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Generalized Li-Haldane correspondence in critical free-fermion systems

Yuxuan Guo1,*, Sheng Yang2,*, and Xue-Jia Yu3,4,5,†

  • *These authors contributed equally to this work.
  • †Contact author: xuejiayu@eitech.edu.cn
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Phys. Rev. Research 8, 023203 – Published 26 May, 2026

DOI: https://doi.org/10.1103/96gp-fq3j

Abstract

Topological phenomena in quantum critical systems have recently attracted growing attention, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, a general framework for identifying such nontrivial phenomena, particularly in higher-dimensional systems, remains insufficiently explored. In this work, we propose a universal fingerprint for detecting nontrivial topology in critical free-fermion systems protected by global on-site symmetries. Specifically, we analytically establish an exact relation between the bulk entanglement spectrum and the boundary energy spectrum at topological criticality in arbitrary dimensions, demonstrating that the degeneracy of edge modes can be extracted from the bulk entanglement spectrum. These findings, further supported by numerical simulations of lattice models, provide a universal fingerprint for identifying nontrivial topology in critical free-fermion systems.

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References (117)

  1. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  2. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  3. T. Senthil, Symmetry-protected topological phases of quantum matter, Annu. Rev. Condens. Matter Phys. 6, 299 (2015).
  4. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  5. Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B 80, 155131 (2009).
  6. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  7. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  8. A. Stern and N. H. Lindner, Topological quantum computation—From basic concepts to first experiments, Science 339, 1179 (2013).
  9. S.-Q. Shen, Topological insulators (Springer, Berlin, 2012), Vol. 174.
  10. B. A. Bernevig, Topological insulators and topological superconductors (Princeton University Press, Princeton, NJ, 2013).
  11. X.-J. Yu, L. Xu, and H.-Q. Lin, Topological physics in quantum critical systems, Phys. Rep. 1160, 1 (2026).
  12. A. Keselman and E. Berg, Gapless symmetry-protected topological phase of fermions in one dimension, Phys. Rev. B 91, 235309 (2015).
  13. T. Scaffidi, D. E. Parker, and R. Vasseur, Gapless symmetry-protected topological order, Phys. Rev. X 7, 041048 (2017).
  14. R. Verresen, N. G. Jones, and F. Pollmann, Topology and edge modes in quantum critical chains, Phys. Rev. Lett. 120, 057001 (2018).
  15. D. E. Parker, T. Scaffidi, and R. Vasseur, Topological Luttinger liquids from decorated domain walls, Phys. Rev. B 97, 165114 (2018).
  16. R. Verresen, Topology and edge states survive quantum criticality between topological insulators, arXiv:2003.05453.
  17. R. Verresen, R. Thorngren, N. G. Jones, and F. Pollmann, Gapless topological phases and symmetry-enriched quantum criticality, Phys. Rev. X 11, 041059 (2021).
  18. R. Thorngren, A. Vishwanath, and R. Verresen, Intrinsically gapless topological phases, Phys. Rev. B 104, 075132 (2021).
  19. C. M. Duque, H.-Y. Hu, Y.-Z. You, V. Khemani, R. Verresen, and R. Vasseur, Topological and symmetry-enriched random quantum critical points, Phys. Rev. B 103, L100207 (2021).
  20. U. Borla, R. Verresen, J. Shah, and S. Moroz, Gauging the Kitaev chain, SciPost Phys. 10, 148 (2021).
  21. X.-J. Yu, R.-Z. Huang, H.-H. Song, L. Xu, C. Ding, and L. Zhang, Conformal boundary conditions of symmetry-enriched quantum critical spin chains, Phys. Rev. Lett. 129, 210601 (2022).
  22. W. Ye, M. Guo, Y.-C. He, C. Wang, and L. Zou, Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality, SciPost Phys. 13, 066 (2022).
  23. Y. Hidaka, S. C. Furuya, A. Ueda, and Y. Tada, Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip, Phys. Rev. B 106, 144436 (2022).
  24. R. Verresen, U. Borla, A. Vishwanath, S. Moroz, and R. Thorngren, Higgs condensates are symmetry-protected topological phases: I. Discrete symmetries, arXiv:2211.01376.
  25. S. Mondal, A. Agarwala, T. Mishra, and A. Prakash, Symmetry-enriched criticality in a coupled spin ladder, Phys. Rev. B 108, 245135 (2023).
  26. R. Wen and A. C. Potter, Bulk-boundary correspondence for intrinsically gapless symmetry-protected topological phases from group cohomology, Phys. Rev. B 107, 245127 (2023).
  27. N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Building models of topological quantum criticality from pivot Hamiltonians, SciPost Phys. 14, 013 (2023).
  28. N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Pivot Hamiltonians as generators of symmetry and entanglement, SciPost Phys. 14, 012 (2023).
  29. R. Thorngren, T. Rakovszky, R. Verresen, and A. Vishwanath, Higgs condensates are symmetry-protected topological phases: II. U(1) gauge theory and superconductors, arXiv:2303.08136.
  30. L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases, Phys. Rev. B 108, 214429 (2023).
  31. X.-J. Yu, S. Yang, H.-Q. Lin, and S.-K. Jian, Universal entanglement spectrum in one-dimensional gapless symmetry protected topological states, Phys. Rev. Lett. 133, 026601 (2024).
  32. L. Li, R.-Z. Huang, and W. Cao, Noninvertible symmetry-enriched quantum critical point, Phys. Rev. B 112, L081113 (2025).
  33. L. Su and M. Zeng, Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup, Phys. Rev. B 109, 245108 (2024).
  34. T. Ando, Gauging on the lattice and gapped/gapless topological phases, arXiv:2402.03566.
  35. S. Prembabu, R. Thorngren, and R. Verresen, Boundary-deconfined quantum criticality at transitions between symmetry-protected topological chains, Phys. Rev. B 109, L201112 (2024).
  36. L. Li, M. Oshikawa, and Y. Zheng, Decorated defect construction of gapless-SPT states, SciPost Phys. 17, 013 (2024).
  37. R. Wen, W. Ye, and A. C. Potter, Topological holography for fermions, arXiv:2404.19004.
  38. S.-J. Huang, Fermionic quantum criticality through the lens of topological holography, Phys. Rev. B 111, 155130 (2025).
  39. R. Wen, String condensation and topological holography for 2+1D gapless SPT, arXiv:2408.05801.
  40. H.-L. Zhang, H.-Z. Li, S. Yang, and X.-J. Yu, Quantum phase transition and critical behavior between the gapless topological phases, Phys. Rev. A 109, 062226 (2024).
  41. W.-H. Zhong, W.-L. Li, Y.-C. Chen, and X.-J. Yu, Topological edge modes and phase transitions in a critical fermionic chain with long-range interactions, Phys. Rev. A 110, 022212 (2024).
  42. X.-J. Yu and W.-L. Li, Fidelity susceptibility at the Lifshitz transition between the noninteracting topologically distinct quantum critical points, Phys. Rev. B 110, 045119 (2024).
  43. X.-J. Yu, S. Yang, S. Liu, H.-Q. Lin, and S.-K. Jian, Gapless symmetry-protected topological states in measurement-only circuits, Phys. Rev. B 113, 134302 (2026).
  44. R. Wen and A. C. Potter, Classification of 1+1D gapless symmetry protected phases via topological holography, Phys. Rev. B 111, 115161 (2025).
  45. W.-H. Zhong, H.-Q. Lin, and X.-J. Yu, Quantum entanglement of fermionic symmetry-enriched quantum critical points in one dimension, Phys. Rev. B 112, 075129 (2025).
  46. R. Wen, Topological holography for 2+1-D gapped and gapless phases with generalized symmetries, arXiv:2503.13685.
  47. L. Bhardwaj, Y. Gai, S.-J. Huang, K. Inamura, S. Schafer-Nameki, A. Tiwari, and A. Warman, Gapless phases in (2+1)d with non-invertible symmetries, arXiv:2503.12699.
  48. L. Li, M. Oshikawa, and Y. Zheng, Intrinsically/purely gapless-SPT from non-invertible duality transformations, SciPost Phys. 18, 153 (2025).
  49. S.-J. Huang and M. Cheng, Topological holography, quantum criticality, and boundary states, SciPost Phys. 18, 213 (2025).
  50. S. Yang, F. Xu, D.-C. Lu, Y.-Z. You, H.-Q. Lin, and X.-J. Yu, Deconfined criticality as intrinsically gapless topological state in one dimension, Phys. Rev. B 113, L201105 (2026).
  51. Z. Tan et al., Exploring nontrivial topology at quantum criticality on a superconducting processor, Commun. Phys. 9, 136 (2026).
  52. L. Zhou, J. Gong, and X.-J. Yu, Topological edge states at Floquet quantum criticality, Commun. Phys. 8, 214 (2025).
  53. G. Cardoso, H.-C. Yeh, L. Korneev, A. G. Abanov, and A. Mitra, Gapless Floquet topology, Phys. Rev. B 111, 125162 (2025).
  54. R. Flores-Calderón, E. J. König, and A. M. Cook, Topological quantum criticality from multiplicative topological phases, Phys. Rev. Lett. 134, 116602 (2025).
  55. S. Yang, H.-Q. Lin, and X.-J. Yu, Gapless topological behaviors in a long-range quantum spin chain, Commun. Phys. 8, 27 (2025).
  56. K.-H. Chou, X.-J. Yu, and P.-Y. Chang, PT symmetry-enriched non-unitary criticality, arXiv:2509.09587.
  57. S. Prembabu and R. Verresen, Multicriticality between purely gapless SPT phases with unitary symmetry, arXiv:2509.20431.
  58. M. Deng, S. Yang, C. Sun, F. Li, and X.-J. Yu, Anomalous dynamical scaling at topological quantum criticality, arXiv:2512.15537.
  59. S. Yang, H.-Q. Lin, and X.-J. Yu, Topological quantum criticality in quasiperiodic Ising chain, arXiv:2602.01223.
  60. Y.-F. Zhao, R. Zhang, L.-J. Zhou, R. Mei, Z.-J. Yan, M. H. W. Chan, C.-X. Liu, and C.-Z. Chang, Zero magnetic field plateau phase transition in higher Chern number quantum anomalous Hall insulators, Phys. Rev. Lett. 128, 216801 (2022).
  61. H. Deng et al., High-temperature quantum anomalous Hall regime in a MnBi2Te4/Bi2Te3 superlattice, Nat. Phys. 17, 36 (2021).
  62. H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L. Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. H. MacDonald, and A. F. Young, Electrical switching of magnetic order in an orbital Chern insulator, Nature (London) 588, 66 (2020).
  63. Y. Saito, J. Ge, L. Rademaker, K. Watanabe, T. Taniguchi, D. A. Abanin, and A. F. Young, Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene, Nat. Phys. 17, 478 (2021).
  64. 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).
  65. A. M. Läuchli, E. J. Bergholtz, J. Suorsa, and M. Haque, Disentangling entanglement spectra of fractional quantum Hall states on torus geometries, Phys. Rev. Lett. 104, 156404 (2010).
  66. N. Regnault, B. A. Bernevig, and F. D. M. Haldane, Topological entanglement and clustering of Jain hierarchy states, Phys. Rev. Lett. 103, 016801 (2009).
  67. R. Thomale, A. Sterdyniak, N. Regnault, and B. A. Bernevig, Entanglement gap and a new principle of adiabatic continuity, Phys. Rev. Lett. 104, 180502 (2010).
  68. L. Fidkowski, Entanglement spectrum of topological insulators and superconductors, Phys. Rev. Lett. 104, 130502 (2010).
  69. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  70. A. M. Turner, Y. Zhang, and A. Vishwanath, Entanglement and inversion symmetry in topological insulators, Phys. Rev. B 82, 241102 (2010).
  71. A. M. Turner, F. Pollmann, and E. Berg, Topological phases of one-dimensional fermions: An entanglement point of view, Phys. Rev. B 83, 075102 (2011).
  72. 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).
  73. G. Y. Cho, K. Shiozaki, S. Ryu, and A. W. W. Ludwig, Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum, J. Phys. A: Math. Theor. 50, 304002 (2017).
  74. A. Chandran, M. Hermanns, N. Regnault, and B. A. Bernevig, Bulk-edge correspondence in entanglement spectra, Phys. Rev. B 84, 205136 (2011).
  75. A. Chandran, V. Khemani, and S. L. Sondhi, How universal is the entanglement spectrum? Phys. Rev. Lett. 113, 060501 (2014).
  76. T. H. Hsieh and L. Fu, Bulk entanglement spectrum reveals quantum criticality within a topological state, Phys. Rev. Lett. 113, 106801 (2014).
  77. M. A. Metlitski and T. Grover, Entanglement entropy of systems with spontaneously broken continuous symmetry, arXiv:1112.5166.
  78. A. B. Kallin, M. B. Hastings, R. G. Melko, and R. R. P. Singh, Anomalies in the entanglement properties of the square-lattice Heisenberg model, Phys. Rev. B 84, 165134 (2011).
  79. Z. Wang, Z. Wang, Y.-M. Ding, B.-B. Mao, and Z. Yan, Bipartite reweight-annealing algorithm of quantum Monte Carlo to extract large-scale data of entanglement entropy and its derivative, Nat. Commun. 16, 5880 (2025).
  80. Z. Wang, S. Yang, B.-B. Mao, M. Cheng, and Z. Yan, Sudden change in the entanglement Hamiltonian: Phase diagram of an Ising entanglement Hamiltonian, Phys. Rev. B 111, 245126 (2025).
  81. Z. Liu, Z. Wang, D.-X. Yao, and Z. Yan, Worldline deconfinement and emergent long-range interaction in entanglement Hamiltonian and entanglement spectrum, Phys. Rev. B 113, 144425 (2026).
  82. B.-B. Mao, Y.-M. Ding, Z. Wang, S. Hu, and Z. Yan, Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamiltonian, Nat. Commun. 16, 2880 (2025).
  83. Y. Zhu, Z. Liu, Z. Wang, Y.-C. Wang, and Z. Yan, Bipartite entanglement and surface criticality, Phys. Rev. Lett. 136, 046501 (2026).
  84. C. Li, X. Li, and Y. N. Zhou, Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders, Quantum Front. 3, 9 (2024).
  85. M. Dalmonte, V. Eisler, M. Falconi, and B. Vermersch, Entanglement Hamiltonians: From field theory to lattice models and experiments, Ann. Phys. (Berlin) 534, 2200064 (2022).
  86. K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech. (2015) P04010.
  87. Y. Kusuki, H. Ooguri, and S. Pal, Universality of Rényi entropy in conformal field theory, Phys. Rev. Lett. 135, 061603 (2025).
  88. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  89. J. Cardy and E. Tonni, Entanglement Hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. (2016) 123103.
  90. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  91. R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017).
  92. M.-C. Chung and I. Peschel, Density-matrix spectra of solvable fermionic systems, Phys. Rev. B 64, 064412 (2001).
  93. S.-A. Cheong and C. L. Henley, Many-body density matrices for free fermions, Phys. Rev. B 69, 075111 (2004).
  94. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A: Math. Theor. 42, 504003 (2009).
  95. We use either periodic or antiperiodic boundary conditions in our actual calculations of the bulk entanglement spectrum to avoid possible numerical issues. For example, (Ĥ11d+Ĥ21d)/2 has two exact zero-energy levels under periodic boundary conditions, which can cause ambiguity in the definition of the ground state by filling half of the lower-lying levels. To this end, we adopt the antiperiodic boundary condition to extract bulk entanglement spectra for 1D models.
  96. Z.-K. Lin, Y. Zhou, B. Jiang, B.-Q. Wu, L.-M. Chen, X.-Y. Liu, L.-W. Wang, P. Ye, and J.-H. Jiang, Measuring entanglement entropy and its topological signature for phononic systems, Nat. Commun. 15, 1601 (2024).
  97. T. V. Zache, C. Kokail, B. Sundar, and P. Zoller, Entanglement spectroscopy and probing the Li-Haldane conjecture in topological quantum matter, Quantum 6, 702 (2022).
  98. C. Kokail, B. Sundar, T. V. Zache, A. Elben, B. Vermersch, M. Dalmonte, R. van Bijnen, and P. Zoller, Quantum variational learning of the entanglement Hamiltonian, Phys. Rev. Lett. 127, 170501 (2021).
  99. M. K. Joshi, C. Kokail, R. van Bijnen, F. Kranzl, T. V. Zache, R. Blatt, C. F. Roos, and P. Zoller, Exploring large-scale entanglement in quantum simulation, Nature (London) 624, 539 (2023).
  100. C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, and P. Zoller, Entanglement Hamiltonian tomography in quantum simulation, Nat. Phys. 17, 936 (2021).
  101. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  102. B. Yan and C. Felser, Topological materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
  103. J. Pixley and J. H. Wilson, Rare regions and avoided quantum criticality in disordered Weyl semimetals and superconductors, Ann. Phys. 435, 168455 (2021).
  104. G. Refael and J. E. Moore, Entanglement entropy of random quantum critical points in one dimension, Phys. Rev. Lett. 93, 260602 (2004).
  105. D. S. Fisher, Random antiferromagnetic quantum spin chains, Phys. Rev. B 50, 3799 (1994).
  106. D. S. Fisher, Critical behavior of random transverse-field Ising spin chains, Phys. Rev. B 51, 6411 (1995).
  107. L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
  108. L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83, 075103 (2011).
  109. J. Wang and Y.-Z. You, Symmetric mass generation, Symmetry 14, 1475 (2022).
  110.  A Su–Schrieffer–Heeger (SSH) model can be constructed from α=2 chain model ĤSSH=i∑γ̂nγ̃̂n+2 and identifying ĉA,n=γ̂2n+iγ̂2n−1 and ĉB,n=γ̃̂2n−1−iγ̃̂2n and the original time reversal symmetry T̂γ̂T̂=γ̂ and T̂γ̃̂T̂=−γ̃̂ acting like a sublattice symmetry on complex fermions, where T̂ĉAT̂=ĉA† and T̂ĉBT̂=−ĉB†.
  111. E. Tang and X.-G. Wen, Interacting one-dimensional fermionic symmetry-protected topological phases, Phys. Rev. Lett. 109, 096403 (2012).
  112. Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev. B 90, 115141 (2014).
  113. As shown in the inset of Fig. 7, the noninteracting model is analogous to the well-known Su-Schrieffer-Heeger model after suitable numbering of the sites. The dual transformation is just a single-site translation of the system along the red solid line, under which Ĥ11d↔Ĥ21d and the chiral-symmetry-preserving interaction term remains unchanged. Here, we do not consider the interaction (ĉA,i†ĉB,i+1+ĉB,i+1†ĉA,i)(n̂A,i+n̂B,i+1), which is equivalent to (ĉA,i†ĉB,i+2+ĉB,i+2†ĉA,i)(n̂A,i+n̂B,i+2) after dual transformation, since it does not touch degrees of freedom of the dangling fermions at the edges.
  114. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  115. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  116. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  117. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).

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Phys. Rev. Research 8, 023203– Published 26 May, 2026
  • Received 28 September 2025
  • Revised 5 December 2025
  • Accepted 6 May 2026
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