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理解强关联电子体系是一个长期的重要目标, 该体系的魅力不仅在于其背后蕴藏着深刻的物理, 还在于其中涌现出的丰富物质态在量子调控、量子计算等领域具有巨大的潜在应用价值. 同时, 理论上非微扰地理解强关联电子体系是极其困难的, 一直充满挑战. 量子蒙特卡罗计算是一类非微扰计算的标准方法, 有助于对强关联电子体系提供非微扰的理解, 因而广泛运用于凝聚态和高能物理领域. 然而, 量子蒙特卡罗计算通常会受到负符号问题的困扰. 本文将具体介绍一些无负符号关联电子模型的设计思路, 并讨论我们近期提出的符号边界理论. 通过设计无负符号或者具有代数符号行为的强关联电子模型, 可以帮助人们研究很多重要的量子多体问题, 包括巡游磁性量子临界行为、非常规超导和磁性序的竞争, 以及莫尔(moiré)量子物质中的关联物相与相变等.Understanding strongly correlated electrons is an important long-term goal, not only for uncovering fundamental physics behind, but also for their emergence of lots of novel states which have potential applications in quantum control and quantum computations. Meanwhile, the strongly correlated electrons are usually extremely hard problems, and it is generally impossible to understand them unbiasedly. Quantum Monte Carlo is a typical unbiased numeric method, which does not depend on any perturbation, and it can help us to exactly understand the strongly correlated electrons, so that it is widely used in high energy and condensed matter physics. However, quantum Monte Carlo usually suffers from the notorious sign problem. In this paper, we introduce general ideas to design sign problem free models and discuss the sign bound theory we proposed recently. In the sign bound theory, we build a direct connection between the average sign and the ground state properties of the system. We find usually the average sign has the conventional exponential decay with system size increasing, leading to exponential complexity; but for some cases it can have algebraic decay, so that quantum Monte Carlo simulation still has polynomial complexity. By designing sign problem free or algebraic sign behaved strongly correlated electron models, we can approach to several long outstanding problems, such as the itinerant quantum criticality, the competition between unconventional superconductivity and magnetism, as well as the recently found correlated phases and phase transitions in moiré quantum matter.
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Keywords:
- sign problem /
- quantum phase transition /
- non-Fermi liquid /
- moiré quantum matter
[1] Imada M, Fujimori A, Tokura Y 1998 Rev. Mod. Phys. 70 1039Google Scholar
[2] Benfatto G, Gallavotti G 1990 J. Stat. Phys. 59 541Google Scholar
[3] Shankar R 1991 Phys. A: Stat. Mech. Appl. 177 530Google Scholar
[4] Polchinski J 1992 arXiv: hep-th/9210046
[5] Shankar R 1994 Rev. Mod. Phys. 66 129Google Scholar
[6] Anderson P W 1987 Science 235 1196Google Scholar
[7] Zhang F C, Rice T M 1988 Phys. Rev. B 37 3759Google Scholar
[8] Lee P A, Nagaosa N, Wen X-G 2006 Rev. Mod. Phys. 78 17Google Scholar
[9] Bednorz J G, Müller K A 1986 Z. Phys. B:Condens. Matter. 64 189Google Scholar
[10] Keimer B, Kivelson S A, Norman M R, Uchida S, Zaanen J 2015 Nature 518 179Google Scholar
[11] Fradkin E, Kivelson S A, Tranquada J M 2015 Rev. Mod. Phys. 87 457Google Scholar
[12] Löhneysen H V, Rosch A, Vojta M, Wölfle P 2007 Rev. Mod. Phys. 79 1015Google Scholar
[13] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar
[14] Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar
[15] Chen G, Jiang L, Wu S, Lyu B, Li H, Chittari B L, Watanabe K, Taniguchi T, Shi Z, Jung J, Zhang Y, Wang F 2019 Nat. Phys. 15 237Google Scholar
[16] Xie Y, Lian B, Jäck B, Liu X, Chiu C L, Watanabe K, Taniguchi T, Bernevig B A, Yazdani A 2019 Nature 572 101Google Scholar
[17] Lu X, Stepanov P, Yang W, Xie M, Aamir M A, Das I, Urgell C, Watanabe K, Taniguchi T, Zhang G, Bachtold A, MacDonald A H, Efetov D K 2019 Nature 574 653Google Scholar
[18] Li T, Jiang S, Li L, Zhang Y, Kang K, Zhu J, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350Google Scholar
[19] Kitaev A 2006 Ann. Phys. 321 2Google Scholar
[20] Troyer M, Wiese U J 2005 Phys. Rev. Lett. 94 170201Google Scholar
[21] Zhang X, Pan G, Xu X Y, Meng Z Y 2021 arXiv: 2112.06139 [cond-mat. str-el]
[22] Blankenbecler R, Scalapino D J, Sugar R L 1981 Phys. Rev. D 24 2278Google Scholar
[23] Hirsch J E 1985 Phys. Rev. B 31 4403Google Scholar
[24] Scalapino D 2007 Handbook of High-Temperature Superconductivity (New York: Springer) pp495–526
[25] LeBlanc J P F, Antipov A E, Becca F, Bulik I W, Chan G K-L, Chung C M, Deng Y, Ferrero M, Henderson T M, Jiménez-Hoyos C A, Kozik E, Liu X W, Millis A J, Prokof’ev N V, Qin M, Scuseria G E, Shi H, Svistunov B V, Tocchio L F, Tupitsyn I S, White S R, Zhang S, Zheng B X, Zhu Z, Gull E 2015 Phys. Rev. X 5 041041Google Scholar
[26] Assaad F F, Herbut I F 2013 Phys. Rev. X 3 031010Google Scholar
[27] Otsuka Y, Yunoki S, Sorella S 2016 Phys. Rev. X 6 011029Google Scholar
[28] Zhou Z, Wang D, Meng Z Y, Wang Y, Wu C 2016 Phys. Rev. B 93 245157Google Scholar
[29] Hohenadler M, Lang T C, Assaad F F 2011 Phys. Rev. Lett 106 100403Google Scholar
[30] Zheng D, Zhang G M, Wu C 2011 Phys. Rev. B 84 205121Google Scholar
[31] He Y Y, Wu H Q, You Y Z, Xu C, Meng Z Y, Lu Z Y 2016 Phys. Rev. B 93 115150Google Scholar
[32] Assaad F F 1999 Phys. Rev. Lett. 83 796Google Scholar
[33] Chen C, Xu X Y, Meng Z Y, Hohenadler M 2019 Phys. Rev. Lett. 122 077601Google Scholar
[34] Zhang Y X, Chiu W T, Costa N C, Batrouni G G, Scalettar R T 2019 Phys. Rev. Lett. 122 077602Google Scholar
[35] Li Z X, Jiang Y F, Yao H 2015 Phys. Rev. B 91 241117Google Scholar
[36] Wang L, Liu Y H, Iazzi M, Troyer M, Harcos G 2015 Phys. Rev. Lett. 115 250601Google Scholar
[37] Assaad F F, Grover T 2016 Phys. Rev. X 6 041049Google Scholar
[38] Gazit S, Randeria M, Vishwanath A 2017 Nat. Phys. 13 484Google Scholar
[39] Xu X Y, Qi Y, Zhang L, Assaad F F, Xu C, Meng Z Y 2019 Phys. Rev. X 9 021022Google Scholar
[40] Berg E, Lederer S, Schattner Y, Trebst S 2019 Annu. Rev. Condens. Matter. Phys. 10 63Google Scholar
[41] Li Z X, Yao H 2019 Annu. Rev. Condens. Matter. Phys. 10 337Google Scholar
[42] Xu X Y, Hong Liu Z, Pan G, Qi Y, Sun K, Meng Z Y 2019 J. Phys. Condens. Matter. 31 463001Google Scholar
[43] Chang C C, Gogolenko S, Perez J, Bai Z, Scalettar R T 2015 Philos. Mag. 95 1260Google Scholar
[44] Loh E Y, Gubernatis J E, Scalettar R T, Sugar R L, White S R 1989 Interacting Electrons in Reduced Dimensions (Boston: Springer US) pp55–60
[45] Assaad F, Evertz H 2008 Computational Many-Particle Physics (Berlin: Springer) pp277–356
[46] Lang G H, Johnson C W, Koonin S E, Ormand W E 1993 Phys. Rev. C 48 1518Google Scholar
[47] Koonin S E, Dean D J, Langanke K 1997 Phys. Rep. 278 1Google Scholar
[48] Hands S, Montvay I, Morrison S, Oevers M, Scorzato L, Skullerud J 2000 Eur. Phys. J. C Part Fields 17 285Google Scholar
[49] Wu C J, Zhang S C 2005 Phys. Rev. B 71 155115Google Scholar
[50] Berg E, Metlitski M A, Sachdev S 2012 Science 338 1606Google Scholar
[51] Xu X Y, Sun K, Schattner Y, Berg E, Meng Z Y 2017 Phys. Rev. X 7 031058Google Scholar
[52] Liu Y, Jiang W, Klein A, Wang Y, Sun K, Chubukov A V, Meng Z Y 2022 Phys. Rev. B 105 L041111Google Scholar
[53] Huffman E F, Chandrasekharan S 2014 Phys. Rev. B 89 111101Google Scholar
[54] Wang L, Corboz P, Troyer M 2014 New J. Phys. 16 103008Google Scholar
[55] Li Z X, Jiang Y F, Yao H 2016 Phys. Rev. Lett. 117 267002Google Scholar
[56] Wei Z C, Wu C, Li Y, Zhang S, Xiang T 2016 Phys. Rev. Lett. 116 250601Google Scholar
[57] [58] Ouyang Y, Xu X Y 2021 Phys. Rev. B 104 L241104Google Scholar
[59] Hertz J A 1976 Phys. Rev. B 14 1165Google Scholar
[60] Millis A J 1993 Phys. Rev. B 48 7183Google Scholar
[61] Moriya T 1985 Spin Fluctuations in Itinerant Electron Magnetism (Berlin: Springer) pp44–81
[62] Chubukov A V 2010 Physics 3 70Google Scholar
[63] Altshuler B L, Ioffe L B, Millis A J 1994 Phys. Rev. B 50 14048Google Scholar
[64] Kim Y B, Furusaki A, Wen X G, Lee P A 1994 Phys. Rev. B 50 17917Google Scholar
[65] Polchinski J 1994 Nucl. Phys. B 422 617Google Scholar
[66] Lee S S 2009 Phys. Rev. B 80 165102Google Scholar
[67] Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075127Google Scholar
[68] Abanov A, Chubukov A V, Schmalian J 2003 Adv. Phys. 52 119Google Scholar
[69] Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075128Google Scholar
[70] Schattner Y, Gerlach M H, Trebst S, Berg E 2016 Phys. Rev. Lett. 117 097002Google Scholar
[71] Schattner Y, Lederer S, Kivelson S A, Berg E 2016 Phys. Rev. X 6 031028Google Scholar
[72] Liu Z H, Xu X Y, Qi Y, Sun K, Meng Z Y 2019 Phys. Rev. B 99 085114Google Scholar
[73] Liu Z H, Pan G, Xu X Y, Sun K, Meng Z Y 2019 Proc. Natl. Acad. Sci. USA 116 16760Google Scholar
[74] Xu X Y, Klein A, Sun K, Chubukov A V, Meng Z Y 2020 npj. Quantum. Mater. 5 65Google Scholar
[75] Kagawa F, Miyagawa K, Kanoda K 2005 Nature 436 534Google Scholar
[76] Lefebvre S, Wzietek P, Brown S, Bourbonnais C, Jérome D, Mézière C, Fourmigué M, Batail P 2000 Phys. Rev. Lett. 85 5420Google Scholar
[77] Arai T, Ichimura K, Nomura K, Takasaki S, Yamada J, Nakatsuji S, Anzai H 2001 Phys. Rev. B 63 104518Google Scholar
[78] Belin S, Behnia K, Deluzet A 1998 Phys. Rev. Lett. 81 4728Google Scholar
[79] De Soto S M, Slichter C P, Kini A M, Wang H H, Geiser U, Williams J M 1995 Phys. Rev. B 52 10364Google Scholar
[80] Kanoda K, Miyagawa K, Kawamoto A, Nakazawa Y 1996 Phys. Rev. B 54 76Google Scholar
[81] Xu X Y, Grover T 2021 Phys. Rev. Lett. 126 217002Google Scholar
[82] Kang J, Vafek O 2019 Phys. Rev. Lett. 122 246401Google Scholar
[83] Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087Google Scholar
[84] Kang J, Vafek O 2018 Phys. Rev. X 8 031088Google Scholar
[85] Po H C, Zou L, Vishwanath A, Senthil T 2018 Phys. Rev. X 8 031089Google Scholar
[86] Xu X Y, Law K T, Lee P A 2018 Phys. Rev. B 98 121406Google Scholar
[87] Zhang X, Pan G, Zhang Y, Kang J, Meng Z Y 2021 Chin. Phys. Lett. 38 077305Google Scholar
[88] Hofmann J S, Khalaf E, Vishwanath A, Berg E, Lee J Y 2022 Phys. Rev. X 12 011061Google Scholar
[89] Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233Google Scholar
[90] Liao Y D, Meng Z Y, Xu X Y 2019 Phys. Rev. Lett. 123 157601Google Scholar
[91] Liao Y D, Kang J, Breiø C N, Xu X Y, Wu H Q, Andersen B M, Fernandes R M, Meng Z Y 2021 Phys. Rev. X 11 011014Google Scholar
[92] Liao Y D, Xu X Y, Meng Z Y, Kang J 2021 Chin. Phys. B 30 017305Google Scholar
[93] Lian B, Song Z D, Regnault N, Efetov D K, Yazdani A, Bernevig B A 2021 Phys. Rev. B 103 205414Google Scholar
[94] Greiter M, Rachel S 2007 Phys. Rev. B 75 184441Google Scholar
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图 1 巡游反铁磁量子临界晶格模型和计算结果[73] (a) 巡游反铁磁量子临界模型示意图; (b) EMUS动量空间网格; (c) 巡游反铁磁量子临界模型的相图
Fig. 1. Itinerant antiferromagnetic quantum critical lattice model and results[73]: (a) Schematic diagram of itinerant antiferromagnetic quantum critical lattice model; (b) momentum mesh for EMUS; (c) phase diagram of itinerant antiferromagnetic quantum critical lattice model.
图 2 巡游铁磁量子临界晶格模型计算结果[74] (a) 巡游铁磁量子临界晶格模型示意图; (b) 巡游铁磁量子临界点上费米子自能虚部和频率的关系; (c) 扣除热涨落的贡献之后, 费米子自能虚部与频率的关系
Fig. 2. Results of itinerant ferromagnetic quantum critical lattice model[74]: (a) Schematic phase diagram of itinerant ferromagnetic quantum critical lattice model; (b) the relation between imaginary part of fermionic self-energy at itinerant ferromagnetic quantum critical point and Matsubara frequency; (c) after deducting the thermal effect, the relation between the imaginary part of the fermionic self-energy at itinerant ferromagnetic quantum critical point and Matsubara frequency.
图 3 d-波超导和反铁磁竞争晶格模型[81] (a) 模型示意图(黑色圆点表示费米子格点, 上面定义了一个仅有最近邻跃迁的哈伯德模型; 菱形点上定义了量子转子模型, 红色和蓝色表示转子与费米子横向和纵向库珀对算符的耦合具有相反的相位); (b) V/t = 0.5时, 模型的平均场相图
Fig. 3. Lattice model with competition of d-wave superconductivity and antiferromagnetic[81]: (a) Schematic diagram of the lattice model (The black dots denote fermion sites, onside is a Hubbard model with only nearest neighbor hopping. A quantum rotor model is defined on diamond points, where red and blue colors denote opposite phase of the coupling between the rotors and the cooper pairing operator along horizontal direction and the vertical direction); (b) the mean-field phase diagram of the model at V/t = 0.5.
图 4 团簇电荷关联模型示意图(a), (c), (e)和相图(b), (d), (f)[92] (a), (b) 单谷(轨道)模型; (c), (d) 双谷(轨道)模型; (e), (f) 修正后的双谷(轨道)模型
Fig. 4. Schematic diagram (a), (c), (e) and phase diagram (b), (d), (f) of cluster charge correlation model[92]: (a), (b) Single-valley (orbital) model; (c), (d) two-valley (orbital) model; (e), (f) modified two-valley (orbital) model.
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[1] Imada M, Fujimori A, Tokura Y 1998 Rev. Mod. Phys. 70 1039Google Scholar
[2] Benfatto G, Gallavotti G 1990 J. Stat. Phys. 59 541Google Scholar
[3] Shankar R 1991 Phys. A: Stat. Mech. Appl. 177 530Google Scholar
[4] Polchinski J 1992 arXiv: hep-th/9210046
[5] Shankar R 1994 Rev. Mod. Phys. 66 129Google Scholar
[6] Anderson P W 1987 Science 235 1196Google Scholar
[7] Zhang F C, Rice T M 1988 Phys. Rev. B 37 3759Google Scholar
[8] Lee P A, Nagaosa N, Wen X-G 2006 Rev. Mod. Phys. 78 17Google Scholar
[9] Bednorz J G, Müller K A 1986 Z. Phys. B:Condens. Matter. 64 189Google Scholar
[10] Keimer B, Kivelson S A, Norman M R, Uchida S, Zaanen J 2015 Nature 518 179Google Scholar
[11] Fradkin E, Kivelson S A, Tranquada J M 2015 Rev. Mod. Phys. 87 457Google Scholar
[12] Löhneysen H V, Rosch A, Vojta M, Wölfle P 2007 Rev. Mod. Phys. 79 1015Google Scholar
[13] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar
[14] Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar
[15] Chen G, Jiang L, Wu S, Lyu B, Li H, Chittari B L, Watanabe K, Taniguchi T, Shi Z, Jung J, Zhang Y, Wang F 2019 Nat. Phys. 15 237Google Scholar
[16] Xie Y, Lian B, Jäck B, Liu X, Chiu C L, Watanabe K, Taniguchi T, Bernevig B A, Yazdani A 2019 Nature 572 101Google Scholar
[17] Lu X, Stepanov P, Yang W, Xie M, Aamir M A, Das I, Urgell C, Watanabe K, Taniguchi T, Zhang G, Bachtold A, MacDonald A H, Efetov D K 2019 Nature 574 653Google Scholar
[18] Li T, Jiang S, Li L, Zhang Y, Kang K, Zhu J, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350Google Scholar
[19] Kitaev A 2006 Ann. Phys. 321 2Google Scholar
[20] Troyer M, Wiese U J 2005 Phys. Rev. Lett. 94 170201Google Scholar
[21] Zhang X, Pan G, Xu X Y, Meng Z Y 2021 arXiv: 2112.06139 [cond-mat. str-el]
[22] Blankenbecler R, Scalapino D J, Sugar R L 1981 Phys. Rev. D 24 2278Google Scholar
[23] Hirsch J E 1985 Phys. Rev. B 31 4403Google Scholar
[24] Scalapino D 2007 Handbook of High-Temperature Superconductivity (New York: Springer) pp495–526
[25] LeBlanc J P F, Antipov A E, Becca F, Bulik I W, Chan G K-L, Chung C M, Deng Y, Ferrero M, Henderson T M, Jiménez-Hoyos C A, Kozik E, Liu X W, Millis A J, Prokof’ev N V, Qin M, Scuseria G E, Shi H, Svistunov B V, Tocchio L F, Tupitsyn I S, White S R, Zhang S, Zheng B X, Zhu Z, Gull E 2015 Phys. Rev. X 5 041041Google Scholar
[26] Assaad F F, Herbut I F 2013 Phys. Rev. X 3 031010Google Scholar
[27] Otsuka Y, Yunoki S, Sorella S 2016 Phys. Rev. X 6 011029Google Scholar
[28] Zhou Z, Wang D, Meng Z Y, Wang Y, Wu C 2016 Phys. Rev. B 93 245157Google Scholar
[29] Hohenadler M, Lang T C, Assaad F F 2011 Phys. Rev. Lett 106 100403Google Scholar
[30] Zheng D, Zhang G M, Wu C 2011 Phys. Rev. B 84 205121Google Scholar
[31] He Y Y, Wu H Q, You Y Z, Xu C, Meng Z Y, Lu Z Y 2016 Phys. Rev. B 93 115150Google Scholar
[32] Assaad F F 1999 Phys. Rev. Lett. 83 796Google Scholar
[33] Chen C, Xu X Y, Meng Z Y, Hohenadler M 2019 Phys. Rev. Lett. 122 077601Google Scholar
[34] Zhang Y X, Chiu W T, Costa N C, Batrouni G G, Scalettar R T 2019 Phys. Rev. Lett. 122 077602Google Scholar
[35] Li Z X, Jiang Y F, Yao H 2015 Phys. Rev. B 91 241117Google Scholar
[36] Wang L, Liu Y H, Iazzi M, Troyer M, Harcos G 2015 Phys. Rev. Lett. 115 250601Google Scholar
[37] Assaad F F, Grover T 2016 Phys. Rev. X 6 041049Google Scholar
[38] Gazit S, Randeria M, Vishwanath A 2017 Nat. Phys. 13 484Google Scholar
[39] Xu X Y, Qi Y, Zhang L, Assaad F F, Xu C, Meng Z Y 2019 Phys. Rev. X 9 021022Google Scholar
[40] Berg E, Lederer S, Schattner Y, Trebst S 2019 Annu. Rev. Condens. Matter. Phys. 10 63Google Scholar
[41] Li Z X, Yao H 2019 Annu. Rev. Condens. Matter. Phys. 10 337Google Scholar
[42] Xu X Y, Hong Liu Z, Pan G, Qi Y, Sun K, Meng Z Y 2019 J. Phys. Condens. Matter. 31 463001Google Scholar
[43] Chang C C, Gogolenko S, Perez J, Bai Z, Scalettar R T 2015 Philos. Mag. 95 1260Google Scholar
[44] Loh E Y, Gubernatis J E, Scalettar R T, Sugar R L, White S R 1989 Interacting Electrons in Reduced Dimensions (Boston: Springer US) pp55–60
[45] Assaad F, Evertz H 2008 Computational Many-Particle Physics (Berlin: Springer) pp277–356
[46] Lang G H, Johnson C W, Koonin S E, Ormand W E 1993 Phys. Rev. C 48 1518Google Scholar
[47] Koonin S E, Dean D J, Langanke K 1997 Phys. Rep. 278 1Google Scholar
[48] Hands S, Montvay I, Morrison S, Oevers M, Scorzato L, Skullerud J 2000 Eur. Phys. J. C Part Fields 17 285Google Scholar
[49] Wu C J, Zhang S C 2005 Phys. Rev. B 71 155115Google Scholar
[50] Berg E, Metlitski M A, Sachdev S 2012 Science 338 1606Google Scholar
[51] Xu X Y, Sun K, Schattner Y, Berg E, Meng Z Y 2017 Phys. Rev. X 7 031058Google Scholar
[52] Liu Y, Jiang W, Klein A, Wang Y, Sun K, Chubukov A V, Meng Z Y 2022 Phys. Rev. B 105 L041111Google Scholar
[53] Huffman E F, Chandrasekharan S 2014 Phys. Rev. B 89 111101Google Scholar
[54] Wang L, Corboz P, Troyer M 2014 New J. Phys. 16 103008Google Scholar
[55] Li Z X, Jiang Y F, Yao H 2016 Phys. Rev. Lett. 117 267002Google Scholar
[56] Wei Z C, Wu C, Li Y, Zhang S, Xiang T 2016 Phys. Rev. Lett. 116 250601Google Scholar
[57] [58] Ouyang Y, Xu X Y 2021 Phys. Rev. B 104 L241104Google Scholar
[59] Hertz J A 1976 Phys. Rev. B 14 1165Google Scholar
[60] Millis A J 1993 Phys. Rev. B 48 7183Google Scholar
[61] Moriya T 1985 Spin Fluctuations in Itinerant Electron Magnetism (Berlin: Springer) pp44–81
[62] Chubukov A V 2010 Physics 3 70Google Scholar
[63] Altshuler B L, Ioffe L B, Millis A J 1994 Phys. Rev. B 50 14048Google Scholar
[64] Kim Y B, Furusaki A, Wen X G, Lee P A 1994 Phys. Rev. B 50 17917Google Scholar
[65] Polchinski J 1994 Nucl. Phys. B 422 617Google Scholar
[66] Lee S S 2009 Phys. Rev. B 80 165102Google Scholar
[67] Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075127Google Scholar
[68] Abanov A, Chubukov A V, Schmalian J 2003 Adv. Phys. 52 119Google Scholar
[69] Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075128Google Scholar
[70] Schattner Y, Gerlach M H, Trebst S, Berg E 2016 Phys. Rev. Lett. 117 097002Google Scholar
[71] Schattner Y, Lederer S, Kivelson S A, Berg E 2016 Phys. Rev. X 6 031028Google Scholar
[72] Liu Z H, Xu X Y, Qi Y, Sun K, Meng Z Y 2019 Phys. Rev. B 99 085114Google Scholar
[73] Liu Z H, Pan G, Xu X Y, Sun K, Meng Z Y 2019 Proc. Natl. Acad. Sci. USA 116 16760Google Scholar
[74] Xu X Y, Klein A, Sun K, Chubukov A V, Meng Z Y 2020 npj. Quantum. Mater. 5 65Google Scholar
[75] Kagawa F, Miyagawa K, Kanoda K 2005 Nature 436 534Google Scholar
[76] Lefebvre S, Wzietek P, Brown S, Bourbonnais C, Jérome D, Mézière C, Fourmigué M, Batail P 2000 Phys. Rev. Lett. 85 5420Google Scholar
[77] Arai T, Ichimura K, Nomura K, Takasaki S, Yamada J, Nakatsuji S, Anzai H 2001 Phys. Rev. B 63 104518Google Scholar
[78] Belin S, Behnia K, Deluzet A 1998 Phys. Rev. Lett. 81 4728Google Scholar
[79] De Soto S M, Slichter C P, Kini A M, Wang H H, Geiser U, Williams J M 1995 Phys. Rev. B 52 10364Google Scholar
[80] Kanoda K, Miyagawa K, Kawamoto A, Nakazawa Y 1996 Phys. Rev. B 54 76Google Scholar
[81] Xu X Y, Grover T 2021 Phys. Rev. Lett. 126 217002Google Scholar
[82] Kang J, Vafek O 2019 Phys. Rev. Lett. 122 246401Google Scholar
[83] Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087Google Scholar
[84] Kang J, Vafek O 2018 Phys. Rev. X 8 031088Google Scholar
[85] Po H C, Zou L, Vishwanath A, Senthil T 2018 Phys. Rev. X 8 031089Google Scholar
[86] Xu X Y, Law K T, Lee P A 2018 Phys. Rev. B 98 121406Google Scholar
[87] Zhang X, Pan G, Zhang Y, Kang J, Meng Z Y 2021 Chin. Phys. Lett. 38 077305Google Scholar
[88] Hofmann J S, Khalaf E, Vishwanath A, Berg E, Lee J Y 2022 Phys. Rev. X 12 011061Google Scholar
[89] Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233Google Scholar
[90] Liao Y D, Meng Z Y, Xu X Y 2019 Phys. Rev. Lett. 123 157601Google Scholar
[91] Liao Y D, Kang J, Breiø C N, Xu X Y, Wu H Q, Andersen B M, Fernandes R M, Meng Z Y 2021 Phys. Rev. X 11 011014Google Scholar
[92] Liao Y D, Xu X Y, Meng Z Y, Kang J 2021 Chin. Phys. B 30 017305Google Scholar
[93] Lian B, Song Z D, Regnault N, Efetov D K, Yazdani A, Bernevig B A 2021 Phys. Rev. B 103 205414Google Scholar
[94] Greiter M, Rachel S 2007 Phys. Rev. B 75 184441Google Scholar
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