- 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

Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states, and finite-temperature topological transitions

Reja H. Wilke1,2,3, Henning Schlömer1,2, Simon M. Linsel1,2, Annabelle Bohrdt1,2, and Fabian Grusdt1,2

PDF

Phys. Rev. B 113, 245131 – Published 15 June, 2026

DOI: https://doi.org/10.1103/5zjl-j7ms

Abstract

Since the discovery of phase transitions driven by topological defects, the classification of phases of matter has been significantly extended beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking (SSB). In particular, intrinsic and symmetry-protected topological (SPT) orders have been discovered in (mostly gapped) quantum many-body ground states. However, these are commonly viewed as zero-temperature phenomena, and their robustness in a gapless ground state or against thermal fluctuations remains challenging to tackle. Here we introduce an explicit construction for SPT-type states with hidden order associated with SSB: They feature (quasi-)long-range correlations along appropriate edges but short-range order in the bulk, ground state degeneracy associated with SSB, and nonlocal string order in the bulk. We apply our construction to predict two types of finite-temperature SPT transitions protected by 1-form symmetries in the Ising and BKT class, respectively, where the usual signs of criticality appear despite the absence of a diverging correlation length in the bulk. While the state featuring hidden Ising order is gapped, the other SPT state associated with the BKT-SPT transition has hidden U(1) or XY order and constitutes an intrinsically gapless SPT state, associated with a gapless Goldstone mode. Specifically, in this work we discuss spins with global Z2 or U(1) symmetry coupled to link variables constituting a loop gas model with a 1-form symmetry. By mapping this system to an Ising-gauge theory, we demonstrate that one of the SPT phases we construct corresponds to the Higgs-SPT phase at T=0 – which we show here to remain stable at finite temperature. Our work paves the way for a more systematic search for hidden order SPT phases, including in gapless systems, and raises the question if a natural (finite-T) spin liquid candidate exists that realizes hidden order in the Higgs-SPT class.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. Phys. JETP 34, 610 (1972).
  2. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
  3. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  4. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, UK, 2011).
  5. X. Chen, Z.-C. Gu, and X.-G. Wen, Complete classification of one-dimensional gapped quantum phases in interacting spin systems, Phys. Rev. B 84, 235128 (2011).
  6. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  7. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  8. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  9. T. Scaffidi, D. E. Parker, and R. Vasseur, Gapless symmetry-protected topological order, Phys. Rev. X 7, 041048 (2017).
  10. R. Thorngren, A. Vishwanath, and R. Verresen, Intrinsically gapless topological phases, Phys. Rev. B 104, 075132 (2021).
  11. R. Verresen, R. Thorngren, N. G. Jones, and F. Pollmann, Gapless topological phases and symmetry-enriched quantum criticality, Phys. Rev. X 11, 041059 (2021).
  12. X. Chen, Y.-M. Lu, and A. Vishwanath, Symmetry-protected topological phases from decorated domain walls, Nat. Commun. 5, 3507 (2014).
  13. P. Mai, J. Zhao, B. E. Feldman, and P. W. Phillips, 1/4 is the new 1/2 when topology is intertwined with Mottness, Nat. Commun. 14, 5999 (2023).
  14. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  15. R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, On thermal stability of topological qubit in Kitaev's 4D model, Open Syst. Inf. Dyn. 17, 1 (2010).
  16. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  17. 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).
  18. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  19. F. Grusdt, Topological order of mixed states in correlated quantum many-body systems, Phys. Rev. B 95, 075106 (2017).
  20. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7, 971 (2011).
  21. A. Uhlmann, Parallel transport and “quantum holonomy” along density operators, Rep. Math. Phys. 24, 229 (1986).
  22. A. Rivas, O. Viyuela, and M. A. Martin-Delgado, Density-matrix Chern insulators: Finite-temperature generalization of topological insulators, Phys. Rev. B 88, 155141 (2013).
  23. J. C. Budich and S. Diehl, Topology of density matrices, Phys. Rev. B 91, 165140 (2015).
  24. C.-E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, and S. Diehl, Probing the topology of density matrices, Phys. Rev. X 8, 011035 (2018).
  25. Z.-M. Huang and S. Diehl, Mixed state topological order parameters for symmetry protected fermion matter, Phys. Rev. Res. 7, 033028 (2025).
  26. Z.-M. Huang, S. Diehl, and X.-Q. Sun, Topological response in open quantum systems with weak symmetries, Phys. Rev. Lett. 135, 213002 (2025).
  27. Z.-M. Huang, L. Colmenarez, M. Müller, and S. Diehl, Coherent information as a mixed-state topological order parameter of fermions, Phys. Rev. Res. 7, 043009 (2025).
  28. R. Ma and C. Wang, Average symmetry-protected topological phases, Phys. Rev. X 13, 031016 (2023).
  29. R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: The decohered, the disordered, and the intrinsic, Phys. Rev. X 15, 021062 (2025).
  30. Z. Zhang, U. Agrawal, and S. Vijay, Quantum communication and mixed-state order in decohered symmetry-protected topological states, Phys. Rev. B 111, 115141 (2025).
  31. Z. Huang and D. P. Arovas, Topological indices for open and thermal systems via Uhlmann's phase, Phys. Rev. Lett. 113, 076407 (2014).
  32. P. Molignini and N. R. Cooper, Topological phase transitions at finite temperature, Phys. Rev. Res. 5, 023004 (2023).
  33. Z.-M. Huang and S. Diehl, Interaction-induced topological phase transition at finite temperature, Phys. Rev. Lett. 134, 053002 (2025).
  34. S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, Symmetry-protected topological order at nonzero temperature, Phys. Rev. A 96, 022306 (2017).
  35. T. Kennedy and H. Tasaki, Hidden Z2×Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45, 304 (1992).
  36. D. V. Else, S. D. Bartlett, and A. C. Doherty, Hidden symmetry-breaking picture of symmetry-protected topological order, Phys. Rev. B 88, 085114 (2013).
  37. M. Levin and Z.-C. Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012).
  38. L.-Y. Hung and X.-G. Wen, Quantized topological terms in weakly coupled gauge theories and their connection to symmetry protected topological phases, Phys. Rev. B 87, 165107 (2013).
  39. D. E. Parker, T. Scaffidi, and R. Vasseur, Topological Luttinger liquids from decorated domain walls, Phys. Rev. B 97, 165114 (2018).
  40. R. Verresen, U. Borla, A. Vishwanath, S. Moroz, and R. Thorngren, Higgs condensates are symmetry-protected topological phases: I. discrete symmetries, arXiv:2211.01376.
  41. H. Sukeno, K. Ikeda, and T.-C. Wei, Bulk and boundary entanglement transitions in the projective gauge-Higgs model, Phys. Rev. B 110, 245102 (2024).
  42. T. Rakovszky and V. Khemani, The physics of (good) LDPC codes I. gauging and dualities, arXiv:2310.16032.
  43. H. Schlömer, A. Bohrdt, and F. Grusdt, Geometric fractionalized Fermi liquids: Hidden antiferromagnetism and pseudogap from fluctuating stripes, PRX Quantum 6, 030342 (2025).
  44. Y. Zhang, E. Demler, and S. Sachdev, Competing orders in a magnetic field: Spin and charge order in the cuprate superconductors, Phys. Rev. B 66, 094501 (2002).
  45. D. Podolsky and E. Demler, Properties and detection of spin nematic order in strongly correlated electron systems, New J. Phys. 7, 59 (2005).
  46. E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19, 3682 (1979).
  47. S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension, Phys. Rev. Lett. 98, 070602 (2007).
  48. F. Wu, Y. Deng, and N. Prokof'ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B 85, 195104 (2012).
  49. M. Ogata and H. Shiba, Bethe-ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model, Phys. Rev. B 41, 2326 (1990).
  50. H. V. Kruis, I. P. McCulloch, Z. Nussinov, and J. Zaanen, Geometry and the hidden order of Luttinger liquids: The universality of squeezed space, Phys. Rev. B 70, 075109 (2004).
  51. X.-G. Wen, Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems, Phys. Rev. B 99, 205139 (2019).
  52. W.-T. Xu, T. Rakovszky, M. Knap, and F. Pollmann, Entanglement properties of gauge theories from higher-form symmetries, Phys. Rev. X 15, 011001 (2025).
  53. S. Sachdev, Topological order, emergent gauge fields, and Fermi surface reconstruction, Rep. Prog. Phys. 82, 014001 (2019).
  54. H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett. 86, 910 (2001).
  55. U. Borla, R. Verresen, J. Shah, and S. Moroz, Gauging the Kitaev chain, SciPost Phys. 10, 148 (2021).
  56. M. Kebrič, U. Schollwöck, and F. Grusdt, Mean-field theory of 1+1D Z2 lattice gauge theory with matter, SciPost Phys. 20, 017 (2024).
  57. J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Quantum simulation of antiferromagnetic spin chains in an optical lattice, Nature (London) 472, 307 (2011).
  58. T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll, E. Demler, I. Bloch, and C. Gross, Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators, Science 357, 484 (2017).
  59. P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015).
  60. A. J. Ferris and G. Vidal, Perfect sampling with unitary tensor networks, Phys. Rev. B 85, 165146 (2012).
  61. M. Buser, U. Schollwöck, and F. Grusdt, Snapshot-based characterization of particle currents and the Hall response in synthetic flux lattices, Phys. Rev. A 105, 033303 (2022).
  62. T. Rakovszky, P. Sala, R. Verresen, M. Knap, and F. Pollmann, Statistical localization: From strong fragmentation to strong edge modes, Phys. Rev. B 101, 125126 (2020).
  63. M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauss, C. Gross, L. Mazza, M. C. Banuls, L. Pollet, I. Bloch, and S. Kuhr, Observation of correlated particle-hole pairs and string order in low-dimensional Mott insulators, Science 334, 200 (2011).
  64. S. M. Linsel, A. Bohrdt, L. Homeier, L. Pollet, and F. Grusdt, Percolation as a confinement order parameter in Z2 lattice gauge theories, Phys. Rev. B 110, L241101 (2024).
  65. G. Dünnweber, S. M. Linsel, A. Bohrdt, and F. Grusdt, Percolation renormalization group analysis of confinement in Z2 lattice gauge theories, Phys. Rev. B 111, 024314 (2025).
  66. S. M. Linsel, L. Pollet, and F. Grusdt, Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices, PRX Quantum 7, 010332 (2026).
  67. R. Peierls, On Ising's model of ferromagnetism, Math. Proc. Camb. Philos. Soc. 32, 477 (1936).
  68. F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259 (1971).
  69. D. Tiwari, S. Bollmann, S. Paeckel, and E. J. König, Quantum restored symmetry protected topological phases, Commun. Phys. 8, 421 (2024).
  70. L. Homeier, A. Bohrdt, S. Linsel, E. Demler, J. C. Halimeh, and F. Grusdt, Realistic scheme for quantum simulation of Z2 lattice gauge theories with dynamical matter in (2 + 1)D, Commun. Phys. 6, 127 (2023).
  71. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  72. D. S. Rokhsar and S. A. Kivelson, Superconductivity and the quantum hard-core dimer gas, Phys. Rev. Lett. 61, 2376 (1988).
  73. R. Verresen, M. D. Lukin, and A. Vishwanath, Prediction of toric code topological order from Rydberg blockade, Phys. Rev. X 11, 031005 (2021).
  74. R. Samajdar, W. W. Ho, H. Pichler, M. D. Lukin, and S. Sachdev, Quantum phases of Rydberg atoms on a kagome lattice, Proc. Natl. Acad. Sci. USA 118, e2015785118 (2021).
  75. G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science 374, 1242 (2021).
  76. Z. Wang and L. Pollet, Renormalized classical spin liquid on the ruby lattice, Phys. Rev. Lett. 134, 086601 (2025).
  77. G. D. Paciani, G. Dünnweber, H. Schlömer, S. Linsel, and F. Grusdt (unpublished).
  78. R. Wilke, H. Schlömer, A. Bohrdt, and F. Grusdt (unpublished).
  79. T. A. Cochran, B. Jobst, E. Rosenberg, Y. D. Lensky, G. Gyawali, N. Eassa, M. Will, D. Abanin, R. Acharya, L. A. Beni, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, B. Ballard, J. C. Bardin, and A. Bengtsson, Visualizing dynamics of charges and strings in (2 + 1)D lattice gauge theories, Nature (London) 642, 315 (2025).
  80. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017).
  81. J. C. Halimeh, L. Homeier, C. Schweizer, M. Aidelsburger, P. Hauke, and F. Grusdt, Stabilizing lattice gauge theories through simplified local pseudogenerators, Phys. Rev. Res. 4, 033120 (2022).

Outline

Information

Phys. Rev. B 113, 245131– Published 15 June, 2026
  • Received 8 June 2025
  • Revised 12 December 2025
  • Accepted 15 May 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