- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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
  • Open Access

From hidden order to skyrmions: Quantum Hall states in an extended Hofstadter-Fermi-Hubbard model

F. J. Pauw1,2, U. Schollwöck1,2, N. Goldman3,4,5, S. Paeckel1,2, and F. A. Palm3,4

PDF

Phys. Rev. B 113, 035142 – Published 22 January, 2026

DOI: https://doi.org/10.1103/cqt2-b327

Abstract

The interplay between topology and strong interactions gives rise to a variety of exotic quantum phases, including fractional quantum Hall states and their lattice analogs—fractional Chern insulators (FCIs). Such topologically ordered states host fractionalized excitations and, for spinful systems, are often accompanied by ferromagnetism and skyrmions. Here, we study a Hofstadter-Hubbard model of spinful fermions on a square lattice, extended by nearest-neighbor interactions. Using large-scale density matrix renormalization group simulations, we demonstrate the emergence of a spin-polarized 1/3-Laughlin-like FCI phase, characterized by a quantized many-body Chern number, a finite charge gap, and hidden off-diagonal long-range order. We further investigate the quantum Hall ferromagnet at ν=1 and its skyrmionic excitations upon doping. In particular, we find that nearest-neighbor repulsion is sufficient to stabilize both particle- and hole-skyrmions in the ground state around ν=1, whereas we do not find such textures around ν=1/3. The diagnostic toolbox presented in this work, based on local densities, correlation functions, and spin-resolved observables, is directly applicable in quantum gas microscopy experiments. Our results open new pathways for experimental exploration of FCIs with spin textures in both ultracold atom and electronic systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (101)

  1. A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall states of atoms in optical lattices, Phys. Rev. Lett. 94, 086803 (2005).
  2. R. N. Palmer and D. Jaksch, High-field fractional quantum Hall effect in optical lattices, Phys. Rev. Lett. 96, 180407 (2006).
  3. M. Hafezi, A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall effect in optical lattices, Phys. Rev. A 76, 023613 (2007).
  4. R. N. Palmer, A. Klein, and D. Jaksch, Optical lattice quantum Hall effect, Phys. Rev. A 78, 013609 (2008).
  5. G. Möller and N. R. Cooper, Composite fermion theory for bosonic quantum Hall states on lattices, Phys. Rev. Lett. 103, 105303 (2009).
  6. G. Möller and N. R. Cooper, Condensed ground states of frustrated Bose-Hubbard models, Phys. Rev. A 82, 063625 (2010).
  7. G. Möller and N. R. Cooper, Fractional Chern insulators in Harper-Hofstadter bands with higher Chern number, Phys. Rev. Lett. 115, 126401 (2015).
  8. D. Bauer, T. S. Jackson, and R. Roy, Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model, Phys. Rev. B 93, 235133 (2016).
  9. D. Hügel, Hugo U. R. Strand, P. Werner, and L. Pollet, Anisotropic Harper-Hofstadter-Mott model: Competition between condensation and magnetic fields, Phys. Rev. B 96, 054431 (2017).
  10. J. Motruk and F. Pollmann, Phase transitions and adiabatic preparation of a fractional Chern insulator in a boson cold-atom model, Phys. Rev. B 96, 165107 (2017).
  11. Y.-C. He, F. Grusdt, A. Kaufman, M. Greiner, and A. Vishwanath, Realizing and adiabatically preparing bosonic integer and fractional quantum Hall states in optical lattices, Phys. Rev. B 96, 201103(R) (2017).
  12. M. Gerster, M. Rizzi, P. Silvi, M. Dalmonte, and S. Montangero, Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks, Phys. Rev. B 96, 195123 (2017).
  13. B. Andrews and G. Möller, Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1, Phys. Rev. B 97, 035159 (2018).
  14. X.-Y. Dong, A. G. Grushin, J. Motruk, and F. Pollmann, Charge excitation dynamics in bosonic fractional Chern insulators, Phys. Rev. Lett. 121, 086401 (2018).
  15. B. Andrews, M. Mohan, and T. Neupert, Abelian topological order of ν = 2/5 and 3/7 fractional quantum Hall states in lattice models, Phys. Rev. B 103, 075132 (2021).
  16. F. A. Palm, M. Buser, J. Léonard, M. Aidelsburger, U. Schollwöck, and F. Grusdt, Bosonic Pfaffian state in the Hofstadter-Bose-Hubbard model, Phys. Rev. B 103, L161101 (2021).
  17. B. Andrews, T. Neupert, and G. Möller, Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model, Phys. Rev. B 104, 125107 (2021).
  18. J. Boesl, R. Dilip, F. Pollmann, and M. Knap, Characterizing fractional topological phases of lattice bosons near the first Mott lobe, Phys. Rev. B 105, 075135 (2022).
  19. B. Wang, X. Dong, and A. Eckardt, Measurable signatures of bosonic fractional Chern insulator states and their fractional excitations in a quantum-gas microscope, SciPost Phys. 12, 095 (2022).
  20. F. A. Palm, S. Mardazad, A. Bohrdt, U. Schollwöck, and F. Grusdt, Snapshot-based detection of ν=12 Laughlin states: Coupled chains and central charge, Phys. Rev. B 106, L081108 (2022).
  21. L. W. Clark, N. Schine, C. Baum, N. Jia, and J. Simon, Observation of Laughlin states made of light, Nature (London) 582, 41 (2020).
  22. J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum Hall state with ultracold atoms, Nature (London) 619, 495 (2023).
  23. C. Wang, F.-M. Liu, M.-C. Chen, H. Chen, X.-H. Zhao, C. Ying, Z.-X. Shang, J.-W. Wang, Y.-H. Huo, C.-Z. Peng, X. Zhu, C.-Y. Lu, and J.-W. Pan, Realization of fractional quantum Hall state with interacting photons, Science 384, 579 (2024).
  24. N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  25. D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum Hall effect in the absence of Landau levels, Nat. Commun. 2, 389 (2011).
  26. T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  27. Y.-L. Wu, B. A. Bernevig, and N. Regnault, Zoology of fractional Chern insulators, Phys. Rev. B 85, 075116 (2012).
  28. F. D. M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51, 605 (1983).
  29. J. Motruk, M. P. Zaletel, Roger S. K. Mong, and F. Pollmann, Density matrix renormalization group on a cylinder in mixed real and momentum space, Phys. Rev. B 93, 155139 (2016).
  30. B. I. Halperin, Theory of the quantized Hall conductance, Helv. Phys. Acta 56, 75 (1983) .
  31. E. H. Rezayi, Reversed-spin excitations of the fractionally quantized Hall effect from finite-size calculations, Phys. Rev. B 36, 5454 (1987).
  32. F. D. M. Haldane and E. H. Rezayi, Spin-singlet wave function for the half-integral quantum Hall effect, Phys. Rev. Lett. 60, 956 (1988).
  33. D.-H. Lee and C. L. Kane, Boson-vortex-skyrmion duality, spin-singlet fractional quantum Hall effect, and spin-1/2 anyon superconductivity, Phys. Rev. Lett. 64, 1313 (1990).
  34. E. H. Rezayi, Wave functions and other properties of spin-reversed quasiparticles at ν = 1/m Landau-level occupation, Phys. Rev. B 43, 5944 (1991).
  35. S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47, 16419 (1993).
  36. H. A. Fertig, L. Brey, R. Côté, and A. H. MacDonald, Charged spin-texture excitations and the Hartree-Fock approximation in the quantum Hall effect, Phys. Rev. B 50, 11018 (1994).
  37. K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Spontaneous interlayer coherence in double-layer quantum Hall systems: Charged vortices and Kosterlitz-Thouless phase transitions, Phys. Rev. B 51, 5138 (1995).
  38. A. H. MacDonald, H. A. Fertig, and L. Brey, Skyrmions without sigma models in quantum Hall ferromagnets, Phys. Rev. Lett. 76, 2153 (1996).
  39. K. Yang, K. Moon, L. Zheng, A. H. MacDonald, S. M. Girvin, D. Yoshioka, and S.-C. Zhang, Quantum ferromagnetism and phase transitions in double-layer quantum Hall systems, Phys. Rev. Lett. 72, 732 (1994).
  40. Z. F. Ezawa and G. Tsitsishvili, Quantum Hall ferromagnets, Rep. Prog. Phys. 72, 086502 (2009).
  41. R. G. Clark, S. R. Haynes, A. M. Suckling, J. R. Mallett, P. A. Wright, J. J. Harris, and C. T. Foxon, Spin configurations and quasiparticle fractional charge of fractional-quantum-Hall-effect ground states in the N = 0 Landau level, Phys. Rev. Lett. 62, 1536 (1989).
  42. J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West, Evidence for a spin transition in the ν=2/3 fractional quantum Hall effect, Phys. Rev. B 41, 7910 (1990).
  43. L. W. Engel, S. W. Hwang, T. Sajoto, D. C. Tsui, and M. Shayegan, Fractional quantum Hall effect at ν=2/3 and 3/5 in tilted magnetic fields, Phys. Rev. B 45, 3418 (1992).
  44. S. Holmes, D. K. Maude, M. L. Williams, J. J. Harris, J. C. Portal, K. W. J. Barnham, and C. T. Foxon, Experimental determination of the transport properties of composite fermions with reduced Lande g-factor, Semicond. Sci. Technol. 9, 1549 (1994).
  45. R. R. Du, A. S. Yeh, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Fractional quantum Hall effect around ν=3/2: Composite fermions with a spin, Phys. Rev. Lett. 75, 3926 (1995).
  46. S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Optically pumped NMR evidence for finite-size skyrmions in GaAs quantum wells near Landau level filling ν = 1, Phys. Rev. Lett. 74, 5112 (1995).
  47. D. R. Leadley, R. J. Nicholas, D. K. Maude, A. N. Utjuzh, J. C. Portal, J. J. Harris, and C. T. Foxon, Fractional quantum Hall effect measurements at zero g factor, Phys. Rev. Lett. 79, 4246 (1997).
  48. D. R. Leadley, R. J. Nicholas, D. K. Maude, A. N. Utjuzh, J. C. Portal, J. J. Harris, and C. T. Foxon, Searches for skyrmions in the limit of zero g-factor, Semicond. Sci. Technol. 13, 671 (1998).
  49. P. Khandelwal, N. N. Kuzma, S. E. Barrett, L. N. Pfeiffer, and K. W. West, Optically pumped NMR in the quantum Hall regimes, Phys. B: Condens. Matter 256, 113 (1998).
  50. I. V. Kukushkin, K. v. Klitzing, and K. Eberl, Spin polarization of composite fermions: Measurements of the Fermi energy, Phys. Rev. Lett. 82, 3665 (1999).
  51. R. K. Kamilla, J. K. Jain, and S. M. Girvin, Fermi-sea-like correlations in a partially filled Landau level, Phys. Rev. B 56, 12411 (1997).
  52. A. Wójs and J. J. Quinn, Spin excitation spectra of integral and fractional quantum Hall systems, Phys. Rev. B 66, 045323 (2002).
  53. R. L. Doretto, M. O. Goerbig, P. Lederer, A. O. Caldeira, and C. Morais Smith, Spin-excitations of the quantum Hall ferromagnet of composite fermions, Phys. Rev. B 72, 035341 (2005).
  54. J. Xia, X. Zhang, X. Liu, Y. Zhou, and M. Ezawa, Universal quantum computation based on nanoscale skyrmion helicity qubits in frustrated magnets, Phys. Rev. Lett. 130, 106701 (2023).
  55. Y. Yang, L. Zhao, D. Yi, T. Xu, Y. Chai, C. Zhang, D. Jiang, Y. Ji, D. Hou, W. Jiang, J. Tang, P. Yu, H. Wu, and T. Nan, Acoustic-driven magnetic skyrmion motion, Nat. Commun. 15, 1018 (2024).
  56. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice, Nature (London) 462, 74 (2009).
  57. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature (London) 467, 68 (2010).
  58. P. M. Preiss, R. Ma, M. E. Tai, J. Simon, and M. Greiner, Quantum gas microscopy with spin, atom-number, and multilayer readout, Phys. Rev. A 91, 041602(R) (2015).
  59. A. Bergschneider, V. M. Klinkhamer, J. H. Becher, R. Klemt, G. Zürn, P. M. Preiss, and S. Jochim, Spin-resolved single-atom imaging of Li6 in free space, Phys. Rev. A 97, 063613 (2018).
  60. A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Nature (London) 545, 462 (2017).
  61. C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys. 17, 1316 (2021).
  62. 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).
  63. F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Phys. Rev. Lett. 122, 086402 (2019).
  64. N. Morales-Durán, N. C. Hu, P. Potasz, and A. H. MacDonald, Nonlocal interactions in Moiré Hubbard systems, Phys. Rev. Lett. 128, 217202 (2022).
  65. J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature (London) 622, 63 (2023).
  66. Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional Chern insulator in Moiré MoTe2, Nature (London) 622, 69 (2023).
  67. H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous Hall effect, Nature (London) 622, 74 (2023).
  68. C. Kuhlenkamp, W. Kadow, A. Imamoğlu, and M. Knap, Chiral pseudospin liquids in Moiré heterostructures, Phys. Rev. X 14, 021013 (2024).
  69. S. M. Girvin and A. H. MacDonald, Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, Phys. Rev. Lett. 58, 1252 (1987).
  70. S. M. Girvin, Off-Diagonal Long-Range Order in the Quantum Hall Effect (Springer US, Boston, MA, 1988), pp. 333.
  71. E. H. Rezayi and F. D. M. Haldane, Off-diagonal long-range order in fractional quantum-Hall-effect states, Phys. Rev. Lett. 61, 1985 (1988).
  72. N. Read, Order parameter and Ginzburg-Landau theory for the fractional quantum Hall effect, Phys. Rev. Lett. 62, 86 (1989).
  73. F. J. Pauw, F. A. Palm, U. Schollwöck, A. Bohrdt, S. Paeckel, and F. Grusdt, Detecting hidden order in fractional Chern insulators, Phys. Rev. Res. 6, 023180 (2024).
  74. R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  75. F. A. Palm, M. Kurttutan, A. Bohrdt, U. Schollwöck, and F. Grusdt, Ferromagnetism and skyrmions in the Hofstadter-Fermi-Hubbard model, New J. Phys. 25, 023021 (2023).
  76. J. K. Ding, L. Yang, W. O. Wang, Z. Zhu, C. Peng, P. Mai, E. W. Huang, B. Moritz, P. W. Phillips, B. E. Feldman, and T. P. Devereaux, Particle-hole asymmetric ferromagnetism and spin textures in the triangular Hubbard-Hofstadter model, Phys. Rev. X 14, 041025 (2024).
  77. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  78. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011), january 2011 Special Issue.
  79. C. Hubig, I. P. McCulloch, U. Schollwöck, and F. A. Wolf, Strictly single-site DMRG algorithm with subspace expansion, Phys. Rev. B 91, 155115 (2015).
  80. C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard, L. Stenzel, A. Swoboda, M. Grundner, S. Mardazad, F. Pauw, and S. Paeckel, The SyTen toolkit, https://syten.eu/.
  81. F. Binanti, N. Goldman, and C. Repellin, Spectroscopy of edge and bulk collective modes in fractional Chern insulators, Phys. Rev. Res. 6, L012054 (2024).
  82. F. N. Ünal, A. Nardin, and N. Goldman, Circular dichroism on the edge of quantum Hall systems: From many-body Chern number to anisotropy measurements Phys. Rev. Lett. 135, 266603 (2025).
  83. P. Streda, Theory of quantised Hall conductivity in two dimensions, J. Phys. C 15, L717 (1982).
  84. R. Tao and D. J. Thouless, Fractional quantization of Hall conductance, Phys. Rev. B 28, 1142 (1983).
  85. E. H. Rezayi and F. D. M. Haldane, Laughlin state on stretched and squeezed cylinders and edge excitations in the quantum Hall effect, Phys. Rev. B 50, 17199 (1994).
  86. C. Repellin, J. Léonard, and N. Goldman, Fractional Chern insulators of few bosons in a box: Hall plateaus from center-of-mass drifts and density profiles, Phys. Rev. A 102, 063316 (2020).
  87. A. Seidel, H. Fu, D.-H. Lee, J. M. Leinaas, and J. Moore, Incompressible quantum liquids and new conservation laws, Phys. Rev. Lett. 95, 266405 (2005).
  88. S. M. Girvin, The quantum Hall effect: Novel excitations and broken symmetries, arXiv:cond-mat/9907002.
  89. E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge, UK, 2013).
  90. A. J. Ferris and G. Vidal, Perfect sampling with unitary tensor networks, Phys. Rev. B 85, 165146 (2012).
  91. F. Pichler, C. Kuhlenkamp, M. Knap, and A. Vishwanath, Microscopic mechanism of anyon superconductivity emerging from fractional Chern insulators, Newton 2, 100340 (2026).
  92. C. Kuhlenkamp, S. Divic, M. P. Zaletel, T. Soejima, and A. Vishwanath, Robust superconductivity upon doping chiral spin liquid and Chern insulators in a Hubbard-Hofstadter model, arXiv:2509.02675.
  93. For an additional discussion of the spin-polarized sector see Appendix pp3-s1.
  94. X. C. Xie, Y. Guo, and F. C. Zhang, Fractional quantum Hall effect with spin reversal, Phys. Rev. B 40, 3487 (1989).
  95. P. A. Maksym, Spin-dependent fractional QHE states in the N =0 Landau level, J. Phys.: Condens. Matter 1, 6299 (1989).
  96. T. Chakraborty, Spin-reversed ground state and energy gap in the fractional quantum Hall effect, Surf. Sci. 229, 16 (1990).
  97. J. K. Jain, Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63, 199 (1989).
  98. M. Dolfi, B. Bauer, S. Keller, and M. Troyer, Pair correlations in doped Hubbard ladders, Phys. Rev. B 92, 195139 (2015).
  99. 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).
  100. A. Sterdyniak, N. Regnault, and B. A. Bernevig, Extracting excitations from model state entanglement, Phys. Rev. Lett. 106, 100405 (2011).
  101. A. Sterdyniak, N. Regnault, and G. Möller, Particle entanglement spectra for quantum Hall states on lattices, Phys. Rev. B 86, 165314 (2012).

Outline

Information

Phys. Rev. B 113, 035142– Published 22 January, 2026
  • Received 18 September 2025
  • Revised 11 December 2025
  • Accepted 2 January 2026
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