Quantum Monte Carlo study of strongly correlated electrons

Xu Xiao-Yan

doi:10.7498/aps.71.20220079

Abstract

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.

Keywords:

Authors and contacts

Xu Xiao-Yan

School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Corresponding author: Xu Xiao-Yan, xiaoyanxu@sjtu.edu.cn

Funds: Project supported by the National Key R&D Program of China (Grant No. 2021YFA1401400), the Shanghai Pujiang Program, China (Grant No. 21PJ1407200), and the Yangyang Development Fund, China.

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References

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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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

[1]	Imada M, Fujimori A, Tokura Y 1998 Rev. Mod. Phys. 70 1039 Google Scholar
[2]	Benfatto G, Gallavotti G 1990 J. Stat. Phys. 59 541 Google Scholar
[3]	Shankar R 1991 Phys. A: Stat. Mech. Appl. 177 530 Google Scholar
[4]	Polchinski J 1992 arXiv: hep-th/9210046
[5]	Shankar R 1994 Rev. Mod. Phys. 66 129 Google Scholar
[6]	Anderson P W 1987 Science 235 1196 Google Scholar
[7]	Zhang F C, Rice T M 1988 Phys. Rev. B 37 3759 Google Scholar
[8]	Lee P A, Nagaosa N, Wen X-G 2006 Rev. Mod. Phys. 78 17 Google Scholar
[9]	Bednorz J G, Müller K A 1986 Z. Phys. B:Condens. Matter. 64 189 Google Scholar
[10]	Keimer B, Kivelson S A, Norman M R, Uchida S, Zaanen J 2015 Nature 518 179 Google Scholar
[11]	Fradkin E, Kivelson S A, Tranquada J M 2015 Rev. Mod. Phys. 87 457 Google Scholar
[12]	Löhneysen H V, Rosch A, Vojta M, Wölfle P 2007 Rev. Mod. Phys. 79 1015 Google Scholar
[13]	Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43 Google 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 80 Google 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. 15 237 Google Scholar
[16]	Xie Y, Lian B, Jäck B, Liu X, Chiu C L, Watanabe K, Taniguchi T, Bernevig B A, Yazdani A 2019 Nature 572 101 Google 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 574 653 Google 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 597 350 Google Scholar
[19]	Kitaev A 2006 Ann. Phys. 321 2 Google Scholar
[20]	Troyer M, Wiese U J 2005 Phys. Rev. Lett. 94 170201 Google 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 2278 Google Scholar
[23]	Hirsch J E 1985 Phys. Rev. B 31 4403 Google 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 041041 Google Scholar
[26]	Assaad F F, Herbut I F 2013 Phys. Rev. X 3 031010 Google Scholar
[27]	Otsuka Y, Yunoki S, Sorella S 2016 Phys. Rev. X 6 011029 Google Scholar
[28]	Zhou Z, Wang D, Meng Z Y, Wang Y, Wu C 2016 Phys. Rev. B 93 245157 Google Scholar
[29]	Hohenadler M, Lang T C, Assaad F F 2011 Phys. Rev. Lett 106 100403 Google Scholar
[30]	Zheng D, Zhang G M, Wu C 2011 Phys. Rev. B 84 205121 Google Scholar
[31]	He Y Y, Wu H Q, You Y Z, Xu C, Meng Z Y, Lu Z Y 2016 Phys. Rev. B 93 115150 Google Scholar
[32]	Assaad F F 1999 Phys. Rev. Lett. 83 796 Google Scholar
[33]	Chen C, Xu X Y, Meng Z Y, Hohenadler M 2019 Phys. Rev. Lett. 122 077601 Google Scholar
[34]	Zhang Y X, Chiu W T, Costa N C, Batrouni G G, Scalettar R T 2019 Phys. Rev. Lett. 122 077602 Google Scholar
[35]	Li Z X, Jiang Y F, Yao H 2015 Phys. Rev. B 91 241117 Google Scholar
[36]	Wang L, Liu Y H, Iazzi M, Troyer M, Harcos G 2015 Phys. Rev. Lett. 115 250601 Google Scholar
[37]	Assaad F F, Grover T 2016 Phys. Rev. X 6 041049 Google Scholar
[38]	Gazit S, Randeria M, Vishwanath A 2017 Nat. Phys. 13 484 Google Scholar
[39]	Xu X Y, Qi Y, Zhang L, Assaad F F, Xu C, Meng Z Y 2019 Phys. Rev. X 9 021022 Google Scholar
[40]	Berg E, Lederer S, Schattner Y, Trebst S 2019 Annu. Rev. Condens. Matter. Phys. 10 63 Google Scholar
[41]	Li Z X, Yao H 2019 Annu. Rev. Condens. Matter. Phys. 10 337 Google Scholar
[42]	Xu X Y, Hong Liu Z, Pan G, Qi Y, Sun K, Meng Z Y 2019 J. Phys. Condens. Matter. 31 463001 Google Scholar
[43]	Chang C C, Gogolenko S, Perez J, Bai Z, Scalettar R T 2015 Philos. Mag. 95 1260 Google 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 1518 Google Scholar
[47]	Koonin S E, Dean D J, Langanke K 1997 Phys. Rep. 278 1 Google Scholar
[48]	Hands S, Montvay I, Morrison S, Oevers M, Scorzato L, Skullerud J 2000 Eur. Phys. J. C Part Fields 17 285 Google Scholar
[49]	Wu C J, Zhang S C 2005 Phys. Rev. B 71 155115 Google Scholar
[50]	Berg E, Metlitski M A, Sachdev S 2012 Science 338 1606 Google Scholar
[51]	Xu X Y, Sun K, Schattner Y, Berg E, Meng Z Y 2017 Phys. Rev. X 7 031058 Google Scholar
[52]	Liu Y, Jiang W, Klein A, Wang Y, Sun K, Chubukov A V, Meng Z Y 2022 Phys. Rev. B 105 L041111 Google Scholar
[53]	Huffman E F, Chandrasekharan S 2014 Phys. Rev. B 89 111101 Google Scholar
[54]	Wang L, Corboz P, Troyer M 2014 New J. Phys. 16 103008 Google Scholar
[55]	Li Z X, Jiang Y F, Yao H 2016 Phys. Rev. Lett. 117 267002 Google Scholar
[56]	Wei Z C, Wu C, Li Y, Zhang S, Xiang T 2016 Phys. Rev. Lett. 116 250601 Google Scholar
[57]	Wei Z C 2017 arXiv: 1712.09412 [cond-mat. str-el]
[58]	Ouyang Y, Xu X Y 2021 Phys. Rev. B 104 L241104 Google Scholar
[59]	Hertz J A 1976 Phys. Rev. B 14 1165 Google Scholar
[60]	Millis A J 1993 Phys. Rev. B 48 7183 Google Scholar
[61]	Moriya T 1985 Spin Fluctuations in Itinerant Electron Magnetism (Berlin: Springer) pp44–81
[62]	Chubukov A V 2010 Physics 3 70 Google Scholar
[63]	Altshuler B L, Ioffe L B, Millis A J 1994 Phys. Rev. B 50 14048 Google Scholar
[64]	Kim Y B, Furusaki A, Wen X G, Lee P A 1994 Phys. Rev. B 50 17917 Google Scholar
[65]	Polchinski J 1994 Nucl. Phys. B 422 617 Google Scholar
[66]	Lee S S 2009 Phys. Rev. B 80 165102 Google Scholar
[67]	Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075127 Google Scholar
[68]	Abanov A, Chubukov A V, Schmalian J 2003 Adv. Phys. 52 119 Google Scholar
[69]	Metlitski M A, Sachdev S 2010 Phys. Rev. B 82 075128 Google Scholar
[70]	Schattner Y, Gerlach M H, Trebst S, Berg E 2016 Phys. Rev. Lett. 117 097002 Google Scholar
[71]	Schattner Y, Lederer S, Kivelson S A, Berg E 2016 Phys. Rev. X 6 031028 Google Scholar
[72]	Liu Z H, Xu X Y, Qi Y, Sun K, Meng Z Y 2019 Phys. Rev. B 99 085114 Google Scholar
[73]	Liu Z H, Pan G, Xu X Y, Sun K, Meng Z Y 2019 Proc. Natl. Acad. Sci. USA 116 16760 Google Scholar
[74]	Xu X Y, Klein A, Sun K, Chubukov A V, Meng Z Y 2020 npj. Quantum. Mater. 5 65 Google Scholar
[75]	Kagawa F, Miyagawa K, Kanoda K 2005 Nature 436 534 Google Scholar
[76]	Lefebvre S, Wzietek P, Brown S, Bourbonnais C, Jérome D, Mézière C, Fourmigué M, Batail P 2000 Phys. Rev. Lett. 85 5420 Google Scholar
[77]	Arai T, Ichimura K, Nomura K, Takasaki S, Yamada J, Nakatsuji S, Anzai H 2001 Phys. Rev. B 63 104518 Google Scholar
[78]	Belin S, Behnia K, Deluzet A 1998 Phys. Rev. Lett. 81 4728 Google 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 10364 Google Scholar
[80]	Kanoda K, Miyagawa K, Kawamoto A, Nakazawa Y 1996 Phys. Rev. B 54 76 Google Scholar
[81]	Xu X Y, Grover T 2021 Phys. Rev. Lett. 126 217002 Google Scholar
[82]	Kang J, Vafek O 2019 Phys. Rev. Lett. 122 246401 Google Scholar
[83]	Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087 Google Scholar
[84]	Kang J, Vafek O 2018 Phys. Rev. X 8 031088 Google Scholar
[85]	Po H C, Zou L, Vishwanath A, Senthil T 2018 Phys. Rev. X 8 031089 Google Scholar
[86]	Xu X Y, Law K T, Lee P A 2018 Phys. Rev. B 98 121406 Google Scholar
[87]	Zhang X, Pan G, Zhang Y, Kang J, Meng Z Y 2021 Chin. Phys. Lett. 38 077305 Google Scholar
[88]	Hofmann J S, Khalaf E, Vishwanath A, Berg E, Lee J Y 2022 Phys. Rev. X 12 011061 Google Scholar
[89]	Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233 Google Scholar
[90]	Liao Y D, Meng Z Y, Xu X Y 2019 Phys. Rev. Lett. 123 157601 Google 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 011014 Google Scholar
[92]	Liao Y D, Xu X Y, Meng Z Y, Kang J 2021 Chin. Phys. B 30 017305 Google Scholar
[93]	Lian B, Song Z D, Regnault N, Efetov D K, Yazdani A, Bernevig B A 2021 Phys. Rev. B 103 205414 Google Scholar
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Vol. 71, Issue 12 - 2022-06-20

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