Correlated physics, charge and magnetic orders in moiré kagomé systems

doi:10.1088/1674-1056/ad9ffa

Abstract Moiré systems have emerged as an ideal platform for exploring interaction effects and correlated states. However, most of the experimental systems are based on either triangular or honeycomb lattices. In this study, based on the self-consistent Hartree-Fock calculation, we investigate the phase diagram of the kagomé lattice in a recently discovered system with two degenerate $\varGamma$ valley orbitals and strong spin-orbit coupling. By focusing on the filling factors of 1/2, 1/3 and 2/3, we identify various symmetry-breaking states by adjusting the screening length and dielectric constant. At the half filling, we discover that the spin-orbit coupling induces Dzyaloshinskii-Moriya interaction and stabilizes a classical magnetic state with $120^\circ$ ordering. Additionally, we observe a transition to a ferromagnetic state with out-of-plane ordering. In the case of 1/3 filling, the system is ferromagnetically ordered due to the lattice frustration. Furthermore, for 2/3 filling, the system exhibits a pinned droplet state and a $120^\circ$ magnetic ordered state at weak and immediate coupling strengths, respectively. For the strong coupling case, when dealing with non-integer filling, the system is always charge ordered with sublattice polarization. Our study serves as a starting point for exploring the effects of correlation in moiré kagomé systems.

Keywords: moiré kagomé system Hartree-Fock spin-orbit coupling

Received: 21 October 2024
Revised: 29 November 2024
Accepted manuscript online: 17 December 2024

PACS:	73.20.-r	(Electron states at surfaces and interfaces)
	73.43.Cd	(Theory and modeling)
	71.70.Ej	(Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Teller effect)
	68.65.-k	(Low-dimensional, mesoscopic, nanoscale and other related systems: structure and nonelectronic properties)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 12350404 and 12174066), the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302600), the National Key Research and Development Program of China (Grant No. 2019YFA0308404), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 23JC1400600 and 2019SHZDZX01).

Corresponding Authors: Jing Wang
E-mail: wjingphys@fudan.edu.c

Cite this article:

Zhaochen Liu(刘兆晨) and Jing Wang(王靖) Correlated physics, charge and magnetic orders in moiré kagomé systems 2025 Chin. Phys. B 34 027304

[1] Carr S, Fang S and Kaxiras E 2020 Nat. Rev. Mater. 5 748
[2] Andrei E Y and MacDonald A H 2020 Nat. Mater. 19 1265
[3] Kennes D M, Claassen M, Xian L, Georges A, Millis A J, Hone J, Dean C R, Basov D, Pasupathy A N and Rubio A 2021 Nat. Phys. 17 155
[4] Andrei E Y, Efetov D K, Jarillo-Herrero P, MacDonald A H, Mak K F, Senthil T, Tutuc E, Yazdani A and Young A F 2021 Nat. Rev. Mater. 6 201
[5] Mak K F and Shan J 2022 Nat. Nanotechnol. 17 686
[6] Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, SanchezYamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C and Jarillo-Herrero P 2018 Nature 556 80
[7] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E and Jarillo-Herrero P 2018 Nature 556 43
[8] Bistritzer R and MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233
[9] Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastner M and Goldhaber-Gordon D 2019 Science 365 605
[10] 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 and Efetov D K 2019 Nature 574 653
[11] Serlin M, Tschirhart C L, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L and Young A F 2020 Science 367 900
[12] Nuckolls K P, Oh M, Wong D, Lian B, Watanabe K, Taniguchi T, Bernevig B A and Yazdani A 2020 Nature 588 610
[13] Xie Y, Pierce A T, Park J M, Parker D E, Khalaf E, Ledwith P, Cao Y, Lee S H, Chen S, Forrester P R, Watanabe K, Taniguchi T, Vishwanath A, Jarillo-Herrero P and Yacoby A 2021 Nature 600 439
[14] Wang L, Shih E M, Ghiotto A, Xian L, Rhodes D A, Tan C, Claassen M, Kennes D M, Bai Y, Kim B, Watanabe K, Taniguchi T, Zhu X, Hone J, Rubio A, Pasupathy A N and Dean C R 2020 Nat. Mater. 19 861
[15] Tang Y, Li L, Li T, Xu Y, Liu S, Barmak K, Watanabe K, Taniguchi T, MacDonald A H, Shan J and Mak K F 2020 Nature 579 353
[16] Regan E C, Wang D, Jin C, Utama M I B, Gao B, Wei X, Zhao S, Zhao W, Zhang Z, Yumigeta K, Blei M, Carlstrom J D, Watanabe K, Taniguchi T, Tongay S, Crommie M, Zettl A and Wang F 2020 Nature 579 359
[17] Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F and Shan J 2020 Nature 587 214
[18] Li T, Jiang S, Shen B, Zhang Y, Li L, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J and Mak K F 2021 Nature 600 641
[19] Li T, Jiang S, Li L, Zhang Y, Kang K, Zhu J, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J and Mak K F 2021 Nature 597 350
[20] Ghiotto A, Shih E M, Pereira G S, Rhodes D A, Kim B, Zang J, Millis A J, Watanabe K, Taniguchi T, Hone J C, Wang L, Dean C R and Pasupathy A N 2021 Nature 597 345
[21] Li H, Li S, Regan E C, Wang D, Zhao W, Kahn S, Yumigeta K, Blei M, Taniguchi T, Watanabe K, Tongay S, Zettl A, Crommie M F and Wang F 2021 Nature 597 650
[22] Tang Y, Gu J, Liu S, Watanabe K, Taniguchi T, Hone J C, Mak K F and Shan J 2022 Nat. Commun. 13 4271
[23] Wang P, Yu G, Kwan Y H, Jia Y, Lei S, Klemenz S, Cevallos F A, Singha R, Devakul T, Watanabe K, Taniguchi T, Sondhi S L, Cava R J, Schoop L M, Parameswaran S A and Wu S 2022 Nature 605 57
[24] Zhao W, Shen B, Tao Z, Han Z, Kang K, Watanabe K, Taniguchi T, Mak K F and Shan J 2023 Nature 616 61
[25] Cai J, Anderson E, Wang C, Zhang X, Liu X, Holtzmann W, Zhang Y, Fan F, Taniguchi T, Watanabe K, Ran Y, Cao T, Fu L, Xiao D, Yao W and Xu X 2023 Nature 622 63
[26] Zeng Y, Xia Z, Kang K, Zhu J, Knuppel P, Vaswani C, Watanabe K, Taniguchi T, Mak K F and Shan J 2023 Nature 622 69
[27] Yu G, Wang P, Uzan-Narovlansky A J, Jia Y, Onyszczak M, Singha R, Gui X, Song T, Tang Y, Watanabe K, et al. 2023 Nat. Commun. 14 7025
[28] Park H, Cai J, Anderson E, Zhang Y, Zhu J, Liu X, Wang C, Holtzmann W, Hu C, Liu Z, Taniguchi T, Watanabe K, haw Chu J, Cao T, Fu L, Yao W, Chang C Z, Cobden D, Xiao D and Xu X 2023 Nature 622 74
[29] Xu F, Sun Z, Jia T, Liu C, Xu C, Li C, Gu Y, Watanabe K, Taniguchi T, Tong B, Jia J, Shi Z, Jiang S, Zhang Y, Liu X and Li T 2023 Phys. Rev. X 13 031037
[30] Wu F, Lovorn T, Tutuc E, Martin I and MacDonald A H 2019 Phys. Rev. Lett. 122 086402
[31] Naik M H and Jain M 2018 Phys. Rev. Lett. 121 266401
[32] Zhang Y, Liu T and Fu L 2021 Phys. Rev. B 103 155142
[33] Angeli M and MacDonald A H 2021 Proc. Natl. Acad. Sci. USA 118 e2021826118
[34] Zhang Y, Yuan N F Q and Fu L 2020 Phys. Rev. B 102 201115
[35] Xian L, Claassen M, Kiese D, Scherer M M, Trebst S, Kennes D M and Rubio A 2021 Nat. Commun. 12 5644
[36] Wu F, Lovorn T, Tutuc E and MacDonald A H 2018 Phys. Rev. Lett. 121 026402
[37] Pan H, Wu F and Das Sarma S 2020 Phys. Rev. B 102 201104
[38] Pan H, Wu F and Das Sarma S 2020 Phys. Rev. Res. 2 033087
[39] Morales-Duran N, Hu N C, Potasz P and MacDonald A H 2022 Phys. Rev. Lett. 128 217202
[40] Hu N C and MacDonald A H 2021 Phys. Rev. B 104 214403
[41] Crepel V and Fu L 2023 Phys. Rev. B 107 L201109
[42] Reddy A P, Devakul T and Fu L 2023 Phys. Rev. Lett. 131 246501
[43] Wang C, Zhang X W, Liu X, He Y, Xu X, Ran Y, Cao T and Xiao D 2024 Phys. Rev. Lett. 132 036501
[44] Qiu W X, Li B, Luo X J and Wu F 2023 Phys. Rev. X 13 041026
[45] Kaushal N and Dagotto E 2023 Phys. Rev. B 107 L201118
[46] Goldman H, Reddy A P, Paul N and Fu L 2023 Phys. Rev. Lett. 131 136501
[47] Song Z, Seifert U F P, Luo Z X and Balents L 2023 Phys. Rev. B 108 155109
[48] Reddy A P, Alsallom F, Zhang Y, Devakul T and Fu L 2023 Phys. Rev. B 108 085117
[49] Motruk J, Rossi D, Abanin D A and Rademaker L 2023 Phys. Rev. Res. 5 L022049
[50] Han J, Lai K, Yu X, Chen J, Guo H and Dai J 2023 Chin. Phys. Lett. 40 067801
[51] Wang R and Song Z 2024 Chin. Phys. Lett. 41 047101
[52] Zhou X and Shen J 2024 Chin. Phys. Lett. 41 047403
[53] Liu Z, Wang H and Wang J 2022 Phys. Rev. B 106 035114
[54] Wang H, Jiang Y, Liu Z and Wang J 2022 arXiv:2209.06524
[condmat.mes-hall]
[55] Claassen M, Xian L, Kennes D M and Rubio A 2022 Nat. Commun. 13 4915
[56] Scheer M G and Lian B 2023 Phys. Rev. B 108 245136
[57] Scheer M G and Lian B 2023 Phys. Rev. Lett. 131 266501
[58] Ramirez A P 1994 Ann. Rev. Mater. Sci. 24 453
[59] Balents L 2010 Nature 464 199
[60] Kita T, Ohashi T and Kawakami N 2013 Phys. Rev. B 87 155119
[61] Guertler S 2014 Phys. Rev. B 90 081105
[62] Higa R and Asano K 2016 Phys. Rev. B 93 245123
[63] Sun R Y and Zhu Z 2021 Phys. Rev. B 104 L121118
[64] Mielke A 1992 J. Phys. A Math. Gen. 25 4335
[65] Koretsune T and Ogata M 2003 J. Phys. Soc. Jpn. 72 2437
[66] Hanisch T, Kleine B, Ritzl A and Muller-Hartmann E 1995 Ann. Phys. 507 303
[67] Tanaka A and Ueda H 2003 Phys. Rev. Lett. 90 067204
[68] Pollmann F, Fulde P and Shtengel K 2008 Phys. Rev. Lett. 100 136404
[69] Liu Q, Yao H and Ma T 2010 Phys. Rev. B 82 045102
[70] Wen J, Ruegg A, Wang C C J and Fiete G A 2010 Phys. Rev. B 82 075125
[71] Nishimoto S, Nakamura M, O’Brien A and Fulde P 2010 Phys. Rev. Lett. 104 196401
[72] Ferhat K and Ralko A 2014 Phys. Rev. B 89 155141
[73] Pollmann F, Roychowdhury K, Hotta C and Penc K 2014 Phys. Rev. B 90 035118
[74] Kim H S, Mishra A and Lee S 2020 Phys. Rev. B 102 155113
[75] Ferrari F, Becca F and Valenti R 2022 Phys. Rev. B 106 L081107
[76] Guo H M and Franz M 2009 Phys. Rev. B 80 113102
[77] Ren Y, Zeng T S, Zhu W and Sheng D N 2018 Phys. Rev. B 98 205146
[78] Ren Y, Jiang H C, Qiao Z and Sheng D 2021 Phys. Rev. Lett. 126 117602
[79] Kiesel M L and Thomale R 2012 Phys. Rev. B 86 121105
[80] Kiesel M L, Platt C and Thomale R 2013 Phys. Rev. Lett. 110 126405
[81] Wang W S, Li Z Z, Xiang Y Y and Wang Q H 2013 Phys. Rev. B 87 115135
[82] Wang W S, Liu Y C, Xiang Y Y and Wang Q H 2016 Phys. Rev. B 94 014508
[83] Schwemmer T, Hohmann H, Durrnagel M, Potten J, Beyer J, Rachel S, Wu Y M, Raghu S, Muller T, Hanke W and Thomale R 2024 Phys. Rev. B 110 024501
[84] Elser V 1989 Phys. Rev. Lett. 62 2405
[85] Marston J B and Zeng C 1991 J. Appl. Phys. 69 5962
[86] Sachdev S 1992 Phys. Rev. B 45 12377
[87] Wang F and Vishwanath A 2006 Phys. Rev. B 74 174423
[88] Ran Y, Hermele M, Lee P A and Wen X G 2007 Phys. Rev. Lett. 98 117205
[89] Yan S, Huse D A and White S R 2011 Science 332 1173
[90] Depenbrock S, McCulloch I P and Schollwock U 2012 Phys. Rev. Lett. 109 067201
[91] Cepas O, Fong C M, Leung P W and Lhuillier C 2008 Phys. Rev. B 78 140405
[92] Messio L, Cepas O and Lhuillier C 2010 Phys. Rev. B 81 064428
[93] Huh Y, Fritz L and Sachdev S 2010 Phys. Rev. B 81 144432
[94] Messio L, Lhuillier C and Misguich G 2011 Phys. Rev. B 83 184401
[95] Indergand M, Lauchli A, Capponi S and Sigrist M 2006 Phys. Rev. B 74 064429
[96] Ko W H, Lee P A and Wen X G 2009 Phys. Rev. B 79 214502
[97] Guertler S and Monien H 2011 Phys. Rev. B 84 174409
[98] Guertler S and Monien H 2013 Phys. Rev. Lett. 111 097204
[99] Jiang H C, Devereaux T and Kivelson S 2017 Phys. Rev. Lett. 119 067002
[100] Peng C, Jiang Y F, Sheng D N and Jiang H C 2021 Adv. Quantum Technol. 4 2000126
[101] Jiang Y F, Yao H and Yang F 2021 Phys. Rev. Lett. 127 187003
[102] Damle A, Lin L and Ying L 2015 J. Chem. Theory Comput. 11 1463
[103] Damle A and Lin L 2018 Multiscale Model. Simul. 16 1392
[104] Damle A, Levitt A and Lin L 2019 Multiscale Model. Simul. 17 167
[105] Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847
[106] Kang J and Vafek O 2018 Phys. Rev. X 8 031088
[107] Koshino M, Yuan N F, Koretsune T, Ochi M, Kuroki K and Fu L 2018 Phys. Rev. X 8 031087
[108] Luo X J, Wang M and Wu F 2023 Phys. Rev. B 107 235127
[109] Xu C, Li J, Xu Y, Bi Z and Zhang Y 2024 Proc. Natl. Acad. Sci. USA 121 e2316749121
[110] Vitale V, Pizzi G, Marrazzo A, Yates J R, Marzari N and Mostofi A A 2020 npj Comput. Mater. 6 66
[111] Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A and Zaletel M P 2020 Phys. Rev. X 10 031034
[112] Bernevig B A, Song Z D, Regnault N and Lian B 2021 Phys. Rev. B 103 205413
[113] Xie F, Kang J, Bernevig B A, Vafek O and Regnault N 2023 Phys. Rev. B 107 075156
[114] Kudin K N, Scuseria G E and Cances E 2002 J. Chem. Phys. 116 8255
[115] Garza A J and Scuseria G E 2012 J. Chem. Phys. 137 054110
[116] MacDonald A H, Girvin S M and Yoshioka D 1988 Phys. Rev. B 37 9753
[117] Nagaoka Y 1966 Phys. Rev. 147 392
[118] Tasaki H 1989 Phys. Rev. B 40 9192
[119] Ciorciaro L, Smolenski T, Morera I, Kiper N, Hiestand S, Kroner M, Zhang Y, Watanabe K, Taniguchi T, Demler E, et al. 2023 Nature 623 509
[120] Morera I, Kanasz-Nagy M, Smolenski T, Ciorciaro L, Imamo glu A and Demler E 2023 Phys. Rev. Res. 5 L022048
[121] Lee K, Sharma P, Vafek O and Changlani H J 2023 Phys. Rev. B 107 235105
[122] Xu M, Kendrick L H, Kale A, Gang Y, Ji G, Scalettar R T, Lebrat M and Greiner M 2023 Nature 620 971
[123] Li C, He M G, Wang C Y and Zhai H 2024 Phys. Rev. B 109 165131
[124] Moessner R, Sondhi S L and Chandra P 2001 Phys. Rev. B 64 144416
[125] O’Brien A, Pollmann F and Fulde P 2010 Phys. Rev. B 81 235115
[126] van de Kraats J, Nielsen K K and Bruun G M 2022 Phys. Rev. B 106 235143
[127] Seifert U F P and Balents L 2024 Phys. Rev. Lett. 132 046501
[128] Zhang Y and Fu L 2023 SciPost Phys. Core 6 038
[129] Davydova M, Zhang Y and Fu L 2023 Phys. Rev. B 107 224420
[130] Tao Z, Zhao W, Shen B, Knuppel P, Watanabe K, Taniguchi T, Shan J and Mak K F 2023 arXiv:2307.12205
[cond-mat.str-el]
[131] Yang Y, Morales M A and Zhang S 2024 Phys. Rev. Lett. 132 076503
[132] Kaushal N, Morales-Duran N, MacDonald A H and Dagotto E 2022 Commun. Phys. 5 289
[133] Tang E, Mei J W and Wen X G 2011 Phys. Rev. Lett. 106 236802
[134] Sun K, Gu Z, Katsura H and Das Sarma S 2011 Phys. Rev. Lett. 106 236803
[135] Neupert T, Santos L, Chamon C and Mudry C 2011 Phys. Rev. Lett. 106 236804

[1]	Effect of the mixing of s-wave and chiral p-wave pairings on electrical shot noise properties of normal metal/superconductor tunnel junctions Yu-Chen Hu(胡雨辰) and Liang-Bin Hu(胡梁宾). Chin. Phys. B, 2024, 33(7): 077202.
[2]	Effect of lattice distortion on spin admixture and quantum transport in organic devices with spin-orbit coupling Ying Wang(王莹), Dan Li(李丹), Xinying Sun(孙新英), Huiqing Zhang(张惠晴), Han Ma(马晗), Huixin Li(李慧欣), Junfeng Ren(任俊峰), Chuankui Wang(王传奎), and Guichao Hu(胡贵超). Chin. Phys. B, 2024, 33(7): 077101.
[3]	Oscillation of Dzyaloshinskii-Moriya interaction driven by weak electric fields Runze Chen(陈润泽), Anni Cao(曹安妮), Xinran Wang(王馨苒), Yang Liu(柳洋), Hongxin Yang(杨洪新), and Weisheng Zhao(赵巍胜). Chin. Phys. B, 2024, 33(2): 027501.
[4]	Spatial electron-spin splitting in single-layered semiconductor microstructure modulated by Dresselhaus spin-orbit coupling Jia-Li Chen(陈嘉丽), Sai-Yan Chen(陈赛艳), Li Wen(温丽), Xue-Li Cao(曹雪丽), and Mao-Wang Lu(卢卯旺). Chin. Phys. B, 2024, 33(11): 118501.
[5]	Bessel vortices in spin-1 Bose-Einstein condensates with Zeeman splitting and spin-orbit coupling Huan-Bo Luo(罗焕波), Xin-Feng Zhang(张鑫锋), Runhua Li(李润华), Yongyao Li(黎永耀), and Bin Liu(刘彬). Chin. Phys. B, 2024, 33(10): 100304.
[6]	Fully relativistic energies, transition properties, and lifetimes of lithium-like germanium Shuang Li(李双), Jing Zhou(周璟), Liu-Hong Zhu(朱柳红), Xiu-Fei Mei(梅秀菲), and Jun Yan(颜君). Chin. Phys. B, 2024, 33(10): 103102.
[7]	Customizing topological phases in the twisted bilayer superconductors with even-parity pairings Conghao Lin(林丛豪), Chuanshuai Huang(黄传帅), and Xiancong Lu(卢仙聪). Chin. Phys. B, 2023, 32(8): 087401.
[8]	Anomalous Josephson effect between d-wave superconductors through a two-dimensional electron gas with both Rashba spin-orbit coupling and Zeeman splitting Bin-Hao Du(杜彬豪), Mou Yang(杨谋), and Liang-Bin Hu(胡梁宾). Chin. Phys. B, 2023, 32(7): 077201.
[9]	Ta thickness effect on field-free switching and spin-orbit torque efficiency in a ferromagnetically coupled Co/Ta/CoFeB trilayer Zhongshu Feng(冯重舒), Changqiu Yu(于长秋), Haixia Huang(黄海侠), Haodong Fan(樊浩东),Mingzhang Wei(卫鸣璋), Birui Wu(吴必瑞), Menghao Jin(金蒙豪), Yanshan Zhuang(庄燕山),Ziji Shao(邵子霁), Hai Li(李海), Jiahong Wen(温嘉红), Jian Zhang(张鉴), Xuefeng Zhang(张雪峰),Ningning Wang(王宁宁), Sai Mu(穆赛), and Tiejun Zhou(周铁军). Chin. Phys. B, 2023, 32(4): 048504.
[10]	Electrical manipulation of a hole ‘spin’-orbit qubit in nanowire quantum dot: The nontrivial magnetic field effects Rui Li(李睿) and Hang Zhang(张航). Chin. Phys. B, 2023, 32(3): 030308.
[11]	Coexistence of giant Rashba spin splitting and quantum spin Hall effect in H-Pb-F Wenming Xue(薛文明), Jin Li(李金), Chaoyu He(何朝宇), Tao Ouyang(欧阳滔), Xiongying Dai(戴雄英), and Jianxin Zhong(钟建新). Chin. Phys. B, 2023, 32(3): 037101.
[12]	Benchmarking calculations of excitation energies and transition properties with spectroscopic accuracy of highly charged ions used for the fusion plasma and astrophysical plasma Chunyu Zhang(张春雨), Kai Wang(王凯), Ran Si(司然), Jinqing Li(李金晴), Changxian Song(宋昌仙), Sijie Wu(吴思捷), Bishuang Yan(严碧霜), and Chongyang Chen(陈重阳). Chin. Phys. B, 2023, 32(11): 113102.
[13]	Low-lying electronic states of osmium monoxide OsO Wen Yan(严汶) and Wenli Zou(邹文利). Chin. Phys. B, 2023, 32(11): 113101.
[14]	Phonon dichroism in proximitized graphene Wen-Yu Shan(单文语). Chin. Phys. B, 2023, 32(10): 106301.
[15]	Perspectives of spin-valley locking devices Lingling Tao(陶玲玲). Chin. Phys. B, 2023, 32(10): 107306.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Correlated physics, charge and magnetic orders in moiré kagomé systems

Cite this article:

Online attention