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Exactly solvable dissipative dynamics and one-form strong-to-weak spontaneous symmetry breaking in interacting two-dimensional spin systems

Lucas Sá1 and Benjamin Béri1,2

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Phys. Rev. B 112, 144311 – Published 24 October, 2025

DOI: https://doi.org/10.1103/1lzb-vp1w

Abstract

We study the dissipative dynamics of a class of interacting “gamma-matrix” spin models coupled to a Markovian environment. For spins on an arbitrary graph, we construct a Lindbladian that maps to a non-Hermitian model of free Majorana fermions hopping on the graph with a background classical Z2 gauge field. We show, analytically and numerically, that the steady states and relaxation dynamics are qualitatively independent of the choice of the underlying graph, in stark contrast to the Hamiltonian case. We also show that the exponentially many steady states provide a concrete example of mixed-state topological order, in the sense of strong-to-weak spontaneous symmetry breaking of a one-form symmetry. While encoding only classical information, the steady states still exhibit long-range quantum correlations. Afterward, we examine the relaxation processes toward the steady state by numerically computing decay rates, which we generically find to be finite, even in the dissipationless limit. However, we identify symmetry sectors where fermion-parity conservation is enhanced to fermion-number conservation, where we can analytically bound the decay rates and prove that they vanish in the limits of both infinitely weak and infinitely strong dissipation. Finally, we show that while the choice of coherent dynamics is very flexible, exact solvability strongly constrains the allowed form of dissipation. Our work establishes an analytically tractable framework to explore nonequilibrium quantum phases of matter and the relaxation mechanisms toward them.

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References (160)

  1. P. W. Anderson, Resonating valence bonds: A new kind of insulator? Mater. Res. Bull. 8, 153 (1973).
  2. P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  3. V. Kalmeyer and R. B. Laughlin, Equivalence of the resonating-valence-bond and fractional quantum Hall states, Phys. Rev. Lett. 59, 2095 (1987).
  4. D. S. Rokhsar and S. A. Kivelson, Superconductivity and the quantum hard-core dimer gas, Phys. Rev. Lett. 61, 2376 (1988).
  5. X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  6. T. Senthil and M. P. A. Fisher, Z2 Gauge theory of electron fractionalization in strongly correlated systems, Phys. Rev. B 62, 7850 (2000).
  7. R. Moessner and S. L. Sondhi, Resonating valence bond phase in the triangular lattice quantum dimer model, Phys. Rev. Lett. 86, 1881 (2001).
  8. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  9. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  10. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  11. 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).
  12. P. A. Lee, An end to the drought of quantum spin liquids, Science 321, 1306 (2008).
  13. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  14. Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
  15. H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nat. Rev. Phys. 1, 264 (2019).
  16. C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  17. J. R. Chamorro, T. M. McQueen, and T. T. Tran, Chemistry of quantum spin liquids, Chem. Rev. 121, 2898 (2021).
  18. L. Clark and A. H. Abdeldaim, Quantum spin liquids from a materials perspective, Annu. Rev. Condens. Matter Phys. 51, 495 (2021).
  19. S. Trebst and C. Hickey, Kitaev materials, Phys. Rep. 950, 1 (2022).
  20. J. Khatua, B. Sana, A. Zorko, M. Gomilšek, K. Sethupathi, M. R. Rao, M. Baenitz, B. Schmidt, and P. Khuntia, Experimental signatures of quantum and topological states in frustrated magnetism, Phys. Rep. 1041, 1 (2023).
  21. M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections, Annu. Rev. Condens. Matter Phys. 9, 17 (2018).
  22. J. Knolle and R. Moessner, A field guide to spin liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  23. J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Dynamics of a two-dimensional quantum spin liquid: Signatures of emergent Majorana fermions and fluxes, Phys. Rev. Lett. 112, 207203 (2014).
  24. G. B. Halász, J. T. Chalker, and R. Moessner, Doping a topological quantum spin liquid: Slow holes in the Kitaev honeycomb model, Phys. Rev. B 90, 035145 (2014).
  25. J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, Dynamics of fractionalization in quantum spin liquids, Phys. Rev. B 92, 115127 (2015).
  26. J. Nasu, M. Udagawa, and Y. Motome, Thermal fractionalization of quantum spins in a Kitaev model: Temperature-linear specific heat and coherent transport of Majorana fermions, Phys. Rev. B 92, 115122 (2015).
  27. J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R. Moessner, Fermionic response from fractionalization in an insulating two-dimensional magnet, Nat. Phys. 12, 912 (2016).
  28. J. Nasu, J. Yoshitake, and Y. Motome, Thermal transport in the Kitaev model, Phys. Rev. Lett. 119, 127204 (2017).
  29. M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann, Dynamics of the Kitaev-Heisenberg model, Phys. Rev. Lett. 119, 157203 (2017).
  30. H. Yao and S. A. Kivelson, Exact chiral spin liquid with non-Abelian anyons, Phys. Rev. Lett. 99, 247203 (2007).
  31. H. Yao, S.-C. Zhang, and S. A. Kivelson, Algebraic spin liquid in an exactly solvable spin model, Phys. Rev. Lett. 102, 217202 (2009).
  32. V. Chua, H. Yao, and G. A. Fiete, Exact chiral spin liquid with stable spin fermi surface on the kagome lattice, Phys. Rev. B 83, 180412(R) (2011).
  33. C. Wu, D. Arovas, and H.-H. Hung, Γ-matrix generalization of the Kitaev model, Phys. Rev. B 79, 134427 (2009).
  34. A. J. Willans, J. T. Chalker, and R. Moessner, Site dilution in the Kitaev honeycomb model, Phys. Rev. B 84, 115146 (2011).
  35. H. Shackleton and M. S. Scheurer, Exactly solvable dissipative spin liquid, Phys. Rev. B 109, 085115 (2024).
  36. J. Gidugu and D. P. Arovas, Dissipative dirac matrix spin model in two dimensions, Phys. Rev. A 109, 022212 (2024).
  37. X.-D. Dai, F. Song, and Z. Wang, Solvable BCS-Hubbard Liouvillians in arbitrary dimensions, Phys. Rev. B 108, 115127 (2023).
  38. K. Hwang, Mixed-State quantum spin liquids and dynamical anyon condensations in Kitaev Lindbladians, Quantum 8, 1412 (2024).
  39. A. Pocklington and A. A. Clerk, Efficient simulation of nontrivial dissipative spin chains via stochastic unraveling, PRX Quantum 6, 030349 (2025).
  40. J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum 9, 1607 (2025).
  41. Z. Wang, X.-D. Dai, H.-R. Wang, and Z. Wang, Topologically ordered steady states in open quantum systems, SciPost Phys. 17, 167 (2024).
  42. Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum 6, 010314 (2025).
  43. Y.-H. Chen and T. Grover, Separability transitions in topological states induced by local decoherence, Phys. Rev. Lett. 132, 170602 (2024).
  44. X.-D. Dai, Z. Wang, H.-R. Wang, and Z. Wang, Steady-state topological order, Phys. Rev. B 111, 115142 (2025).
  45. R. Sohal and A. Prem, Noisy approach to intrinsically mixed-state topological order, PRX Quantum 6, 010313 (2025).
  46. S. Sang and T. H. Hsieh, Stability of mixed-state quantum phases via finite Markov length, Phys. Rev. Lett. 134, 070403 (2025).
  47. Z. Li, D. Lee, and B. Yoshida, How much entanglement is needed for topological codes and mixed states with anomalous symmetry?, Phys. Rev. X 15, 021090 (2025).
  48. T.-T. Wang, M. Song, Z. Y. Meng, and T. Grover, Analog of topological entanglement entropy for mixed states, PRX Quantum 6, 010358 (2025).
  49. L. A. Lessa, S. Sang, T.-C. Lu, T. H. Hsieh, and C. Wang, Higher-form anomaly and long-range entanglement of mixed states, arXiv:2503.12792.
  50. S. Sang, Y. Zou, and T. H. Hsieh, Mixed-State quantum phases: Renormalization and quantum error correction, Phys. Rev. X 14, 031044 (2024).
  51. Z. Li and R. S. K. Mong, Replica topological order in quantum mixed states and quantum error correction, Phys. Rev. B 111, 125106 (2025).
  52. T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum 6, 010315 (2025).
  53. Y. Bao, R. Fan, A. Vishwanath, and E. Altman, Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions, arXiv:2301.05687.
  54. R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diagnostics of mixed-state topological order and breakdown of quantum memory, PRX Quantum 5, 020343 (2024).
  55. J. Kim, E. Altman, and J. Y. Lee, Error threshold of SYK codes from strong-to-weak parity symmetry breaking, arXiv:2410.24225.
  56. J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023).
  57. 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).
  58. P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open systems: Purification perspective, Phys. Rev. B 110, 155150 (2024).
  59. L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-Weak spontaneous symmetry breaking in mixed quantum states, PRX Quantum 6, 010344 (2025).
  60. C. Zhang, Y. Xu, J.-H. Zhang, C. Xu, Z. Bi, and Z.-X. Luo, Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order, Phys. Rev. B 111, 115137 (2025).
  61. R. Ma and A. Turzillo, Symmetry-Protected topological phases of mixed states in the doubled space, PRX Quantum 6, 010348 (2025).
  62. A. Moharramipour, L. A. Lessa, C. Wang, T. H. Hsieh, and S. Sahu, Symmetry-Enforced entanglement in maximally mixed states, PRX Quantum 5, 040336 (2024).
  63. D. Gu, Z. Wang, and Z. Wang, Spontaneous symmetry breaking in open quantum systems: Strong, weak, and strong-to-weak, arXiv:2406.19381.
  64. Z. Liu, L. Chen, Y. Zhang, S. Zhou, and P. Zhang, Diagnosing strong-to-weak symmetry breaking via Wightman correlators, Commun. Phys. 8, 274 (2025).
  65. Z. Weinstein, Efficient detection of strong-to-weak spontaneous symmetry breaking via the rényi-1 Correlator, Phys. Rev. Lett. 134, 150405 (2025).
  66. O. Ogunnaike, J. Feldmeier, and J. Y. Lee, Unifying emergent hydrodynamics and lindbladian low-energy spectra across symmetries, constraints, and long-range interactions, Phys. Rev. Lett. 131, 220403 (2023).
  67. Y. Xu and C.-M. Jian, Average-exact mixed anomalies and compatible phases, Phys. Rev. B 111, 125128 (2025).
  68. X. Huang, M. Qi, J.-H. Zhang, and A. Lucas, Hydrodynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking, Phys. Rev. B 111, 125147 (2025).
  69. J.-H. Zhang, C. Xu, and Y. Xu, Fluctuation-dissipation theorem and information geometry in open quantum systems, arXiv:2409.18944.
  70. Y. Guo and S. Yang, Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order, Phys. Rev. B 111, L201108 (2025).
  71. T. Ando, S. Ryu, and M. Watanabe, Gauge theory and mixed state criticality, arXiv:2411.04360.
  72. L. Chen, N. Sun, and P. Zhang, Strong-to-weak symmetry breaking and entanglement transitions, Phys. Rev. B 111, L060304 (2025).
  73. N. Sun, P. Zhang, and L. Feng, Scheme to detect the strong-to-weak symmetry breaking via randomized measurements, Phys. Rev. Lett. 135, 090403 (2025).
  74. T. Orito, Y. Kuno, and I. Ichinose, Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum ising model under multiple decoherence, Phys. Rev. B 111, 054106 (2025).
  75. S. Lu, P. Zhu, and Y.-M. Lu, Bilayer construction for mixed state phenomena with strong, weak symmetries and symmetry breakings, arXiv:2411.07174.
  76. X. Feng, Z. Cheng, and M. Ippoliti, Hardness of observing strong-to-weak symmetry breaking, arXiv:2504.12233.
  77. T. Prosen, Third quantization: A general method to solve master equations for quadratic open fermi systems, New J. Phys. 10, 043026 (2008).
  78. T. Prosen and I. Pižorn, Quantum phase transition in a far-from-equilibrium steady state of an XY spin chain, Phys. Rev. Lett. 101, 105701 (2008).
  79. T. Prosen, Spectral theorem for the lindblad equation for quadratic open fermionic systems, J. Stat. Mech. (2010) P07020.
  80. T. Prosen and B. Žunkovič, Exact solution of markovian master equations for quadratic fermi systems: Thermal baths, open XY spin chains and non-equilibrium phase transition, New J. Phys. 12, 025016 (2010).
  81. B. Žunkovič and T. Prosen, Explicit solution of the lindblad equation for nearly isotropic boundary driven XY spin 1/2 chain, J. Stat. Mech. (2010) P08016.
  82. T. Prosen and T. H. Seligman, Quantization over boson operator spaces, J. Phys. A 43, 392004 (2010).
  83. M. Žnidarič, A matrix product solution for a nonequilibrium steady state of an XX chain, J. Phys. A 43, 415004 (2010).
  84. T. Prosen and E. Ilievski, Nonequilibrium phase transition in a periodically driven XY spin chain, Phys. Rev. Lett. 107, 060403 (2011).
  85. B. Žunkovič, Closed hierarchy of correlations in markovian open quantum systems, New J. Phys. 16, 013042 (2014).
  86. M. Žnidarič, Exact solution for a diffusive nonequilibrium steady state of an open quantum chain, J. Stat. Mech. (2010) L05002.
  87. M. Žnidarič, Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation, Phys. Rev. E 83, 011108 (2011).
  88. K. Temme, M. M. Wolf, and F. Verstraete, Stochastic exclusion processes versus coherent transport, New J. Phys. 14, 075004 (2012).
  89. V. Eisler, Crossover between ballistic and diffusive transport: The quantum exclusion process, J. Stat. Mech. (2011) P06007.
  90. T. Prosen, Open XXZ spin chain: Nonequilibrium steady state and a strict bound on ballistic transport, Phys. Rev. Lett. 106, 217206 (2011).
  91. T. Prosen, Exact nonequilibrium steady state of a strongly driven open XXZ chain, Phys. Rev. Lett. 107, 137201 (2011).
  92. T. Prosen and E. Ilievski, Families of quasilocal conservation laws and quantum spin transport, Phys. Rev. Lett. 111, 057203 (2013).
  93. D. Karevski, V. Popkov, and G. M. Schütz, Exact matrix product solution for the boundary-driven lindblad XXZ chain, Phys. Rev. Lett. 110, 047201 (2013).
  94. T. Prosen, Exact nonequilibrium steady state of an open hubbard chain, Phys. Rev. Lett. 112, 030603 (2014).
  95. V. Popkov and T. Prosen, Infinitely dimensional lax structure for the one-dimensional Hubbard model, Phys. Rev. Lett. 114, 127201 (2015).
  96. T. Prosen, Matrix product solutions of boundary driven quantum chains, J. Phys. A 48, 373001 (2015).
  97. B. Buča and T. Prosen, Strongly correlated non-equilibrium steady states with currents–quantum and classical picture, Eur. Phys. J. Spec. Top. 227, 421 (2018).
  98. V. Popkov, T. Prosen, and L. Zadnik, Exact nonequilibrium steady state of open XXZ/XYZ Spin-1/2 chain with Dirichlet boundary conditions, Phys. Rev. Lett. 124, 160403 (2020).
  99. M. V. Medvedyeva, F. H. L. Essler, and T. Prosen, Exact bethe ansatz spectrum of a tight-binding chain with dephasing noise, Phys. Rev. Lett. 117, 137202 (2016).
  100. D. A. Rowlands and A. Lamacraft, Noisy spins and the Richardson-Gaudin model, Phys. Rev. Lett. 120, 090401 (2018).
  101. N. Shibata and H. Katsura, Dissipative spin chain as a non-hermitian Kitaev ladder, Phys. Rev. B 99, 174303 (2019).
  102. N. Shibata and H. Katsura, Dissipative quantum Ising chain as a non-Hermitian Ashkin-Teller model, Phys. Rev. B 99, 224432 (2019).
  103. A. A. Ziolkowska and F. H. L. Essler, Yang-Baxter integrable Lindblad equations, SciPost Phys. 8, 044 (2020).
  104. P. Ribeiro and T. Prosen, Integrable quantum dynamics of open collective spin models, Phys. Rev. Lett. 122, 010401 (2019).
  105. J. M. Torres, Closed-form solution of Lindblad master equations without gain, Phys. Rev. A 89, 052133 (2014).
  106. M. Nakagawa, N. Kawakami, and M. Ueda, Exact Liouvillian spectrum of a one-dimensional dissipative Hubbard model, Phys. Rev. Lett. 126, 110404 (2021).
  107. B. Buča, C. Booker, M. Medenjak, and D. Jaksch, Bethe ansatz approach for dissipation: Exact solutions of quantum many-body dynamics under loss, New J. Phys. 22, 123040 (2020).
  108. L. Sá, P. Ribeiro, and T. Prosen, Integrable nonunitary open quantum circuits, Phys. Rev. B 103, 115132 (2021).
  109. M. de Leeuw, C. Paletta, and B. Pozsgay, Constructing integrable Lindblad superoperators, Phys. Rev. Lett. 126, 240403 (2021).
  110. M. de Leeuw and C. Paletta, The Bethe ansatz for a new integrable open quantum system, arXiv:2207.14193.
  111. L. Su and I. Martin, Integrable nonunitary quantum circuits, Phys. Rev. B 106, 134312 (2022).
  112. M. Žnidarič, U. Duh, and L. Zadnik, Integrability is generic in homogeneous U(1)-invariant nearest-neighbor qubit circuits, Phys. Rev. B 112, L020302 (2025).
  113. K. Wang, F. Piazza, and D. J. Luitz, Hierarchy of relaxation timescales in local random Liouvillians, Phys. Rev. Lett. 124, 100604 (2020).
  114. O. E. Sommer, F. Piazza, and D. J. Luitz, Many-body hierarchy of dissipative timescales in a quantum computer, Phys. Rev. Res. 3, 023190 (2021).
  115. J. L. Li, D. C. Rose, J. P. Garrahan, and D. J. Luitz, Random matrix theory for quantum and classical metastability in local Liouvillians, Phys. Rev. B 105, L180201 (2022).
  116. N. D. Hartmann, J. L. Li, and D. J. Luitz, Fate of dissipative hierarchy of timescales in the presence of unitary dynamics, Phys. Rev. B 109, 054203 (2024).
  117. V. Popkov and C. Presilla, Full spectrum of the Liouvillian of open dissipative quantum systems in the Zeno limit, Phys. Rev. Lett. 126, 190402 (2021).
  118. Y.-N. Zhou, L. Mao, and H. Zhai, Rényi entropy dynamics and lindblad spectrum for open quantum systems, Phys. Rev. Res. 3, 043060 (2021).
  119. L. Sá, P. Ribeiro, and T. Prosen, Lindbladian dissipation of strongly-correlated quantum matter, Phys. Rev. Res. 4, L022068 (2022).
  120. A. M. García-García, L. Sá, J. J. M. Verbaarschot, and J. P. Zheng, Keldysh wormholes and anomalous relaxation in the dissipative Sachdev-Ye-Kitaev model, Phys. Rev. D 107, 106006 (2023).
  121. T. Mori, Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit, Phys. Rev. B 109, 064311 (2024).
  122. T. Yoshimura and L. Sá, Robustness of quantum chaos and anomalous relaxation in open quantum circuits, Nat. Commun. 15, 9808 (2024).
  123. T. Yoshimura and L. Sá, Theory of irreversibility in quantum many-body systems, Phys. Rev. E 111, 064135 (2025).
  124. B. Buča and T. Prosen, A note on symmetry reductions of the lindblad equation: Transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  125. O. Gamayun and O. Lychkovskiy, Out-of-equilibrium dynamics of the Kitaev model on the bethe lattice via coupled Heisenberg equations, SciPost Phys. 12, 175 (2022).
  126. A. Teretenkov and O. Lychkovskiy, Exact dynamics of quantum dissipative XX models: Wannier-Stark localization in the fragmented operator space, Phys. Rev. B 109, L140302 (2024).
  127. A. A. Belavin, B. Y. Zeldovich, A. M. Perelomov, and V. S. Popov, Relaxation of quantum systems with equidistant spectra, Sov. Phys. JETP 29, 145 (1969).
  128. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  129. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  130. X.-G. Wen, Topological orders in rigid states, Int. J. Mod. Phys. B 04, 239 (1990).
  131. X.-G. Wen, Topological orders and edge excitations in fractional quantum Hall states, Adv. Phys. 44, 405 (1995).
  132. X.-G. Wen, Quantum orders in an exact soluble model, Phys. Rev. Lett. 90, 016803 (2003).
  133. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  134. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  135. S. H. Simon, Topological Quantum (Oxford University Press, Oxford, 2023).
  136. M. B. Hastings and X.-G. Wen, Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72, 045141 (2005).
  137. Z. Nussinov and G. Ortiz, A symmetry principle for topological quantum order, Ann. Phys. 324, 977 (2009).
  138. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  139. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  140. S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, New J. Phys. 11, 043029 (2009).
  141. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
  142. T. Prosen, Ruelle resonances in quantum many-body dynamics, J. Phys. A 35, L737 (2002).
  143. T. Prosen, Ruelle resonances in kicked quantum spin chain, Physica D 187, 244 (2004).
  144. T. Prosen, Chaos and complexity of quantum motion, J. Phys. A 40, 7881 (2007).
  145. M. Žnidarič, Momentum-dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems, Phys. Rev. E 110, 054204 (2024).
  146. J. A. Jacoby, D. A. Huse, and S. Gopalakrishnan, Spectral gaps of local quantum channels in the weak-dissipation limit, Phys. Rev. B 111, 104303 (2025).
  147. C. Zhang, L. Nie, and C. von Keyserlingk, Thermalization rates and quantum Ruelle-Pollicott resonances: Insights from operator hydrodynamics, arXiv:2409.17251.
  148. I. Bendixson, Sur les racines d'une équation fondamentale, Acta Math. 25, 359 (1902).
  149. L. Mirsky, An Introduction to Linear Algebra (Clarendon, Oxford, 1955).
  150. E. H. Lieb, Flux phase of the half-filled band, Phys. Rev. Lett. 73, 2158 (1994).
  151. K. Klocke, D. Simm, G.-Y. Zhu, S. Trebst, and M. Buchhold, Entanglement dynamics in monitored Kitaev circuits: Loop models, symmetry classification, and quantum Lifshitz scaling, Phys. Rev. B 111, 224301 (2025).
  152. D. Bernard and A. LeClair, A classification of non-Hermitian random matrices, in Statistical Field Theories, edited by A. Cappelli and G. Mussardo (Springer, Dordrecht, 2002), pp. 207–214.
  153. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  154. H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  155. L. Sá, P. Ribeiro, and T. Prosen, Symmetry classification of many-body Lindbladians: Tenfold way and beyond, Phys. Rev. X 13, 031019 (2023).
  156. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Symmetry of open quantum systems: Classification of dissipative quantum chaos, PRX Quantum 4, 030328 (2023).
  157. A. M. García-García, L. Sá, J. J. M. Verbaarschot, and C. Yin, Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes, Phys. Rev. D 109, 105017 (2024).
  158. S. Lieu, M. McGinley, and N. R. Cooper, Tenfold way for quadratic Lindbladians, Phys. Rev. Lett. 124, 040401 (2020).
  159. A. Altland, M. Fleischhauer, and S. Diehl, Symmetry classes of open fermionic quantum matter, Phys. Rev. X 11, 021037 (2021).
  160. M. Kawasaki, K. Mochizuki, and H. Obuse, Topological phases protected by shifted sublattice symmetry in dissipative quantum systems, Phys. Rev. B 106, 035408 (2022).

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Phys. Rev. B 112, 144311– Published 24 October, 2025
  • Received 27 May 2025
  • Accepted 29 September 2025
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