V ∗ 1 V2 Dνc,νf − γv′ ⋆ Dνc,νf � R,|k|<Λc � α,η,s eik·R−|k|2λ2/2 √NM �� V ∗ 2 f † R,αηsck,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � − V ∗ 2 V1 � + NMγv′ ⋆V ∗ 2 V1 Dνc,νf + � α,α′, η,η′,s,s′ NM Dνc,νf � − γ2Of αηs,αη′s′Oc′1 α′η′s′,αηs − � γv′ ⋆Of αηs,α′η′s′Oc′,2 α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + � |k|<Λc � α,α′, η,η′,s,s′ � Of αηs,α′η′s′e−|k|2λ2 : c† k,a′η′s′ck,aηs : δα,aδα′,a′ + � γv′ ⋆Of αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2 : c† k,a′η′sck,aηs : +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� + � R � α,α′, η,η′,s,s′ � : f † R,αηsfR,α′η′s′ : Oc′,1 α′η′s′,αηs + � γv′ ⋆ : f † R,αηsfR′,α′η′s′ : Oc′,2 α′η′s′,αηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' �� (S41) V1, V2 describes the Fock contribution that characterize the hybridization between f- and Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Of, Oc′,1, Oc′,2 are the mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 20 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field decoupling of ˆHJ We now perform a mean-field decoupling of the ferromagnetic exchange coupling term [1] ˆHJ ≈ ˆHMF J = − J 2NM � R � αα′ηη′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ss′ � |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k′|<Λc ei(k−k′)·R(ηη′ + (−1)α+α′) � ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟩⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs1⟩c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ − f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs1ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨: f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ :⟩⟨: c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs :⟩+ : f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ : ⟨: c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs :⟩ + ⟨: f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ :⟩ : c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs : � (S42) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock term The Fock term takes the form of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − J 2NM � R � αα′ηη′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ss′ � |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k′|<Λc ei(k−k′)·R(ηη′ + (−1)α+α′) � ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟩⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs1⟩c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ − f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs1ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟨c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ � = − J � R � ξ=± � ⟨ � |k′|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs ei(−k′)·R √NM δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟩⟨ � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′(−1)α′+1 eik·R √NM c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs ei(−k′)·R √NM δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟨ � |k′|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′(−1)α′+1 eik·R √NM � α′η′s′ c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′⟩ − ⟨ � k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs ei(−k′)·R √NM δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs⟩ � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ eik·R √NM δξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′(−1)α′+1c† k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η′s′fR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ � (S43) We then introduce the following mean-fields V3 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 NM √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ V4 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 NM √NM ⟨Ψ|ηf † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ (S44) and assume the ground state is translational invariant such that � |k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ = 1 NM � R � |k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ = V3 � |k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ = 1 NM � R � |k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ = V4 (S45) Then the Fock term can be written as F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − JNM[|V3|2 + |V4|2] + J � R,|k|<Λc,αηs �ei(−k′)·R √NM � δ1,η(−1)α+1f † R,αηsck′,α+2,ηsV ∗ 3 + δ−1,η(−1)α+1f † R,αηsck′,α+2,ηsV ∗ 4 + � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � (S46) 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Hartree term The Hartree term takes the form of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − J 2NM � R � αα′ηη′,ss′ � |k|,|k′|<Λc ei(k−k′)·R(ηη′ + (−1)α+α′) � − ⟨: f † R,αηsfR,α′η′s′ :⟩⟨: c† k′,α′+2,η′s′ck,α+2,ηs :⟩ + : f † R,αηsfR,α′η′s′ : ⟨: c† k′,α′+2,η′s′ck,α+2,ηs :⟩ + ⟨: f † R,αηsfR,α′η′s′ :⟩ : c† k′,α′+2,η′s′ck,α+2,ηs : � (S47) We introduce the following density matric which has also been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1] Oc′′ aηs,a′η′s′ = 1 NM � |k|<Λc ⟨Ψ| : c† k,a+2ηsck,a′+2η′s′ : |Ψ⟩, a, a′ ∈ {1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S48) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48, the Hartree term becomes H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' = − JNM 2 � αα′ηη′ss′ (ηη′ + (−1)α+α′)Of αηs,α′η′s′Oc′′ α′η′s′,αηs + J 2 � Rc,αα′ηη′ss′ : f † R,αηsfR,α′η′s′ : Oc′′ α′η′s′,αηs + J 2 � |k|<Λc,αα′ηη′ss′ Of αηs,α′η′s′ : c† k′,α′+2,η′s′ck,α+2,ηs : (S49) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Fock and Hartree terms Combing Hartree and Fock terms (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S46 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S49), we have ˆHJ ≈ ˆHMF J = − JNM � ξ=± |V3|2 + |V4|2 + J � R,|k|<Λc,αηs �ei(−k′)·R √NM � δ1,η(−1)α+1f † R,αηsck′,α+2,ηsV ∗ 3 + δ−1,η(−1)α+1f † R,αηsck′,α+2,ηsV ∗ 4 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' � − JNM 2 � αα′ηη′ss′ (ηη′ + (−1)α+α′)Of αηs,α′η′s′Oc′′ α′η′s′,αηs + J 2 � Rc,αα′ηη′ss′ : f † R,αηsfR,α′η′s′ : Oc′′ α′η′s′,αηs + J 2 � |k|<Λc,αα′ηη′ss′ Of αηs,α′η′s′ : c† k′,α′+2,η′s′ck,α+2,ηs : (S50) V3, V4 describes the Fock contribution that characterize the hybridization between f- and Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Of, Oc′′ are the mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution and have also been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Filling constraints and mean-field equations We note that in the Kondo model the filling of f electrons is fixed to be νf at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To simplify the calculation, we take a common approximation that only requires the average filling of f-electron to be νf [3, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In other words, we only require 1 NM � R,αηs⟨Ψ| : f † R,αηsfR,αηs : |Ψ⟩ = νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then add the following term to the Hamiltonian ˆHλf = � R,αηs λf � : f † R,αηsfR,αηs : −νf � (S51) and determine the Langrangian multiplier λf from the following equation 1 NM � R,αηs ⟨Ψ| : f † R,αηsfR,αηs : |Ψ⟩ = νf (S52) 22 In practice, we perform calculations at fixed total filling ν = νf + νc, where νf and νc are the average fillings of f and c electrons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since νf is also fixed in the Kondo model, we will self-consistently determine the chemical potential µc (in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S28) by requiring 1 NM � |k|<Λc,aηs ⟨Ψ| : c† k,aηsck,aηs : |Ψ⟩ = νc = ν − νf (S53) Finally, our mean-field Hamiltonian takes the form of ˆHMF = ˆHc + ˆHcc + ˆHMF K + ˆHMF J + ˆHλf + ˆHµc (S54) and we determine V1, V2, V3, V4, Of, Oc′,1, Oc′,2, Oc′′, λf, µc from the self-consistent equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' During the self-consistent solution, at each step, we will adjust λf, µc according to the current filling of f- and c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We use νi f and νi c denote the filling of f and c at i-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For the i + 1-th step, we will update λf, µc as λf → λf + r(νi f − νf), µc → µc − r(νi c − νc), where r(> 0) will be manually adjusted to improve the convergence (in practice, we take r ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of the symmetric Kondo state We focus on the symmetric Kondo phase without any symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, we require our density matrix of f- and c- electrons (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48) to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We can then utilize symmetry to simplify the self-consistent equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We first consider the U(1)v symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S32, a U(1)v symmetric solution satisfies Of αηs,α′η′s′ = Of αηs,α′η′s′e−iθν(η−η′) ⇒ Of αηs,α−ηs′ = 0 Oc′,1 aηs,a′η′s′ = Oc′,1 aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′,1 aηs,a′−η′s′ = 0 Oc′,2 aηs,α′η′s′ = Oc′,2 aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′,2 aηs,α′−η′s′ = 0 Oc′′ aηs,a′η′s′ = Oc′′ aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′′ aηs,a′−ηs′ = 0 (S55) and V1, V2, V3, V4 are invariant under U(1)v transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S55, Of, Oc′,1, Oc′,2, Oc′′ are block diagonalized in valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next consider a SU(2)η transformation acting on the valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ [ei � µ θη µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='sOf αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′ηs′[ei � µ θη µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ 2 = Of αηs2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′ηs′ 2 ⇒ Of aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ [ei � µ θη µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='sOc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′[ei � µ θη µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ 2 = Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ⇒ Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ [ei � µ θη µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='sOc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′[ei � µ θη µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ 2 = Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ⇒ Oc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2 aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ � s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ [ei � µ θη µσµ]s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='sOc′′ aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′[ei � µ θη µσµ]s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ 2 = Oc′′ aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ⇒ Oc′′ aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′ηs′ ∝ Is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s′ (S56) where I is an 2 × 2 identical matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, V1, V2, V3, V4 are invariant under SU(2)η transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S55 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S56, the density matrices Of, Oc′,1, Oc′′ are diagonalized in valley and spin incdies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then introduce 2 × 2 matrices, of,η, oc′,1,η, oc′,2,η, oc′′,η, such that Of αηs,α′η′s′ = of,η α,α′δη,η′δs,s′, Oc′,1 aηs,a′η′s′ = oc′,1,η a,a′ δη,η′δs,s′, Oc′,2 aηs,a′η′s′ = oc′,2,η a,a′ δη,η′δs,s′, Oc′′ aηs,a′η′s′ = oc′′,η a,a′ δη,η′δs,s′ (S57) We now consider the effect of discrete symmetries in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S31 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S57, we find T : (of,η)∗ = of,−η, (oc′,1,η)∗ = oc′,1,−η, (oc′,2,η)∗ = oc′,2,−η, (oc′′,η)∗ = oc′′,−η C3z : ei 2πη 3 σzof,ηe−i 2πη 3 σz = of,η, ei 2πη 3 σzoc′,1,ηe−i 2πη 3 σz = oc′,1,η, ei η2π 3 oc′,2,ηe−i η2π 2 σz = oc′,2,η, oc′′,η = oc′′,η C2x : σxof,ησx = of,η, σxoc′,1,ησx = oc′,1,η, σxoc′,2,ησx = oc′,2,η, σxoc′′,ησx = oc′′,η C2zT : (σxof,ησx)∗ = of,η, (σxoc′,1,ησx)∗ = oc′,1,η, (σxoc′,2,ησx)∗ = −oc′,2,−η, (σxoc′′,ησx)∗ = oc′′,η (S58) 23 From the definition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38), Of, Oc′,1, Oc′′ are Hermitian matrices and then of, oc′,1, oc′′ are also Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Com- bining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S58 and the Hermitian properties, we can introduce real numbers χf 0, χc′,1 0 , χc′′ 0 , χc′′ 1 and then of, oc′, oc′′ take the following structure of,η = of,−η = χf 0σ0, oc′,1,η = oc′,−η = χc′,1 0 σ0, oc′,2,η = 0, oc′′,η = χc′′ 0 σ0 + χc′′ 1 σx (S59) where σ0, σx,y,z are identity and Pauli matrices respectively with row and column indices α = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since the filling of f- and Γ1 ⊕ Γ2 c- electrons are νf = Tr[Of], νc′′ = Tr[Oc′′] respectively, we find χf 0 = νf/8, χc′′ 0 = νc′′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S59 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S57, for the symmetric solution, we finally have Of αηs,α′η′s′ = δα,α′δη,η′δs,s′νf/8 Oc′,1 aηs,a′η′s′ = δa,a′δη,η′δs,s′χc′,1 0 , Oc′,2 aηs,a′η′s′ = 0, a, a′ ∈ {1, 2} Oc′′ aηs,a′η′s′ = δη,η′δs,s′(δa,a′νc′′/8 + δa,3−a′χc′′ x ), a, a′ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2} (S60) As for the hybridization fields, we find discrete symmetries will not impose constraints on V1, V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for V3, V4, we have T : V3 = V ∗ 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C3z : V3 = ei2π/3V3, V4 = ei2π/3V4 C2x : V3 = V4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C2zT : V3 = V ∗ 4 (S61) therefore V3 = V4 = 0 (S62) In summary, instead of solving self-consistent equations of Of, Oc′,1, Oc′′, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48), we can use νf = 1 NM � R,αηs ⟨Ψ| : f † R,αηsfR,αηs : |Ψ⟩ χc′,1 0 = 1 8NM � |k|<Λc,a=1,2,ηs ⟨Ψ|e−|k|2λ2 : c† k,aηsck,aηs : |Ψ⟩ νc′′ = 1 NM � |k|<Λc,a=3,4,ηs ⟨Ψ| : c† k,aηsck,aηs : |Ψ⟩ χc′′ 1 = 1 8NM � |k|<Λc,ηs ⟨Ψ|c† k,3ηsck,4ηs + c† k,4ηsck,3ηs|Ψ⟩ V3 = V4 = 0 (S63) and obtain Of, Oc, Oc′′ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that the first equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S63 is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In summary, combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S53 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S63, we have a complete set of mean-field self-consistent equations for the symmetric Kondo state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We comment that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S63 are the same mean-field equations as we derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S4 A, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S4 B and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S4 C, but with additional symmetry requirement, that is the ground states satisfy all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We mention that, at ν = νf = νc = 0, we have Of αηs,α′η′s′ = 0, Oc′ αηs,α′η′s′ = 0 and the Hartree term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S41 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now prove the Hartree term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S50 also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note the only non-zero components of Oc′′ are Oc′′ 1ηs,2ηs, Oc′′ 2ηs,1ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S49, the Hartree term takes the form of (with Of = 0) − J 2NM � Rs1s2 � αα′ηη′ � |k1|,|k2|<Λc ei(k1−k2)·R(ηη′ + (−1)α+α′) � : f † Rαηs1fRα′η′s2 : Oc′′ α′η′s2,αηs1 � = − J 2NM � Rs � αη � |k1|,|k2|<Λc ei(k1−k2)·R(ηη + (−1)α+3−α) � : f † RαηsfRα′ηs : Oc′′ 3−αηs,αηs � = − J 2NM � Rs � αη � |k1|,|k2|<Λc ei(k1−k2)·R(0) � : f † RαηsfRα′ηs : Oc′′ 3−αηs,αηs � =0 (S64) and hence vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In summary, at ν = 0, we only need to consider V1, V2, and other mean fields vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Properties of the symmetric Kondo state We solve the self-consistent equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S52, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S53 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S63 at integer filling ν = 0, −1, −2 with νf = ν, νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We identify the symmetric Kondo (SK) states at ν = 0, −1, −2 which are characterized by V1 ̸= 0 (|γ2V1/Dνf ,νc| = 95meV, 111meV, 209meV at ν = 0, −1, −2 respectively), V2 ̸= 0 (|v′ ⋆γV2/Dνf ,νc| = 80meV, 97meV, 197meV at ν = 0, −1, −2 respectively) and V3 = 0, V4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Even if we allow non-zero V3, V4 and ini- tialize the mean-field calculations with non-zero V3, V4, V3, V4, we still find V3 = V4 = 0 after self-consistent calculations (amplitudes smaller than 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This is because ˆHJ describes ferromagnetic interactions and disfavors the development of non- zero V3, V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also comment that the non-zero V1, V2, introduce an effective f-c hybridization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S37) and characterize the Kondo physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next discuss the topological feature of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since the SK states preserve all the symmetries, it is sufficient to only consider the bands in valley + and spin ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find at ν = 0, −1, −2, the representations formed by flat bands at Γ, K, M are Γ1 ⊕ Γ2, K2K3, M1 ⊕ M2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that the representations formed by flat bands here are equivalent to that of the non-interacting THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus, the flat bands form a fragile topology at ν = −1, −2 and a stable topology at ν = 0 due to the additional particle-hole symmetry at ν = 0 [1, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, we also calculate the Wilson loop of the flat bands (valley + spin ↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the calculation of Wilson loop, we let k1 ∈ { i N }i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',N−1, k2 = { j N }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',N−1 and k1 ∈ { i N }i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',N−1, k2 = { j N }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',N−1 and k = k1bM,1 + k2bM,2, where bM,1 = 4π 3aM ( √ 3, 0), bM,2 = 4π 3aM ( √ 3 2 , 3 2) are two moir´e reciprocal lattice vectors and aM is the moir´e lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then let |un,k⟩ denote the n-th eigenvectors of the single-particle Hamiltonian H(k) (of valley + spin ↑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We focus on the subset of the bands, which we denote with band indices n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='., nband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, we take the flat bands as the subset of the bands that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then define the matrix Uk as a matrix formed by the eigenvectors of the flat bands Uk = [|u1,k⟩, |u2,k⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=', |unband,k⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Wilson loop [85] along the k2 direction is defined as W(k1) = U † k1,k2=0 N−1 � j=1 � Uk1,k2= 2πj N U † k1,k2= 2π(j+1) N � V (k1=0,k2=2π)Uk1,k2=2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S65) where V G is defined as H(k+G) = V GH(k)V G,†, G = nbM,1 +mbM,2, n, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We mention that c-electrons are defined in the momentum space that can be larger than the first MBZ (depending on the momentum cutoff Λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus we introduce V G that maps ck to ck+G to restore the periodic condition H(k + G) = V GH(k)V G,†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The corresponding Wilson loop Hamiltonian [85] is H(k1) = −i ln(W(k1)) (S66) We plot the Wilson loop spectrum (eigenvalues of H(k1)) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5, where we observe the Wilson loop has winding number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [85], in the presence of additional particle-hole symmetry at ν = 0 [1, 85], (−1)n with n the winding number of Wilson loop is a stable topological index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We conclude that at ν = 0, the symmetric Kondo state has a stable topology that is characterized by the odd winding number of the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5, we observe the behaviors of the Wilson loop are similar at different fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We check the overlapping of the flat-band wavefunctions between different bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We let {|uν i,k⟩}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',nband denote the wavefunction of flat bands at filling ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We define the overlapping between wavefunctions at fillings ν and ν′ as Overlap(ν, ν′) = 1 N � k � i,j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=',nband} ⟨uν i,k|uν′ j,k⟩⟨uν′ j,k|uν i,k⟩ (S67) We find Overlap(0, −1) = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1%, Overlap(−1, −2) = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The large overlapping of wavefunctions between different fillings indicates similar behaviors of the Wilson loop at different fillings as we showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we analyze the mean-field Hamiltonian of the symmetric Kondo state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The mean-field single-particle Hamiltonian of valley η and spin ↑ (spin ↑ and spin ↓ are equivalent) of the Kondo symmetric state can be approximately written as ˜H(η)(k) = � ˜H(f,η)(k) ˜H(fc,η)(k) ˜H(fc,η),†(k) ˜H(c,η)(k) � (S68) ˜H(f,η)(k) = EfI2×2 ˜H(c,η)(k) = � EcI2×2 v⋆(ηkxσ0 + ikyσz) v⋆(ηkxσ0 − ikyσz) Ec′′I2×2 � ˜H(fc,η)(k) = � ˜γσ0 + ˜v′⋆(ηkxσx + kyσy) 02×2 � (S69) 25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Wilson loop spectrum of flat bands of SK states at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' where ˜H(f,η), ˜H(c,η), ˜H(fc,η) denote the single-particle Hamiltonian of the f-block, c-block and fc-block respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Ef, Ec, Ec′′ denote the energy shifting induced by the Hartree term, one-body scattering term ˆHcc and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Ef, Ec can be k dependent and we only keep its k-independent part which makes dominant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Ec′′ comes from the Hartree contribution of ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S49, which is relatively small and we set Ec′′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also set M = 0, since it is small compared to the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ˜γ, ˜v⋆ ′ denote the renormalized f-c hybridization emerged from Kondo interactions (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also drop the damping factor e−|k|2λ2/2 to simplify the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In practice, we find |˜γ| = 1 Dνc,νf |γ2V ∗ 1 + γv′ ⋆V ∗ 2 | ≈ 175meV, 209meV, 406meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the chiral limit v′ ⋆ = 0, we also have ˜v′⋆ = 0 ( ˜v′⋆ = v′ ⋆ = γv′ ⋆V ∗ 1 /Dνc,νf ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that |˜v′ ⋆||k| can reach a similar amplitude as the k-independent hybridization |˜γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, we expect in most regions of MBZ, |˜γ| makes the dominant contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We, therefore, drop the set ˜v′⋆ = 0 or equivalently v′ ⋆ = 0 as an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By setting v′ ⋆ = 0, we can further separate ˜H(η)(k) into two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The first block corresponds to the row and column indices 1, 3, 5 with electron operators,fk,1ηs, ck,1ηs, ck,3ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The second block corresponds to the row and column indices 2, 4, 6 with electron operators,fk,2ηs, ck,2ηs, ck,4ηs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We focus on the first block whose single-particle Hamiltonian is ˜h(η)(k) = � � Ef ˜γ 0 ˜γ Ec v⋆(ηkx + iky) 0 v⋆(ηkx − iky) 0 � � (S70) We next analyze the eigensystems of ˜h(η)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that ˜γ provides the largest energy scales near ΓM point and will gap out f- and Γ3 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To observe this, we first consider the first 2 × 2 block of ˜h(η)(k) which describes the single-particle Hamiltonian of f- and Γ3 c-electrons � Ef ˜γ ˜γ Ec � (S71) The eigenvalues and eigenvectors are E1 = Ec + Ef 2 − � ˜γ2 + (Ec − Ef)2 4 , E2 = Ec + Ef 2 + � ˜γ2 + (Ec − Ef)2 4 v1 = 1 � 2Efc(Efc − E3) �E3 − Efc ˜γ�T , v2 = 1 � 2Efc(Efc + E3) �E3 + Efc ˜γ�T (S72) where E3 = Ef −Ec 2 , Efc = � ˜γ2 + E2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Since ˜γ is larger than Ec, Ef, Γ3 c-electrons and f-electrons are gapped out by the hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Consequently, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Numerically, we indeed find the orbital weights of Γ1 ⊕ Γ2 c-electrons are large (71%, 77%, 89% at ν = 0, −1, −2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next treat v⋆ perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find the dispersion of the flat band becomes Eflat k ≈ −Ef|k|2(v⋆)2 E1E2 = Ef|k|2(v⋆)2 ˜γ2 − EcEf (S73) At ν = 0 with particle-hole symmetry, Ef = Ec = 0 and Eflat k ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, at ν = −1, −2, where Ef ̸= 0, Ec ̸= 0, flat bands become dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We observe that |Ec| is much smaller than |˜γ| at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Ef increases as we change from ν = 0 to ν = −2, because we are doping more holes to the f-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = −2, Ef can reach ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='5|˜γ|, but at ν = −1, Ef ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1|˜γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Approximately, the dispersion of the flat band is Eflat k ≈ (v⋆)2Ef/˜γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = −1, we have Ek ≈ 13meV · ˚A2|k|2 and, at 26 ν = −2, we have Ek ≈ 45meV · ˚A2|k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This indicates a larger dispersion at ν = −2, which is consistent with our numerical result shown in the main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next analyze the wavefunctions of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The corresponding electron operator of the flat band d† flat,k is d† flat,k ≈ 1 Ak � c† k,3ηs + v⋆(ηkx − iky) EcEf − ˜γ2 � − Efc† k,1ηs + ˜γf † k,1ηs �� (S74) where the normalization factor Ak = � 1 + |v⋆|2|k|2(E2 f + ˜γ2) (EcEf − ˜γ2)2 (S75) We observe that in the large |˜γ| limit, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons (c† k,3ηs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also provide the Berry curvature derived from the wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S74 Ω(k) = −2(EcEf − ˜γ2)2(E2 f + ˜γ2)v2 ⋆ � (EcEf − ˜γ2)2 + (E2 f + ˜γ2)v2⋆|k|2 �2 (S76) We next calculate the Wilson loop from the wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The wavefunctions of d† flat,k is u(k) = 1 Ak � 1 v⋆ηkx−iky EcEf −˜γ2 (−Ef) v⋆ηkx−iky EcEf −˜γ2 ˜γ �T (S77) where the first, second and third rows denote c† k,3ηs, f † k,1ηs, c† k,1ηs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We then parametrize the momentum as k = x1aMbM,1 + x2aMbM,2, bM,1 = 4π 3aM ( √ 3, 0), bM,2 = 4π 3aM ( √ 3 2 , 3 2) (S78) x1, x2 ∈ [−1 2, 1 2] 1 aM (S79) and define |u(x1, x2)⟩ as |u(k)⟩ with k = x1aMbM,1 + x2aMbM,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Wilson loop can be written as W(x1) = N−1 � j=0 ⟨u(x1, x2 = x2,i)|u(x1, x2 = x2,j+1⟩⟨u(x1, x2,N)|u(x1, x2,0)⟩, x2,i = − 1 2aM + 1 aM i N (S80) The spectrum of the Wilson loop is N(x1) = −i ln(W(x1)) = −i � 1 2aM − 1 2aM ⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, 1/(2aM))|u(x1, −1/(2aM)⟩) (S81) In the continuous limit with aM ��� 0, we find N(x1) = −i � ∞ −∞ ⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, ∞)|u(x1, −∞⟩) (S82) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S77 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S82, we find N(x1) =π � 1 + v2 ⋆x1 � x2 1v2⋆ + (˜γ2−EcEf )2 E2 f +˜γ2 � (S83) Even though Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S82 is calculated from the perturbative wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S77, it qualitatively captures the behaviors of the Wilson loop shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We observe that N(−∞) = 0 and N(∞) = 2π, which indicates a 2π winding at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also mention that current calculations correspond to one of the two flat bands for each valley and each spin, because we only pick one block of the single-particle Hamiltonian as we discussed near Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The other flat band can be derived in the same manner and has similar behaviors, since it has a similar single-particle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 27 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' MEAN-FIELD SOLUTIONS OF THE TOPOLOGICAL HEAVY-FERMION MODEL We now discuss the mean-field equations of topological heavy-fermion mode in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We use a similar Hartree-Fock ap- proximation as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, we decouple ˆHJ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The mean-field expectation values we considered are Of αηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ = 1 NM � R ⟨Ψ| : f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ : |Ψ⟩ Oc′ aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ = 1 NM � |k|<Λc ⟨Ψ| : c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' a ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2} Oc′′ aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′η′s′ = 1 NM � |k|<Λc ⟨Ψ| : c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a+2ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a′+2η′s′ : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' a′ ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2} Oc′f aηs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′ = 1 √ NN � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R e−ik·R⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α′η′s′|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' a ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2} V3 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 NM √NM ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ V4 = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='|k|<Λc � αη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='s eik·Rδ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='η(−1)α+1 NM √NM ⟨Ψ|ηf † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηs|Ψ⟩ (S84) where Of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Oc′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' V3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' V4 have also been used in the Kondo lattice mean-field calculations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, THF model also has a chemical potential term ˆHµ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S21) and we determine µ by requiring the total filling of f- and c-electrons to be ν: ν = Tr[Of] + Tr[Oc′] + Tr[Oc′′] (S85) where we note that the filling of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons are νf = Tr[Of], νc′ = Tr[Oc′], νc′′ = Tr[Oc′′] , (S86) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We discuss the difference and similarities between the mean-field equations of the KL model and that of the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For the THF model, we introduce mean fields Of, Oc′, Oc′′, Oc′f, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S84) (for a generic state without enforcing any symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for KL model, we introduce mean fields V1, V2, Of, Oc′,1, Oc′,2, Oc′′, V3, V4 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S35, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44) (for a generic state without enforcing any symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For both models, Oc′′, Of, V3, V4 are part of mean fields and contribute the mean-field decoupling of ˆHJ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the THF model, we introduce a chemical potential term µ that couples to both the f-electron density operators and c- electron density operator (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We enforce the total filling of f- and c-electrons to be ν by tuning µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the KL model, we introduce a Lagrangian multiplier λf (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S51) that couples to f-electron density operators, and a chemical potential µc (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S28) that couples to the c-electron density operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We enforce the fillings of f-electrons and c-electrons to be νf and νc respectively by tuning λf and µc in the KL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also mention that Oc′ in THF model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S84) and Oc′,1 in the KL model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38) are different, where the latter one has included an additional damping factor e−|k|2λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the THF model, we do not need hybridization fields V1, V2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S35), since both come from the decoupling of Kondo interactions that only appear in the KL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of fully symmetric state We next discuss the solution of the symmetric state The fully symmetric state is characterized by density matrices Of, Oc′, Oc′′, Oc′f and hybridization fields V3, V4 that satisfy all symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The structures of Of, Oc′′ in the fully sym- metric state are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also prove that V3 = V4 = 0 in a fully symmetric state (near Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now discuss 28 the symmetry properties of Oc′, Oc′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S32, a U(1)v symmetric solution satisfies Oc′ aηs,α′η′s′ = Oc′ aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′ aηs,α−ηs′ = 0 Oc′f aηs,α′η′s′ = Oc′f aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′f aηs,α−ηs′ = 0 (S87) Then, Oc′, Oc′f is block diagonalized in valley indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next consider a SU(2)η transformation acting on the valley η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' It indicates � s,s′ [ei � µ θη µσµ]s2,sOc′ αηs,α′ηs′[ei � µ θη µσµ]s′,s′ 2 = Oc′ αηs2,α′ηs′ 2 ⇒ Oc′ αηs,α′ηs′ � s,s′ [ei � µ θη µσµ]s2,sOc′f αηs,α′ηs′[ei � µ θη µσµ]s′,s′ 2 = Oc′f αηs2,α′ηs′ 2 ⇒ Oc′f αηs,α′ηs′ (S88) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S87 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S88, we can introduce 2 × 2 matrices oc′,η, oc′f,η, such that Oc′ αηs,α′η′s′ = oc′ α,α′δs,s′δη,η′, Oc′f αηs,α′η′s′ = oc′f α,α′δs,s′δη,η′ (S89) We now consider the effect of discrete symmetries in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S31 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S89, we find T : (oc′,η)∗ = oc′,−η, (oc′f,η)∗ = oc′f,−η C3z : ei2π/3ησzoc′,ηe−i2πη/3σz = oc′,η, ei2π/3ησzoc′f,ηe−i2πη/3σz = oc′f,η C2x : σxoc′,ησx = oc′,η, σxoc′f,ησx = oc′f,η C2zT : σx(oc′,η)∗σx = oc′,η, σx(oc′f,η)∗σx = oc′f,η (S90) Then we can introduce a single real number χc′ 0 , χc′f 0 to characterize the density matrices Oc′ αηs,α′η′s′ = χc′ 0 δα,α′δη,η′, Oc′f αηs,α′η′s′ = χc′f 0 δα,α′δη,η′δs,s′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S91) Since the filling of Γ3 c-electrons is νc′ = Tr[Oc′] = 8χc′ 0 , we let χc′ 0 = νc′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, instead of calculating the original density matrices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S84, we can calculate the following quantities νf = 1 NM � R,αηs ⟨Ψ| : f † R,αηsfR,αηs : |Ψ⟩ νc′ = 1 NM � |k|<Λc,a=1,2,ηs ⟨Ψ| : c† k,aηsck,aηs : |Ψ⟩ νc′′ = 1 NM � |k|<Λc,a=3,4,ηs ⟨Ψ| : c† k,aηsck,aηs : |Ψ⟩ χc′f = 1 8NM √NM � |k|<Λc,R,αηs e−ik·R⟨Ψ|c† k,αηsfR,αηs|Ψ⟩ (S92) and construct density matrices via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S60 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, the filling constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S85 becomes ν = νf + νc′ + νc′′ (S93) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S92 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S93, we have a complete set of the mean-field self-consistent equations of symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here we discuss the differences and similarities between the symmetric solution in the KL model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S4 E) and the sym- metric solution in the THF model as introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Both Kondo symmetric (KS) state in KL model and the symmetric state in the THF model preserve all the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The KS state is adiabatically connected to the symmetric state in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The mean-field solutions are exact at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 29 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Wilson loop spectrum of symmetric state in the THF model at ν = 0, −1, −2 To obtain a more precise description of the Kondo state, we need to introduce a Gutzwiller projector to our symmetric- state wavefunction in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The Gutzwiller projector will suppress the charge fluctuations of f-electrons and is expected to further lower the energy of the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also comment that, in the THF model, the flat bands are mainly formed by f-electrons with f-electron orbital weights 80%, 85% and 87% at ν = 0, −1, −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, in the KL model, the flat bands are mainly formed by Γ1 ⊕Γ2 electrons as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This is because, in the KL model, we observed an enhanced f-c hybridization driven by Kondo interactions which is absent in the symmetric state solution of THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We expect the enhanced hybridization will be recovered after introducing the Gutzwiller projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find the spectrum of the Wilson loop in the symmetric state of the THF model has winding number one and three crossing points(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, in the SK state of the KL model, the spectrum of the Wilson loop spectrum has winding number one and one crossing point (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, at ν = 0, for both the THF model and the KL model, the symmetric state has a stable topology with an odd Winding number of Wilson loop spectrum [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also mention the difference in the Wilson loop spectrum comes from the absence of enhanced f-c hybridization in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Mean-field equations of the symmetric state in the presence of strain We now discuss the mean-field solution of the symmetric state in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We add the following term to the Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S12) ˆHstrain = α � R,ηs (f † R,1ηsfR,2ηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=') (S94) We note that ˆHstrain only breaks C3z symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To show this, we rewrite the ˆHstrain as ˆHstrain = � R,αηs,α′η′s′ αf † R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′ hstrain = σxτ0ς0 (S95) where the matrix structure of ˆHstrain is denoted by hstrainσxτ0ς0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now show it commutes with Df(T), Df(C2x), Df(C2zT), Df(gSU(2)η(θµ η )), Df(gU(1)v((θv)), Df(gU(1)c((θc)) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S31, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S32) and hence ˆHstrain preserves all symmetries except for C3z [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(T)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' σ0τxε0] = 0 [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(C2x)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' σxτ0ς0] = 0 [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(C2zT)] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' σxτ0ς0] = 0 [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(gU(1)c((θc))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' e−iθcσ0τ0ς0] = 0 [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(gU(1)v((θv))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' σ0e−iθvτzς0] = 0 [hstrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Df(gSU(2)η(θµ η ))] = [σxτ0ς0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' σ0e−i � µ θη µ τ0+ητz 4 ςµ] = 0 30 In the presence of strain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' the symmetric state is defined as the state that preserves all the symmetries except for C3z which is broken by the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the presence of C3z-breaking strain, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S57 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S89 still hold, because the system still has U(1)c × U(1)v × SU(2)+ × SU(2)− symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S58 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S90, we only need to consider T, C2x, C2zT and we find Of αηs,α′η′s′ = � χf 0σ0 + χf 1σx � α,α′δη,η′δs,s′ Oc′ αηs,α′η′s′ = � χc′ 0 σ0 + χc′ 1 σx � α,α′ δη,η′δs,s′, Oc′′ αηs,α′η′s′ = � χc′′ 0 σ0 + χc′′ 1 σx � α,α′ δη,η′δs,s′, Oc′f αηs,α′η′s′ = � χc′f 0 σ0 + χc′f 1 σx � α,α′δη,η′δs,s′ (S96) where χf 0, χf 1, χc′ 0 , χc′ 1 , χc′′ 0 , χc′′ 1 , χc′f 0 , χc′f 1 are real numbers tha characterize the density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S38, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S48 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S96,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' we find χf 0 = 1 8NM � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αetas ⟨Ψ| : f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χf 1 = 1 8NM � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs ⟨Ψ|f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηs + f † R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηs|Ψ⟩ χc′ 0 = 1 8NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs ⟨Ψ| : c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηsc† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′ 1 = 1 8NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs ⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηs + c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηs|Ψ⟩ χc′′ 0 = 1 8NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs ⟨Ψ| : c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηsc† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aηs : |Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′′ 1 = 1 8NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs ⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4ηs + c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4ηsck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3ηs|Ψ⟩ χc′f 0 = 1 8NM √NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs e−ik·R⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′f 1 = 1 8NM √NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs e−ik·R⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηs + c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηs|Ψ⟩ χc′′f 0 = 1 8NM √NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs e−ik·R⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='α+2ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='αηs|Ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′′f 1 = 1 8NM √NM � |k|<Λc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='ηs e−ik·R⟨Ψ|c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2ηs + c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4ηsfR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3ηs|Ψ⟩ (S97) where we also have χf 0 = νf/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′ 0 = νc′/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' χc′′ 0 = νc′′/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As for V3, V4, we only consider the T, C2x, CzT of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S61, which indicates V3 = V4 = V ∗ 3 = V ∗ 4 (S98) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S44, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S93, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S97 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S98 with filling constraints ν = νf + νc′ + νc′′, we have a complete set of the mean-field self-consistent equations of symmetric state in the presence of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We perform calculations with non-zero strain at ν = 0, −1, −2, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We initialize the calculations with the fully symmetric solutions derived at zero strain, and the procedure converges within 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The results are illustrated and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effect of doping We now discuss the effect of doping at zero strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For hole doping at ν = 0, −1, −2, −3 and electron doping at ν = 0, we mainly dope electrons to the light bands that are mostly formed by c-electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Consequently, the energy difference between the symmetric state and the ordered state decreases since we have more conduction c-electrons near the Fermi energy, and the system favors the symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now point out the complexity of electron dopings at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Doping electrons at ν = −1, −2 is equivalent to dope electrons to the heavy bands that are mostly formed by f-electrons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The heavy (flat) bands become closer to the Fermi energy, and hence, the energy cost of putting f-electrons into flat bands will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then we can fill the heavy (flat) bands with a small energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By filling the heavy (flat) bands, the type of orders formed by f-electrons can change a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To observe 31 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Dipsersions of KIVC state at ν = 0, KIVC+VP state at ν = −1, KIVC state at ν = −2 and VP state at ν = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The color represents the weight of f-(yellow) and c-(blue) electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Evolution of order parameter as a function of doping at ν = 0, −1, −2 the change of the ordered states, we consider the following order parameters Ox = 1 NM � R,αηs,α′η′s′ f † R,αηs[ox]αηs,α′η′s′fR,α′η′s′, x ∈ {KIV C, Sz, Vz, Vy} oKIV C = σyτyς0, oSz = σ0τ0ςz, oVz = σ0τzς0, oVy = σ0τyς0 (S99) We measure the expectation values of Ox=KIV C,Sz,Vz,Vy with respect to the ordered states at each filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S8, we show the evolution of ⟨Ox⟩ as a function of doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find for hole doping at ν = 0, −1, −2 and electron doping at ν = 0 (where carriers go to light bands in both cases), the system stays in the same ordered states (compared to the integer filling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, for electron doping at ν = −1, −2, we can observe the changes of the order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This is because we are mainly dope f-electrons for electron doping at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We thus conclude that electron doping at ν = −1, −2 will introduce sizeable changes of the order parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Both the change of order parameters and the doping effect will affect the energy competition between the symmetric state and the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Finally, we comment on the ν = −3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = −3, the actual ground state might be a CDW state which breaks the translational symmetry [145] and is beyond our current consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, at ν = −3, even for the valley polarized state we currently considered, electron doping is equivalent to doping both heavy and light bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S7), which is different from ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We leave the detailed study of ν = −3 for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Effect of strain We next analyze the effect of the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We first note that as we increase α (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S95), the strain will gradually suppress the KIVC order (OKIV C, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This can be observed from {oKIV C, hstrain} = {σyτyς0, σxτ0ς0} = 0 (S100) Heuristically, the anti-commuting nature indicates the competition between oKIV C and hstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus, as we increase hstrain, oKIV C will be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also find the spin-polarization OSz and valley polarization OVz commute with hstrain [oSz, hstrain] = [σ0τ0ςz, σxτ0ς0] = 0, [oVz, hstrain] = [σ0τzς0, σxτ0ς0] = 0 (S101) 32 Heuristically, this indicates the valley and spin polarization do not directly compete with hstrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, as we will show in this section, a sufficiently large strain could still destroy the valley and spin polarization in the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' For future convenience, we also introduce the eigenstates of the strain Hamiltonian ˆHstrain (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S95) d† R,1ηs = 1 √ 2(f † R,1ηs − f † R,2ηs), d† R,2ηs = 1 √ 2(f † R,1ηs + f † R,2ηs) (S102) We will call d† R,1ηs and d† R,2ηs as d1 and d2 electrons (orbitals), respectively, for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We mention that d1, d2 are f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The strain Hamiltonian can then be written as ˆHstrain = α � R,αηs (−d† R,1ηsdR,1ηs + d† R,2ηsdR,2ηs) (S103) Thus, for a positive strain amplitude α > 0, the energy of d1 electrons will be lowered and the energy of d2 electrons will be raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We introduce ⟨hstrain⟩ to characterize the population imbalance between d1 and d2 electrons ⟨hstrain⟩ = ⟨ 1 NM � R,αηs,α′η′s′ f † R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′⟩ = 1 NM � R,αηs ⟨d† R,1ηsdR,1ηs − d† R,2ηsdR,2ηs⟩ (S104) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9, we plot the evolution of various order parameters and also |⟨hstrain⟩| where the expectation value is taken with respect to the ordered state solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In all cases, |⟨hstrain⟩| increases as we increase α, since α linearly coupled to hstrain term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The KIVC order will be suppressed and fully destroyed at sufficient strong strain at ν = 0, −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At ν = 0, after the destruction of the KIVC order, self-consistent calculation produces a symmetric ground state that only breaks C3z symmetry, even though we initialize the mean-field calculation with an ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, at ν = −1, −2, after the destruction of KIVC order, the spin polarization and valley polarization still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By further increasing the strain, the ordered states will finally become unstable (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9), which means the mean-field calculations that are initialized with ordered solutions converge to a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We next analyze the transition from an ordered state to a symmetric state at a large strain at ν = −1, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ν = −1 We first consider the ν = −1 with 4meV ≲ α ≲ 18meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In this parameter region, the KIVC order is destroyed but valley and spin polarization persist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10 (a) (b), we plot the band structures in this parameter region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that flat bands that are mostly formed by f-electrons (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10) move towards the Fermi level, as we increase strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Near the transition point to the symmetric state, the flat bands are very close to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' This signals an instability of the ordered states since we can fill the flat band without any energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By diagonalizing the mean-field Hamiltonian, we find the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10) correspond to d1 electrons (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' By filling the flat bands, we have more populations in d1 orbitals, which increase |⟨hstrain⟩| (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S104) and drive the system to a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We now estimate the critical value of strain αc at which a transition from an ordered state to a symmetric state happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At αc, the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10) are very close to the Fermi energy and induce the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To estimate αc, we calculate the excitation gap of the flat bands (marked by red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10): ∆Eflat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then we have ∆Eflat ���� α≈αc = 0 (S105) We estimate ∆Eflat using the zero-hybridization limit [2] of the model, where γ = 0, v′ ⋆ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In addition, we also set J = 0 to simplify the calculation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The zero-hybridization model with non-zero strains are ˆHzero-hyb = ˆHU + ˆHW + ˆHV + ˆHstain + ˆHc + ˆHµ (S106) where ˆHU, ˆHW , ˆHV , ˆHstrain, ˆHc, ˆHµ are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S18, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S95, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S13 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S12 respectively, and ˆHV are treated with mean-field methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In the zero-hybridization model, the filling of f-electrons νf and c-electrons-νc are good quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We solve the zero-hybridization model at fixed total filling ν with the assumption that the ground state does not break translational symmetry (fillings of f-electrons are uniform) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To estimate the excitation gap of the flat bands, we calculate the energy cost of adding one dR,1ηs electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We mention that, in our mean-field calculations with finite f-c hybridization, the relevant flat bands (marked by red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10) correspond to d1 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We let |Ψzero-hyb⟩ denote the ground state of the zero-hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The state with one-more dR,1ηs is |Ψexct zero-hyb⟩ = d† R,1ηs|Ψzero-hyb⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' (S107) 33 We next calculate ∆Eflat = ⟨Ψexct zero-hyb| ˆHzero-hyb|Ψexct zero-hyb⟩ − ⟨Ψzero-hyb| ˆHzero-hyb|Ψzero-hyb⟩ (S108) The energy loss from Hubbard interaction term is ∆EU = ⟨Ψexct zero-hyb| ˆHU|Ψexct U ⟩ − ⟨Ψzero-hyb| ˆHU|Ψzero-hyb⟩ = U 2 (νf + 1)2 − U 2 ν2 f = U(νf + 1 2) The energy loss from ˆHW term is ∆EW =⟨Ψexct zero-hyb| ˆHW |Ψexct zero-hyb⟩ − ⟨Ψzero-hyb| ˆHW |Ψzero-hyb⟩ = � a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a(νf + 1) − � a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='aνf = � a=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a (S109) where νc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a denotes the filling of c-electrons in orbital a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The energy change from ˆHV is ∆EV = ⟨Ψexct zero-hyb| ˆHV |Ψexct zero-hyb⟩ − ⟨Ψzero-hyb| ˆHV |Ψzero-hyb⟩ = 0 (S110) The energy change from ˆHstrain is ∆Estrain = ⟨Ψexct zero-hyb| ˆHstrain|Ψexct zero-hyb⟩ − ⟨Ψzero-hyb| ˆHstrain|Ψzero-hyb⟩ = −α (S111) The energy change from chemical potential ˆHµ is ∆Eµ = ⟨Ψexct zero-hyb| ˆHµ|Ψexct zero-hyb⟩ − ⟨Ψzero-hyb| ˆHµ|Ψzero-hyb⟩ − µ (S112) Then the excitation energy of adding one dR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='1ηs electron is ∆Eflat = ∆EU + ∆EW + ∆strain + ∆Eµ = U 2 (νf + 1/2) + � a Waνc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='a − µ − α (S113) We further take the following approximation: W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 = W = 47meV (the difference between W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='4 is about 15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then ∆Eflat ≈ U 2 (νf + 1/2) + Wνc − µ − α (S114) At ν = −1 and 0meV ≤ α ≤ 43meV, the ground state of the zero-hybridization model has νf = −1, νc = ν − νf = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then ∆Eflat = −U 2 − µ − α (S115) We next determine chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Chemical potential µ is determined by requiring the c-electrons filling to be νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The single-particle Hamiltonian of c-electron in the zero-hybridization limit takes the form of ˆHc,zero-hyb = ˆHc + � k,aηs (Wνf + V (0) Ω0 νc − µ)c† k,aηsck,aηs (S116) where we have set W1,2,3,4 = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that when Wνf − V (0) Ω0 νc − µ = 0 (S117) ˆHc,zero-hyb = ˆHc and we have νc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, µ = Wνf + V (0) Ω0 νc = −W (S118) 34 where we take νc = 0, νf = −1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S115 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S118, we find ∆Eflat = W − U 2 − α (S119) Then the flat bands reach Fermi energy when ∆Eflat = 0, which leads to ∆Eflat = 0 ⇒ αc = W − U 2 = 18meV (S120) which is close to the value (also around α = 18meV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9 (b)) from self-consistent calculations of the finite- hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Here, the finite-hybridization model refers to the original THF model with finite γ, v′ ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, we conclude the transition from an ordered state to a symmetric state happens at α = αc ≈ 18meV at ν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We also discuss the solutions of the zero-hybridization model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S11 (a), we show the ground state properties of the zero-hybridization model at various strains and ν = −1, where νf 1 = 1 NM � R,ηs : d† R,1ηsdR,1ηs :, νf 2 = 1 NM � R,ηs : d† R,2ηsdR,2ηs : (S121) denotes the filling of d1 and d2 electrons respectively with νf = νf 1 + νf 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We find a transition happens at α ≈ 25meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that this transition is described by ��lling one more dR,1ηs electrons at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' After the transition, there will be 4 f-electrons filling d1 orbitals, and zero f-electrons filling the d1 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus for d1 orbitals, all the valleys and spins are filled, but for d2 orbitals all the valleys and spins are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Therefore, there is no room to develop order and the ground state is a symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We note that the transition in the zero-hybridization limit and the transition in the finite-hybridization model (at ν = −1, α ≈ 16meV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9) share the same origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' They are both driven by filling electrons in d1 orbitals (in the finite-hybridization model, we fill the flat bands) and, after the transition, both ground states are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Thus, the results between zero-hybridization and finite-hybridization models are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, the critical values αc for the two models are different, since we have finite f-c hybridization in the finite-hybridization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' ν = −2 We next discuss the transition from an ordered state to a symmetric state at ν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' We focus on the parameter region 10meV ≲ α ≲ 45meV, where the KIVC order is destroyed but valley and spin polarization exist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10 (c) (d), we plot the band structures in this parameter region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As we increase strain, we note that flat bands (marked with red circles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10), move towards the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Similar to the ν = −1 case, when the flat bands reach the Fermi energy, a transition to the symmetric state happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' However, at ν = −2, we need a much larger strain to destroy the ordered state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S9 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' To understand this, we start from the zero-hybridization limit of the model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S114, the excitation energy of the relevant flat bands (marked by red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S10) ∆Eflat = U 2 (νf + 1/2) + Wνc − µ − α (S122) By solving the zero-hybridization model, we find the ground states have νf = −1 and νc = −1 in the parameter region we focused 4meV ≲ α ≲ 18meV, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' Then ∆Eflat = −U 2 − W − µ − α (S123) We now calculate the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' µ is determined by requiring the filling of c-electrons to be νc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The single- particle Hamiltonian of c-electron in the zero-hybridization limit (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S116) takes the form of ˆHc,zero-hyb = ˆHc + � k,aηs (Wνf + V (0) Ω0 νc − µ)c† k,aηsck,aηs (S124) where we have set W1,2,3,4 = W, and take the mean-field treatment of ˆHV (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' S20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' At M = 0 limit (M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content='697meV, which is relatively small), the dispersion of c-electrons are Ek = ±v⋆|k| − Ec, where we define Ec = Wνf + V (0) Ω0 νc − µ (S125) 35 Then all the c-states with energy smaller than 0 will be filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE3T4oBgHgl3EQftAvj/content/2301.04673v1.pdf'} +page_content=' The corresponding Fermi momentum kF is |v⋆kF | = Ec ⇒ kF = 1 |v⋆|Ec (S126) Then the filling of c-electrons is νc = − 8 AMBZ � |k|