- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Kagome chiral spin liquid in transition metal dichalcogenide moiré bilayers

Johannes Motruk1,*, Dario Rossi1, Dmitry A. Abanin1,2, and Louk Rademaker1,3

  • 1Department of Theoretical Physics, University of Geneva, Quai Ernest-Ansermet 24, 1205 Geneva, Switzerland
  • 2Google Research, Mountain View, California, USA
  • 3Department of Quantum Matter Physics, University of Geneva, Quai Ernest-Ansermet 24, 1205 Geneva, Switzerland

  • *johannes.motruk@unige.ch
PDF

Phys. Rev. Research 5, L022049 – Published 5 June, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L022049

Abstract

At n=3/4 filling of the moiré flat band, transition metal dichalcogenide (TMD) moiré bilayers will develop kagome charge order. We derive an effective spin model for the resulting localized spins and find that its further neighbor spin interactions can be much less suppressed than the corresponding electron hopping strength. Using density matrix renormalization group simulations, we study its phase diagram and, for realistic model parameters relevant for WSe2/WS2, we show that this material can realize the exotic chiral spin liquid phase and the highly debated kagome spin liquid. Our work thus demonstrates that the frustration and strong interactions present in TMD heterobilayers provide an exciting platform to study spin liquid physics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (80)

  1. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  2. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  3. L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nat. Phys. 16, 725 (2020).
  4. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  5. K. F. Mak and J. Shan, Semiconductor moiré materials, Nat. Nanotechnol. 17, 686 (2022).
  6. Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan, and K. F. Mak, Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices, Nature (London) 579, 353 (2020).
  7. E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta, M. Blei, J. D. Carlström, K. Watanabe, T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and F. Wang, Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices, Nature (London) 579, 359 (2020).
  8. L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim, K. Watanabe, T. Taniguchi, X. Zhu, J. Hone, A. Rubio, A. N. Pasupathy, and C. R. Dean, Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nat. Mater. 19, 1 (2020).
  9. T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K. Watanabe, T. Taniguchi, D. Chowdhury, L. Fu, J. Shan, and K. F. Mak, Continuous Mott transition in semiconductor moiré superlattices, Nature (London) 597, 350 (2021).
  10. A. Ghiotto, E.-M. Shih, G. S. S. G. Pereira, D. A. Rhodes, B. Kim, J. Zang, A. J. Millis, K. Watanabe, T. Taniguchi, J. C. Hone, L. Wang, C. R. Dean, and A. N. Pasupathy, Quantum criticality in twisted transition metal dichalcogenides, Nature (London) 597, 345 (2021).
  11. T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. Devakul, K. Watanabe, T. Taniguchi, L. Fu, J. Shan, and K. F. Mak, Quantum anomalous Hall effect from intertwined moiré bands, Nature (London) 600, 641 (2021).
  12. Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated insulating states at fractional fillings of moiré superlattices, Nature (London) 587, 214 (2020).
  13. X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, S.-F. Shi, and Y.-T. Cui, Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice, Nat. Phys. 17, 715 (2021).
  14. H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, Imaging two-dimensional generalized Wigner crystals, Nature (London) 597, 650 (2021).
  15. E. Liu, T. Taniguchi, K. Watanabe, N. M. Gabor, Y.-T. Cui, and C. H. Lui, Excitonic and Valley-Polarization Signatures of Fractional Correlated Electronic Phases in a WSe2/WS2 Moiré Superlattice, Phys. Rev. Lett. 127, 037402 (2021).
  16. S. Miao, T. Wang, X. Huang, D. Chen, Z. Lian, C. Wang, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, Z. Wang, D. Xiao, Y.-T. Cui, and S.-F. Shi, Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moiré superlattice, Nat. Commun. 12, 3608 (2021).
  17. X. Wang, C. Xiao, H. Park, J. Zhu, C. Wang, T. Taniguchi, K. Watanabe, J. Yan, D. Xiao, D. R. Gamelin, W. Yao, and X. Xu, Light-induced ferromagnetism in moiré superlattices, Nature (London) 604, 468 (2022).
  18. Y. Tang, K. Su, L. Li, Y. Xu, S. Liu, K. Watanabe, T. Taniguchi, J. Hone, C.-M. Jian, C. Xu, K. F. Mak, and J. Shan, Evidence of frustrated magnetic interactions in a Wigner–Mott insulator, Nat. Nanotechnol. 18, 233 (2023).
  19. E. Anderson, F.-R. Fan, J. Cai, W. Holtzmann, T. Taniguchi, K. Watanabe, D. Xiao, W. Yao, and X. Xu, Programming correlated magnetic states via gate controlled moiré geometry, arXiv:2303.17038.
  20. F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands, Phys. Rev. Lett. 121, 026402 (2018).
  21. Y. Zhang, N. F. Q. Yuan, and L. Fu, Moiré quantum chemistry: Charge transfer in transition metal dichalcogenide superlattices, Phys. Rev. B 102, 201115(R) (2020).
  22. L. Rademaker, Spin-orbit coupling in transition metal dichalcogenide heterobilayer flat bands, Phys. Rev. B 105, 195428 (2022).
  23. Y. Zhang, T. Devakul, and L. Fu, Spin-textured Chern bands in AB-stacked transition metal dichalcogenide bilayers, Proc. Natl. Acad. Sci. USA 118, e2112673118 (2021).
  24. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2016).
  25. J. Knolle and R. Moessner, A field guide to spin liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  26. V. Kalmeyer and R. B. Laughlin, Equivalence of the Resonating-Valence-Bond and Fractional Quantum Hall States, Phys. Rev. Lett. 59, 2095 (1987).
  27. D. F. Schroeter, E. Kapit, R. Thomale, and M. Greiter, Spin Hamiltonian for which the Chiral Spin Liquid is the Exact Ground State, Phys. Rev. Lett. 99, 097202 (2007).
  28. H. C. Jiang, Z. Y. Weng, and D. N. Sheng, Density Matrix Renormalization Group Numerical Study of the Kagome Antiferromagnet, Phys. Rev. Lett. 101, 117203 (2008).
  29. S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground state of the S = 1/2 kagome Heisenberg antiferromagnet, Science 332, 1173 (2011).
  30. H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless Spin-Liquid Ground State in the S=1/2 Kagome Antiferromagnet, Phys. Rev. Lett. 118, 137202 (2017).
  31. Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Signatures of Dirac Cones in a DMRG Study of the Kagome Heisenberg Model, Phys. Rev. X 7, 031020 (2017).
  32. A. M. Läuchli, J. Sudan, and R. Moessner, S=12 kagome Heisenberg antiferromagnet revisited, Phys. Rev. B 100, 155142 (2019).
  33. Y. Iqbal, F. Ferrari, A. Chauhan, A. Parola, D. Poilblanc, and F. Becca, Gutzwiller projected states for the J1−J2 Heisenberg model on the Kagome lattice: Achievements and pitfalls, Phys. Rev. B 104, 144406 (2021).
  34. S. R. White, Density Matrix Formulation for Quantum Renormalization Groups, Phys. Rev. Lett. 69, 2863 (1992).
  35. I. P. McCulloch, Infinite size density matrix renormalization group, revisited, arXiv:0804.2509.
  36. Y. Zhou, D. N. Sheng, and E.-A. Kim, Quantum Phases of Transition Metal Dichalcogenide Moiré Systems, Phys. Rev. Lett. 128, 157602 (2022).
  37. A. Szasz, J. Motruk, M. P. Zaletel, and J. E. Moore, Chiral Spin Liquid Phase of the Triangular Lattice Hubbard Model: A Density Matrix Renormalization Group Study, Phys. Rev. X 10, 021042 (2020).
  38. D. Kiese, Y. He, C. Hickey, A. Rubio, and D. M. Kennes, TMDs as a platform for spin liquid physics: A strong coupling study of twisted bilayer WSe2, APL Mater. 10, 031113 (2022).
  39. C. Kuhlenkamp, W. Kadow, A. İmamoğlu, and M. Knap, Tunable topological order of pseudo spins in semiconductor heterostructures, arXiv:2209.05506.
  40. H. Pan, F. Wu, and S. Das Sarma, Quantum phase diagram of a moiré-Hubbard model, Phys. Rev. B 102, 201104(R) (2020).
  41. Y. Tan, P. K. H. Tsang, V. Dobrosavljević, and L. Rademaker, Doping a Wigner-Mott insulator: Electron slush in transition-metal dichalcogenide moiré heterobilayers, arXiv:2210.07926.
  42. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L022049 for details on the ab initio calculations of the model parameters, mean-field theory of the charge order, derivation of the effective spin model, DMRG parameters, nature of the gauge transformation, characterization of magnetic states, and distinguishing the KSL from the q=(0,0)* order, which includes Refs. [42, 43].
  43. A. H. MacDonald, S. M. Girvin, and D. Yoshioka, tU expansion for the Hubbard model, Phys. Rev. B 37, 9753 (1988).
  44. F. Kolley, S. Depenbrock, I. P. McCulloch, U. Schollwöck, and V. Alba, Phase diagram of the J1−J2 Heisenberg model on the kagome lattice, Phys. Rev. B 91, 104418 (2015).
  45. N. Morales-Durán, N. C. Hu, P. Potasz, and A. H. MacDonald, Nonlocal Interactions in Moiré Hubbard Systems, Phys. Rev. Lett. 128, 217202 (2022).
  46. N. Morales-Durán, P. Potasz, and A. H. MacDonald, Magnetism and Quantum Melting in Moiré-Material Wigner Crystals, arXiv:2210.15168.
  47. M. Takahashi, Half-filled Hubbard model at low temperature, J. Phys. C: Solid State Phys. 10, 1289 (1977).
  48. H. Pan, F. Wu, and S. Das Sarma, Band topology, Hubbard model, Heisenberg model, and Dzyaloshinskii-Moriya interaction in twisted bilayer WSe2, Phys. Rev. Res. 2, 033087 (2020).
  49. J. Zang, J. Wang, J. Cano, and A. J. Millis, Hartree-Fock study of the moiré Hubbard model for twisted bilayer transition metal dichalcogenides, Phys. Rev. B 104, 075150 (2021).
  50. A. Wietek, J. Wang, J. Zang, J. Cano, A. Georges, and A. Millis, Tunable stripe order and weak superconductivity in the moiré Hubbard model, Phys. Rev. Res. 4, 043048 (2022).
  51. S.-S. Gong, W. Zhu, L. Balents, and D. N. Sheng, Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice, Phys. Rev. B 91, 075112 (2015).
  52. Y.-C. He, D. N. Sheng, and Y. Chen, Chiral Spin Liquid in a Frustrated Anisotropic Kagome Heisenberg Model, Phys. Rev. Lett. 112, 137202 (2014).
  53. S.-S. Gong, W. Zhu, and D. N. Sheng, Emergent chiral spin liquid: Fractional quantum Hall effect in a kagome Heisenberg model, Sci. Rep. 4, 6317 (2014).
  54. Y.-C. He and Y. Chen, Distinct Spin Liquids and Their Transitions in Spin-1/2 XXZ Kagome Antiferromagnets, Phys. Rev. Lett. 114, 037201 (2015).
  55. A. Wietek, A. Sterdyniak, and A. M. Läuchli, Nature of chiral spin liquids on the kagome lattice, Phys. Rev. B 92, 125122 (2015).
  56. A. B. Harris, C. Kallin, and A. J. Berlinsky, Possible néel orderings of the kagomé antiferromagnet, Phys. Rev. B 45, 2899 (1992).
  57. R. R. P. Singh and D. A. Huse, Three-Sublattice Order in Triangular- and Kagomé-Lattice Spin-Half Antiferromagnets, Phys. Rev. Lett. 68, 1766 (1992).
  58. S. Sachdev, Kagome′- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons, Phys. Rev. B 45, 12377 (1992).
  59. L. Messio, C. Lhuillier, and G. Misguich, Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets, Phys. Rev. B 83, 184401 (2011).
  60. C. Zeng and V. Elser, Numerical studies of antiferromagnetism on a Kagomé net, Phys. Rev. B 42, 8436 (1990).
  61. X. G. Wen, Gapless boundary excitations in the quantum Hall states and in the chiral spin states, Phys. Rev. B 43, 11025 (1991).
  62. H. Li and F. D. M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101, 010504 (2008).
  63. X.-L. Qi, H. Katsura, and A. W. W. Ludwig, General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States, Phys. Rev. Lett. 108, 196402 (2012).
  64. R. R. P. Singh and D. A. Huse, Ground state of the spin-1/2 kagome-lattice Heisenberg antiferromagnet, Phys. Rev. B 76, 180407(R) (2007).
  65. H.-C. Jiang, Z. Wang, and L. Balents, Identifying topological order by entanglement entropy, Nat. Phys. 8, 902 (2012).
  66. J.-W. Mei, J.-Y. Chen, H. He, and X.-G. Wen, Gapped spin liquid with Z2 topological order for the kagome Heisenberg model, Phys. Rev. B 95, 235107 (2017).
  67. H. J. Changlani, D. Kochkov, K. Kumar, B. K. Clark, and E. Fradkin, Macroscopically Degenerate Exactly Solvable Point in the Spin-1/2 Kagome Quantum Antiferromagnet, Phys. Rev. Lett. 120, 117202 (2018).
  68. S. Jiang, P. Kim, J. H. Han, and Y. Ran, Competing spin liquid phases in the s=12 Heisenberg model on the kagome lattice, SciPost Phys. 7, 006 (2019).
  69. A. Wietek and A. M. Läuchli, Valence bond solid and possible deconfined quantum criticality in an extended kagome lattice Heisenberg antiferromagnet, Phys. Rev. B 102, 020411(R) (2020).
  70. A. G. Salvador, C. Kuhlenkamp, L. Ciorciaro, M. Knap, and A. İmamoğlu, Optical Signatures of Periodic Magnetization: The Moiré Zeeman Effect, Phys. Rev. Lett. 128, 237401 (2022).
  71. O. Kazakova, R. Puttock, C. Barton, H. Corte-León, M. Jaafar, V. Neu, and A. Asenjo, Frontiers of magnetic force microscopy, J. Appl. Phys. 125, 060901 (2019).
  72. R. Wiesendanger, Spin mapping at the nanoscale and atomic scale, Rev. Mod. Phys. 81, 1495 (2009).
  73. S. Chatterjee, J. F. Rodriguez-Nieva, and E. Demler, Diagnosing phases of magnetic insulators via noise magnetometry with spin qubits, Phys. Rev. B 99, 104425 (2019).
  74. D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled Spin and Valley Physics in Monolayers of MoS2 and Other Group-VI Dichalcogenides, Phys. Rev. Lett. 108, 196802 (2012).
  75. K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS2 by optical helicity, Nat. Nanotechnol. 7, 494 (2012).
  76. S. Banerjee, W. Zhu, and S.-Z. Lin, Electromagnetic signatures of chiral quantum spin liquid, arXiv:2304.08635.
  77. M. Claassen, L. Xian, D. M. Kennes, and A. Rubio, Ultra-strong spin–orbit coupling and topological moiré engineering in twisted ZrS2 bilayers, Nat. Commun. 13, 4915 (2022).
  78. A. P. Reddy, T. Devakul, and L. Fu, Moiré alchemy: Artificial atoms, Wigner molecules, and emergent Kagome lattice, arXiv:2301.00799.
  79. J. Motruk, D. Rossi, and L. Rademaker, Kagome chiral spin liquid in transition metal dichalcogenide moiré bilayers [Data set], Université de Genève, Yareta, https://doi.org/10.26037/yareta:y635nolybbdfjjvwovjwm3dnmm.
  80. J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes, 5 (2018); code available from https://github.com/tenpy/tenpy.

Outline

Information

Phys. Rev. Research 5, L022049– Published 5 June, 2023
  • Received 5 December 2022
  • Revised 27 April 2023
  • Accepted 2 May 2023
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation