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Essay: Generalized Landau Paradigm for Quantum Phases and Phase Transitions

Xie Chen

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Phys. Rev. Lett. 135, 250001 – Published 17 December, 2025

DOI: https://doi.org/10.1103/tmvy-vsqd

Abstract

The Landau paradigm is a central dogma for understanding phase and phase transitions in condensed matter systems, yet for decades, it has been known that a variety of quantum phases exist beyond the framework. Is there a more general framework that provides a systematic understanding of phases and phase transitions in quantum many-body systems? In this Essay, we discuss how the recently developed notions of generalized symmetry and generalized gauging point to a way to extend the Landau paradigm. In the new framework, beyond-Landau phases and transitions are attributed to the breaking of generalized symmetries, often induced by the generalized gauging procedure facilitated through the symmetry topological field theory formalism. We discuss what needs to be understood to make the generalized Landau paradigm useful in the study of quantum phase and phase transitions.

Part of a series of Essays which concisely present author visions for the future of their field.

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PRL Forward-Looking Essays

To welcome work from areas across the physical sciences and inspire the new generation of physicists, this series of forward-looking Essays is designed to envision future directions for a given topic .

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References (107)

  1. L. D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz. 7, 19 (1937), http://archive.ujp.bitp.kiev.ua/files/journals/53/si/53SI08p.pdf.
  2. K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).
  3. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  4. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  5. D. M. Hofman and N. Iqbal, Goldstone modes and photonization for higher form symmetries, SciPost Phys. 6, 006 (2019).
  6. L. V. Delacrétaz, D. M. Hofman, and G. Mathys, Superfluids as higher-form anomalies, SciPost Phys. 8, 047 (2020).
  7. N. Iqbal and J. McGreevy, Toward a 3d Ising model with a weakly-coupled string theory dual, SciPost Phys. 9, 019 (2020).
  8. H. Moradi, S. F. Moosavian, and A. Tiwari, Topological holography: Towards a unification of Landau and beyond-Landau physics, SciPost Phys. Core 6, 066 (2023).
  9. L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, Categorical Landau paradigm for gapped phases, Phys. Rev. Lett. 133, 161601 (2024).
  10. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  11. P. Curie, Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique, J. Phys. Theor. Appl. 3, 393 (1894).
  12. D. ter Haar, On the theory of superconductivity, in Collected Papers of L. D. Landau (Pergamon, New York, 1965), pp. 546–568.
  13. X. G. Wen, Topological orders in rigid states, Int. J. Mod. Phys. B 04, 239 (1990).
  14. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).
  15. Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B 80, 155131 (2009).
  16. F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
  17. E. Ardonne, P. Fendley, and E. Fradkin, Topological order and conformal quantum critical points, Ann. Phys. (Amsterdam) 310, 493 (2004).
  18. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined quantum critical points, Science 303, 1490 (2004).
  19. Y.-M. Lu and D.-H. Lee, Quantum phase transitions between bosonic symmetry-protected topological phases in two dimensions: Emergent QED3 and anyon superfluid, Phys. Rev. B 89, 195143 (2014).
  20. S.-H. Shao, What’s done cannot be undone: TASI lectures on non-invertible symmetries, arXiv:2308.00747.
  21. S. Schäfer-Nameki, iCTP lectures on (non-)invertible generalized symmetries, Phys. Rep. 1063, 1 (2024).
  22. J. Fuchs, I. Runkel, and C. Schweigert, TFT construction of RCFT correlators I: Partition functions, Nucl. Phys. B 646, 353 (2002).
  23. N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7, 203 (2016).
  24. L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, J. High Energy Phys. 03 (2018) 189.
  25. L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Symmetric Hamiltonians and matrix product operator intertwiners, PRX Quantum 4, 020357 (2023).
  26. L. Lootens, C. Delcamp, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Topological sectors, PRX Quantum 5, 010338 (2024).
  27. J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Gauging quantum states: From global to local symmetries in many-body systems, Phys. Rev. X 5, 011024 (2015).
  28. L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry, Phys. Rev. Res. 2, 043086 (2020).
  29. D. Gaiotto and J. Kulp, Orbifold groupoids, J. High Energy Phys. 02 (2021) 132.
  30. A. Chatterjee and X.-G. Wen, Holographic theory for continuous phase transitions: Emergence and symmetry protection of gaplessness, Phys. Rev. B 108, 075105 (2023).
  31. S.-J. Huang and M. Cheng, Topological holography, quantum criticality, and boundary states, SciPost Phys. 18, 213 (2025).
  32. L. Kong, Z.-H. Zhang, J. Zhao, and H. Zheng, Higher condensation theory, arXiv:2403.07813.
  33. L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, Gapped phases with non-invertible symmetries: (1+1)D, SciPost Phys. 18, 032 (2025).
  34. S. Seifnashri, S.-H. Shao, and X. Yang, Gauging non-invertible symmetries on the lattice, SciPost Phys. 19, 063 (2025).
  35. L. Kong, X.-G. Wen, and H. Zheng, Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, arXiv:1502.01690.
  36. W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2, 033417 (2020).
  37. L. Bhardwaj, Y. Lee, and Y. Tachikawa, SL(2,Z) action on QFTS with Z2 symmetry and the Brown-Kervaire invariants, J. High Energy Phys. 11 (2020) 141.
  38. J. Pulmann, P. Ševera, and F. Valach, A nonabelian duality for (higher) gauge theories, Adv. Theor. Math. Phys. 25, 241 (2021).
  39. T. Lichtman, R. Thorngren, N. H. Lindner, A. Stern, and E. Berg, Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states, Phys. Rev. B 104, 075141 (2021).
  40. L. Kong and H. Zheng, A mathematical theory of gapless edges of 2d topological orders. Part I, J. High Energy Phys. 02 (2020) 150.
  41. L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, Classification of topological phases with finite internal symmetries in all dimensions, J. High Energy Phys. 09 (2020) 093.
  42. F. Apruzzi, F. Bonetti, I. García Etxebarria, S. S. Hosseini, and S. Schäfer-Nameki, Symmetry TFTs from string theory, Commun. Math. Phys. 402, 895 (2023).
  43. A. Chatterjee and X.-G. Wen, Symmetry as a shadow of topological order and a derivation of topological holographic principle, Phys. Rev. B 107, 155136 (2023).
  44. D. S. Freed, G. W. Moore, and C. Teleman, Topological symmetry in quantum field theory, arXiv:2209.07471.
  45. Y.-H. Lin, M. Okada, S. Seifnashri, and Y. Tachikawa, Asymptotic density of states in 2d CFTs with non-invertible symmetries, J. High Energy Phys. 03 (2023) 094.
  46. L. Kong and H. Zheng, Gapless edges of 2d topological orders and enriched monoidal categories, Nucl. Phys. B927, 140 (2018).
  47. L. Kong and H. Zheng, A mathematical theory of gapless edges of 2d topological orders. Part II, Nucl. Phys. B966, 115384 (2021).
  48. L. Kong, X.-G. Wen, and H. Zheng, One dimensional gapped quantum phases and enriched fusion categories, J. High Energy Phys. 03 (2022) 022.
  49. L. Kong and H. Zheng, Categories of quantum liquids I, J. High Energy Phys. 08 (2022) 070.
  50. L. Kong and H. Zheng, Categories of quantum liquids II, Commun. Math. Phys. 405, 203 (2024).
  51. R. Xu and Z.-H. Zhang, Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries, Phys. Rev. B 110, 155106 (2024).
  52. L. Lootens, C. Delcamp, and F. Verstraete, Entanglement and the density matrix renormalisation group in the generalised Landau paradigm, Nat. Phys. 21, 1657 (2025).
  53. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).

  54. The Jordan-Wigner transformation differs from gauging by a self-duality of the Ising chain that exchanges the phases on either side of the transition, without changing the nature of the transition.

  55. P. Jordan and E. Wigner, Über das paulische äquivalenzverbot, Z. Phys. 47, 631 (1928).

  56. On a periodic chain, the Ising Hamiltonian differs from the transformed Majorana Hamiltonian by a boundary term. We ignore this subtlety here.

  57. P. Ginsparg, Applied conformal field theory, arXiv:hep-th/9108028.
  58. S. B. Bravyi and A. Y. Kitaev, Quantum codes on a lattice with boundary, arXiv:quant-ph/9811052.
  59. J. Lou, C. Shen, C. Chen, and L.-Y. Hung, A (Dummy’s) guide to working with gapped boundaries via (fermion) condensation, J. High Energy Phys. 02 (2021) 171.
  60. R. Wen, W. Ye, and A. C. Potter, Topological holography for fermions, arXiv:2404.19004.
  61. S.-J. Huang, Fermionic quantum criticality through the lens of topological holography, Phys. Rev. B 111, 155130 (2025).
  62. L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Lattice models for phases and transitions with non-invertible symmetries, arXiv:2405.05964.
  63. L. Bhardwaj, D. Pajer, S. Schafer-Nameki, A. Tiwari, A. Warman, and J. Wu, Gapped phases in (2+1)d with non-invertible symmetries: Part I, SciPost Phys. 19, 056 (2025).
  64. L. Bhardwaj, S. Schafer-Nameki, A. Tiwari, and A. Warman, Gapped phases in (2+1)d with non-invertible symmetries: Part II, arXiv:2502.20440.
  65. R. Wen, Topological holography for 2+1-d gapped and gapless phases with generalized symmetries, arXiv:2503.13685.
  66. K. Inamura, S.-J. Huang, A. Tiwari, and S. Schafer-Nameki, (2+1)d lattice models and tensor networks for gapped phases with categorical symmetry, arXiv:2506.09177.
  67. A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98, 160409 (2007).
  68. T. Johnson-Freyd, On the classification of topological orders, Commun. Math. Phys. 393, 989 (2022).
  69. T. Lan, L. Kong, and X.-G. Wen, Classification of (3+1)D bosonic topological orders: The case when pointlike excitations are all bosons, Phys. Rev. X 8, 021074 (2018).
  70. T. Lan and X.-G. Wen, Classification of 3+1D bosonic topological orders (II): The case when some pointlike excitations are fermions, Phys. Rev. X 9, 021005 (2019).
  71. F. Apruzzi, F. Bedogna, and N. Dondi, SymTh for non-finite symmetries, arXiv:2402.14813.
  72. A. Antinucci and F. Benini, Anomalies and gauging of u(1) symmetries, Phys. Rev. B 111, 024110 (2025).
  73. T. D. Brennan and Z. Sun, A SymTFT for continuous symmetries, J. High Energy Phys. 12 (2024) 100.
  74. F. Bonetti, M. D. Zotto, and R. Minasian, SymTFTs for continuous non-Abelian symmetries, Phys. Lett. B 871, 140010 (2025).
  75. S. D. Pace, Ö. M. Aksoy, and H. T. Lam, Spacetime symmetry-enriched SymTFT: From LSM anomalies to modulated symmetries and beyond, arXiv:2507.02036.
  76. T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains, Commun. Math. Phys. 147, 431 (1992).
  77. T. Kennedy and H. Tasaki, Hidden z2×z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45, 304 (1992).
  78. C. Zhang and M. Levin, Exactly solvable model for a deconfined quantum critical point in 1d, Phys. Rev. Lett. 130, 026801 (2023).
  79. C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological defect lines and renormalization group flows in two dimensions, J. High Energy Phys. 01 (2019) 026.
  80. R. Thorngren and Y. Wang, Fusion category symmetry I: Anomaly in-flow and gapped phases, J. High Energy Phys. 04 (2024) 132.
  81. Y. Choi, C. Córdova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, Noninvertible duality defects in 3+1 dimensions, Phys. Rev. D 105, 125016 (2022).
  82. Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao, Remarks on boundaries, anomalies, and noninvertible symmetries, Phys. Rev. D 108, 125005 (2023).
  83. J. Kaidi, E. Nardoni, G. Zafrir, and Y. Zheng, Symmetry TFTs and anomalies of non-invertible symmetries, J. High Energy Phys. 10 (2023) 053.
  84. C. Córdova, P.-S. Hsin, and C. Zhang, Anomalies of non-invertible symmetries in (3+1)D, SciPost Phys. 17, 131 (2024).
  85. C. Copetti, L. Córdova, and S. Komatsu, Noninvertible symmetries, anomalies, and scattering amplitudes, Phys. Rev. Lett. 133, 181601 (2024).
  86. I. G. Etxebarria and N. Iqbal, A Goldstone theorem for continuous non-invertible symmetries, J. High Energy Phys. 09 (2023) 145.
  87. J. A. Damia, R. Argurio, and S. Chaudhuri, When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries, J. High Energy Phys. 03 (2024) 042.
  88. L. Tsui and X.-G. Wen, Lattice models that realize zn−1 symmetry-protected topological states for even n, Phys. Rev. B 101, 035101 (2020).
  89. S. Seifnashri and S.-H. Shao, Cluster state as a noninvertible symmetry-protected topological phase, Phys. Rev. Lett. 133, 116601 (2024).
  90. C. Fechisin, N. Tantivasadakarn, and V. V. Albert, Noninvertible symmetry-protected topological order in a group-based cluster state, Phys. Rev. X 15, 011058 (2025).
  91. W. Cao, M. Yamazaki, and L. Li, Duality viewpoint of noninvertible symmetry protected topological phases, arXiv:2502.20435.
  92. K. G. Wilson, Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971).
  93. K. G. Wilson, Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior, Phys. Rev. B 4, 3184 (1971).
  94. N. Iqbal and J. McGreevy, Mean string field theory: Landau-Ginzburg theory for 1-form symmetries, SciPost Phys. 13, 114 (2022).
  95. W. Ji and X. Chen, Topological defects of 2+1d systems from line excitations in 3+1d bulk, arXiv:2407.02488.
  96. J. Zhao, Z. Yan, M. Cheng, and Z. Y. Meng, Higher-form symmetry breaking at Ising transitions, Phys. Rev. Res. 3, 033024 (2021).
  97. Y.-T. Oh, S. D. Pace, J. H. Han, Y. You, and H.-Y. Lee, Aspects of Zn rank-2 gauge theory in (2+1) dimensions: Construction schemes, holonomies, and sublattice one-form symmetries, Phys. Rev. B 107, 155151 (2023).
  98. Y. Choi, Y. Sanghavi, S.-H. Shao, and Y. Zheng, Non-invertible and higher-form symmetries in 2+1D lattice gauge theories, SciPost Phys. 18, 008 (2025).
  99. Z. Nussinov and G. Ortiz, Sufficient symmetry conditions for topological quantum order, Proc. Natl. Acad. Sci. U.S.A. 106, 16944 (2009).
  100. Z. Nussinov and G. Ortiz, A symmetry principle for topological quantum order, Ann. Phys. (Amsterdam) 324, 977 (2009).
  101. A. Kovner and B. Rosenstein, New look at QED4: The photon as a Goldstone boson and the topological interpretation of electric charge, Phys. Rev. D 49, 5571 (1994).
  102. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  103. A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds, J. Math. Phys. (N.Y.) 55, 092305 (2014).
  104. R. Kobayashi and M. Barkeshli, Soft symmetries of topological orders, arXiv:2501.03314.
  105. J. Zhao, J.-Q. Lou, Z.-H. Zhang, L.-Y. Hung, L. Kong, and Y. Tian, String condensations in 3+1D and Lagrangian algebras, Adv. Theor. Math. Phys. 27, 583 (2023).
  106. Z.-X. Luo, Gapped boundaries of (3+1)-dimensional topological order, Phys. Rev. B 107, 125425 (2023).
  107. W. Ji, N. Tantivasadakarn, and C. Xu, Boundary states of three dimensional topological order and the deconfined quantum critical point, SciPost Phys. 15, 231 (2023).

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Phys. Rev. Lett. 135, 250001– Published 17 December, 2025
  • Received 8 October 2025
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