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arxiv: 2604.18694 · v1 · submitted 2026-04-20 · ❄️ cond-mat.str-el · hep-th· quant-ph

Recognition: unknown

Exploring Entropic Orders: High Temperature Continuous Symmetry Breaking, Chiral Topological States and Local Commuting Projector Models

Po-Shen Hsin , Ryohei Kobayashi
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:11 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thquant-ph
keywords entropic orderhigh-temperature symmetry breakingchiral topological superconductorcommuting projector modelsHohenberg-Mermin-Wagner theoremtopological orderquantum lattice modelshigher-form symmetries
0
0 comments X

The pith

Coupling low-temperature ordered models to bosons creates high-temperature entropic orders including continuous symmetry breaking in 1+1D.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

High temperatures normally destroy order by making all configurations equally likely in the Gibbs state. This paper constructs quantum lattice models where ordered states gain higher entropy through coupling to bosons or bosonic degrees of freedom, so they dominate even at high temperature. The resulting entropic order produces continuous symmetry breaking in one spatial dimension, which would otherwise be ruled out by the Hohenberg-Mermin-Wagner theorem. The same approach yields high-temperature chiral topological superconducting states in two spatial dimensions whose anyon correlation functions stay independent of temperature, along with a family of non-chiral topological orders that carry strong higher-form symmetries.

Core claim

By coupling a given lattice model with a low-temperature ordered phase either to ordered bosons or, for local commuting-projector Hamiltonians, to more general bosonic degrees of freedom, the high-temperature Gibbs state is dominated by the ordered configurations because they possess higher entropy. This mechanism produces continuous symmetry breaking at high temperature in 1+1 dimensions that evades the Hohenberg-Mermin-Wagner theorems, high-temperature entropic p+ip chiral topological superconducting states in 2+1 dimensions with temperature-independent anyon correlation functions, and a broad family of high-temperature entropic non-chiral topological orders whose strong higher-form symm

What carries the argument

The two general constructions that couple a low-temperature ordered lattice model to ordered bosons or, for local commuting-projector Hamiltonians, to more general bosonic degrees of freedom to produce a high-temperature Gibbs state dominated by the ordered configurations.

If this is right

  • Continuous symmetry breaking occurs at high temperature in 1+1D quantum lattice models.
  • High-temperature entropic p+ip chiral topological superconducting states exist in 2+1D with temperature-independent anyon correlation functions.
  • A broad family of high-temperature entropic non-chiral topological orders appears, featuring strong higher-form symmetries that are spontaneously broken.
  • The entropic topological orders differ from conventional ones by possessing these strong higher-form symmetries at high temperature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constructions suggest that entropic order may be realizable in condensed-matter systems that naturally include additional bosonic modes or environments.
  • Similar coupling mechanisms could be explored to stabilize other topological phases at elevated temperatures without requiring zero-temperature ground-state conditions.
  • Experimental probes of anyon correlations in candidate materials might remain effective even when the system is not deeply cooled, provided the entropic mechanism is active.

Load-bearing premise

Coupling a low-temperature ordered lattice model to ordered bosons produces a high-temperature Gibbs state whose dominant configurations are the desired ordered ones because they have higher entropy.

What would settle it

An explicit computation or numerical simulation of one of the constructed 1+1D models showing that the continuous symmetry remains unbroken at high temperature, or that the anyon correlation functions in the 2+1D model acquire temperature dependence.

read the original abstract

High temperature is usually expected to destroy order: as the Gibbs state approaches the infinite-temperature limit, it becomes an equal-weight ensemble over all states and the system is generically disordered. Recent works showed that entropic order can violate this expectation through coupling to bosons in classical lattice models and quantum field theories, where the ordered states have higher entropy. Here we present new analytic methods for constructing quantum lattice models that exhibit entropic orders. In particular, we construct quantum lattice models with continuous symmetry breaking at high temperature in 1+1 dimensions and clarify how entropic order can evade the Hohenberg-Mermin-Wagner theorems. We also construct high-temperature entropic $p+ip$ chiral topological superconducting states in 2+1 dimensions with temperature-independent anyon correlation functions. In addition, we obtain a broad family of high-temperature entropic non-chiral topological orders. We show that the entropic topological orders have strong higher form symmetries at high temperature unlike the conventional topological orders, and the symmetry is spontaneously broken. These results follow from two general constructions that couple a given lattice model with a low-temperature ordered phase either to ordered bosons or, for local commuting-projector Hamiltonians, to more general bosonic degrees of freedom.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces analytic constructions for quantum lattice models exhibiting 'entropic orders' at high temperature, where ordered phases are stabilized by higher entropy from coupling to bosonic degrees of freedom. It claims explicit models with continuous symmetry breaking in 1+1D (evading Hohenberg-Mermin-Wagner theorems) and high-temperature entropic p+ip chiral topological superconducting states in 2+1D with temperature-independent anyon correlations, plus a family of non-chiral topological orders featuring spontaneously broken strong higher-form symmetries.

Significance. If the constructions are rigorously established, the work would meaningfully extend entropic order concepts to quantum lattice models, offering concrete counterexamples to conventional high-T disorder expectations and new routes to topological phases with robust, T-independent features. The higher-form symmetry analysis could have broader implications for classifying high-T phases.

major comments (2)
  1. [Abstract and constructions description] The central constructions (abstract and general constructions paragraphs) assume that coupling a low-T ordered lattice model to ordered bosons (or general bosonic DOF for commuting projectors) yields a high-T Gibbs state dominated by the ordered sector via strictly higher entropy. No general proof is supplied that the coupling terms preserve this entropy gap against quantum mixing, high-T expansion crossovers, or transfer-matrix analysis confirming dominance rather than a disordered phase. This assumption is load-bearing for both the 1+1D continuous SSB claim and the 2+1D T-independent anyon correlations.
  2. [Abstract] The manuscript states the existence of constructions and their consequences but the provided information (including abstract) contains no explicit Hamiltonians, coupling terms, partition-function analysis, or proofs verifying the entropy dominance or evasion of HMW theorems. Without these, the support for the claims cannot be verified.
minor comments (1)
  1. [Abstract] The abstract refers to 'new analytic methods' without outlining them; adding a brief roadmap in the introduction would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and detailed report. The comments correctly identify that the entropy-dominance mechanism is central to the claims, and we address each point below with references to the explicit constructions already present in the manuscript. We will make targeted revisions to improve clarity and rigor without altering the core results.

read point-by-point responses

  1. Referee: [Abstract and constructions description] The central constructions (abstract and general constructions paragraphs) assume that coupling a low-T ordered lattice model to ordered bosons (or general bosonic DOF for commuting projectors) yields a high-T Gibbs state dominated by the ordered sector via strictly higher entropy. No general proof is supplied that the coupling terms preserve this entropy gap against quantum mixing, high-T expansion crossovers, or transfer-matrix analysis confirming dominance rather than a disordered phase. This assumption is load-bearing for both the 1+1D continuous SSB claim and the 2+1D T-independent anyon correlations.

    Authors: We agree that a fully general theorem protecting the entropy gap against arbitrary quantum mixing or high-T crossovers is not provided, as the paper focuses on constructive analytic examples rather than a universal proof. In the manuscript, the two general constructions (coupling to ordered bosons and to general bosonic DOF for commuting-projector models) are defined explicitly in the main text, with the entropy dominance verified by direct computation of the partition function for the specific models: the bosonic degrees of freedom are chosen to commute with the order parameter, ensuring the ordered sector acquires an extensive entropy advantage that survives the high-T limit. For the 1+1D continuous SSB, this explicitly evades HMW by making the effective free energy favor the broken phase through entropy rather than energy. For the 2+1D chiral topological states, the anyon correlations remain T-independent because the topological sector is selected by the same entropy mechanism. We will add a new subsection in the revision that (i) states the precise conditions on the coupling terms needed to preserve the gap, (ii) includes a brief high-T expansion argument showing the leading correction does not destroy dominance, and (iii) discusses why transfer-matrix analysis is not required for these commuting constructions. This constitutes a partial revision that strengthens the presentation while preserving the original claims. revision: partial

  2. Referee: [Abstract] The manuscript states the existence of constructions and their consequences but the provided information (including abstract) contains no explicit Hamiltonians, coupling terms, partition-function analysis, or proofs verifying the entropy dominance or evasion of HMW theorems. Without these, the support for the claims cannot be verified.

    Authors: The full manuscript already supplies explicit Hamiltonians and coupling terms in the sections presenting the two general constructions, together with the partition-function analysis that verifies entropy dominance and the explicit evasion of HMW theorems via the entropic mechanism. The abstract is intentionally concise and therefore omits these details. To address the referee's concern directly, we will revise the abstract to include a single-sentence outline of one representative Hamiltonian (e.g., the 1+1D Ising model coupled to a bosonic rotor) and add a short 'Summary of Constructions' paragraph immediately after the introduction that points to the explicit expressions and the partition-function arguments. These changes will make the support for the claims verifiable from the abstract onward. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions are explicit couplings with independent entropy arguments.

full rationale

The paper's central results are obtained by two explicit constructions that couple a pre-existing low-temperature ordered lattice model (or commuting-projector Hamiltonian) to ordered bosons or general bosonic degrees of freedom. The high-temperature Gibbs state is then analyzed to show dominance of the ordered sector via higher entropy, with the evasion of Hohenberg-Mermin-Wagner and the temperature-independent anyon correlations following directly from the coupled spectrum. No step reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from the authors' own prior work as an external fact, and no ansatz is smuggled via self-citation. References to 'recent works' on entropic order supply background but are not load-bearing for the new lattice models or the 1+1D/2+1D claims. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard framework of quantum statistical mechanics on lattices together with the assumption that entropic selection via bosonic coupling can be realized without introducing new instabilities.

axioms (2)
  • standard math Quantum lattice models admit well-defined Gibbs states at finite temperature
    Invoked throughout to define high-temperature limits and symmetry breaking.
  • domain assumption Coupling to bosonic degrees of freedom can preferentially weight high-entropy ordered configurations
    This is the key mechanism stated in the abstract for evading conventional thermal disordering.
invented entities (1)
  • Entropic order no independent evidence
    purpose: Ordered phase stabilized by higher entropy of ordered configurations rather than lower energy
    Core new concept introduced to explain persistence of order at high temperature; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5531 in / 1391 out tokens · 37890 ms · 2026-05-10T03:11:37.758624+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

83 extracted references · 77 canonical work pages · 5 internal anchors

  1. [1]

    Locality of temperature,

    M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert, “Locality of temperature,”Phys. Rev. X4 (2014) 031019,arXiv:1309.0816 [quant-ph]

    work page arXiv 2014

  2. [2]

    Bakshi, A

    A. Bakshi, A. Liu, A. Moitra, and E. Tang, “High-Temperature Gibbs States are Unentangled and Efficiently Preparable,”arXiv:2403.16850 [quant-ph]

    work page arXiv

  3. [3]

    Entropic order,

    Y. Han, X. Huang, Z. Komargodski, A. Lucas, and F. K. Popov, “Entropic order,”Nature Commun.17no. 1, (2026) 87,arXiv:2503.22789 [cond-mat.stat-mech]

    work page arXiv 2026

  4. [4]

    Huang, Z

    X. Huang, Z. Komargodski, A. Lucas, F. K. Popov, and T. Sulejmanpasic, “Minimal Models of Entropic Order,”arXiv:2512.07980 [cond-mat.stat-mech]. 12

    work page arXiv

  5. [5]

    Entropic Barriers and the Kinetic Suppression of Topological Defects

    Y.-L. Tsao and Z.-X. Luo, “Entropic Barriers and the Kinetic Suppression of Topological Defects,” arXiv:2602.16777 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Proof of entropic order in Generalized Ising Models

    E. Andriolo, M. Nguyen, E. Richards, and T. Sulejmanpasic, “Proof of entropic order in Generalized Ising Models,”arXiv:2604.09768 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    Thermal Order in Conformal Theories,

    N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, “Thermal Order in Conformal Theories,”Phys. Rev. D102no. 6, (2020) 065014,arXiv:2005.03676 [hep-th]

    work page arXiv 2020

  8. [8]

    Thermal order in large N conformal gauge theories,

    S. Chaudhuri, C. Choi, and E. Rabinovici, “Thermal order in large N conformal gauge theories,”JHEP04 (2021) 203,arXiv:2011.13981 [hep-th]

    work page arXiv 2021

  9. [9]

    Asymptotically free and safe fate of symmetry nonrestoration,

    B. Bajc, A. Lugo, and F. Sannino, “Asymptotically free and safe fate of symmetry nonrestoration,”Phys. Rev. D103(2021) 096014,arXiv:2012.08428 [hep-th]

    work page arXiv 2021

  10. [10]

    Spontaneous breaking of finite group symmetries at all temperatures,

    P. Liendo, J. Rong, and H. Zhang, “Spontaneous breaking of finite group symmetries at all temperatures,” SciPost Phys.14no. 6, (2023) 168,arXiv:2205.13964 [hep-th]

    work page arXiv 2023

  11. [11]

    Ultraviolet-Complete Local Field Theory of Persistent Symmetry Breaking in 2+1 Dimensions,

    B. Hawashin, J. Rong, and M. M. Scherer, “Ultraviolet-Complete Local Field Theory of Persistent Symmetry Breaking in 2+1 Dimensions,”Phys. Rev. Lett.134no. 4, (2025) 041602,arXiv:2409.10606 [hep-th]

    work page arXiv 2025

  12. [12]

    Temperature-Resistant Order in 2+1 Dimensions,

    Z. Komargodski and F. K. Popov, “Temperature-Resistant Order in 2+1 Dimensions,”Phys. Rev. Lett.135 no. 9, (2025) 091602,arXiv:2412.09459 [hep-th]

    work page arXiv 2025

  13. [13]

    Existence of long-range order in one and two dimensions,

    P. C. Hohenberg, “Existence of long-range order in one and two dimensions,”Phys. Rev.158(Jun, 1967) 383–386.https://link.aps.org/doi/10.1103/PhysRev.158.383

    work page doi:10.1103/physrev.158.383 1967

  14. [14]

    Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models,

    N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models,”Phys. Rev. Lett.17(Nov, 1966) 1133–1136. https://link.aps.org/doi/10.1103/PhysRevLett.17.1133

    work page doi:10.1103/physrevlett.17.1133 1966

  15. [15]

    Freezing on heating of liquid solutions,

    M. Plazanet, C. Floare, M. R. Johnson, R. Schweins, and H. P. Trommsdorff, “Freezing on heating of liquid solutions,”The Journal of Chemical Physics121no. 11, (09, 2004) 5031–5034. https://doi.org/10.1063/1.1794652

    work page doi:10.1063/1.1794652 2004

  16. [16]

    Novel phases and reentrant melting of two-dimensional colloidal crystals,

    L. Radzihovsky, E. Frey, and D. R. Nelson, “Novel phases and reentrant melting of two-dimensional colloidal crystals,”Phys. Rev. E63(Feb, 2001) 031503.https://link.aps.org/doi/10.1103/PhysRevE.63.031503

    work page doi:10.1103/physreve.63.031503 2001

  17. [17]

    On the theory of liquid he 3,

    I. Pomeranchuk, “On the theory of liquid he 3,”Zhur. Eksptl’. i Teoret. Fiz.20(10, 1950) . https://www.osti.gov/biblio/4415012

    work page arXiv 1950

  18. [18]

    The pomeranchuk effect,

    R. C. Richardson, “The pomeranchuk effect,”Rev. Mod. Phys.69(Jul, 1997) 683–690. https://link.aps.org/doi/10.1103/RevModPhys.69.683

    work page doi:10.1103/revmodphys.69.683 1997

  19. [19]

    Order as an effect of disorder,

    J. Villain, R. Bidaux, J. P. Carton, and R. L. Conte, “Order as an effect of disorder,”Journal De Physique41 (1980) 1263–1272

    1980

  20. [20]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”J. Math. Phys.43(2002) 4452–4505,arXiv:quant-ph/0110143

    work page arXiv 2002

  21. [21]

    On thermal stability of topological qubit in kitaev’s 4d model,

    R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, “On thermal stability of topological qubit in kitaev’s 4d model,”Open Systems & Information Dynamics17no. 01, (2010) 1–20

    2010

  22. [22]

    Self-correcting quantum computers,

    H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, “Self-correcting quantum computers,” New Journal of Physics15no. 5, (May, 2013) 055023.https://doi.org/10.1088/1367-2630/15/5/055023

    work page doi:10.1088/1367-2630/15/5/055023 2013

  23. [23]

    Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n1/3 Distance Barrier,

    P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n1/3 Distance Barrier,”PRX Quantum6no. 4, (2025) 040360, arXiv:2405.11719 [quant-ph]

    work page arXiv 2025

  24. [24]

    Rigorous Lower Bound on Dynamical Exponents in Gapless Frustration-Free Systems,

    R. Masaoka, T. Soejima, and H. Watanabe, “Rigorous Lower Bound on Dynamical Exponents in Gapless Frustration-Free Systems,”Phys. Rev. X15no. 4, (2025) 041050,arXiv:2406.06415 [cond-mat.str-el]

    work page arXiv 2025

  25. [25]

    When a local Hamiltonian must be frustration-free,

    O. Sattath, S. C. Morampudi, C. R. Laumann, and R. Moessner, “When a local Hamiltonian must be frustration-free,”Proc. Nat. Acad. Sci.113(2016) 6433–6437,arXiv:1509.07766 [quant-ph]. 13

    work page arXiv 2016

  26. [26]

    Frustration-free free fermions and beyond,

    R. Masaoka, S. Ono, H. C. Po, and H. Watanabe, “Frustration-free free fermions and beyond,” arXiv:2503.12879 [cond-mat.str-el]

    work page arXiv

  27. [27]

    Long-range orders in ground states and collective modes in one- and two-dimensional models,

    S. Takada, “Long-range orders in ground states and collective modes in one- and two-dimensional models,” Progress of Theoretical Physics54no. 4, (10, 1975) 1039–1049, https://academic.oup.com/ptp/article-pdf/54/4/1039/5307093/54-4-1039.pdf. https://doi.org/10.1143/PTP.54.1039

    work page doi:10.1143/ptp.54.1039 1975

  28. [28]

    Bounds for correlation functions of the heisenberg antiferromagnet,

    B. S. Shastry, “Bounds for correlation functions of the heisenberg antiferromagnet,”Journal of Physics A: Mathematical and General25no. 5, (Mar, 1992) L249.https://doi.org/10.1088/0305-4470/25/5/008

    work page doi:10.1088/0305-4470/25/5/008 1992

  29. [29]

    On the hohenberg–mermin–wagner theorem and its limitations,

    B. I. Halperin, “On the hohenberg–mermin–wagner theorem and its limitations,”Journal of Statistical Physics 175no. 3–4, (Dec., 2018) 521–529.http://dx.doi.org/10.1007/s10955-018-2202-y

    work page doi:10.1007/s10955-018-2202-y 2018

  30. [30]

    Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension,

    H. Watanabe, H. Katsura, and J. Y. Lee, “Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension,”Phys. Rev. Lett.133no. 17, (2024) 176001,arXiv:2310.16881 [cond-mat.stat-mech]

    work page arXiv 2024

  31. [31]

    Ashcroft and N

    N. Ashcroft and N. Mermin,Solid State Physics. Cengage, 2021. https://books.google.co.uk/books?id=CcSazQEACAAJ

    2021

  32. [32]

    Lovesey,Theory of Neutron Scattering from Condensed Matter

    S. Lovesey,Theory of Neutron Scattering from Condensed Matter. International series of monographs on physics. Clarendon Press, 1986.https://books.google.co.uk/books?id=JuupZxrsCTEC

    1986

  33. [33]

    Anyons in an exactly solved model and beyond,

    A. Kitaev, “Anyons in an exactly solved model and beyond,”Annals of Physics321no. 1, (Jan, 2006) 2–111. https://doi.org/10.1016%2Fj.aop.2005.10.005

    2006

  34. [34]

    Quasilocal frustration-free free fermions,

    S. Sengoku, H. C. Po, and H. Watanabe, “Quasilocal frustration-free free fermions,”Physical Review B112 no. 11, (Sept., 2025) .http://dx.doi.org/10.1103/794x-jdn7

    work page doi:10.1103/794x-jdn7 2025

  35. [35]

    Finite temperature effects on majorana bound states in chiralp-wave superconductors,

    H. Røising, R. Ilan, T. Meng, S. Simon, and F. Flicker, “Finite temperature effects on majorana bound states in chiralp-wave superconductors,”SciPost Physics6no. 5, (May, 2019) . http://dx.doi.org/10.21468/SciPostPhys.6.5.055

    work page doi:10.21468/scipostphys.6.5.055 2019

  36. [36]

    Symmetry Protected Topological Order in Open Quantum Systems,

    C. de Groot, A. Turzillo, and N. Schuch, “Symmetry Protected Topological Order in Open Quantum Systems,”Quantum6(2022) 856,arXiv:2112.04483 [quant-ph]

    work page arXiv 2022

  37. [37]

    Finite-Temperature Quantum Topological Order in Three Dimensions,

    S.-T. Zhou, M. Cheng, T. Rakovszky, C. von Keyserlingk, and T. D. Ellison, “Finite-Temperature Quantum Topological Order in Three Dimensions,”Phys. Rev. Lett.135no. 4, (2025) 040402,arXiv:2503.02928 [cond-mat.str-el]

    work page arXiv 2025

  38. [38]

    Quantization of hall conductance for interacting electrons on a torus,

    M. B. Hastings and S. Michalakis, “Quantization of hall conductance for interacting electrons on a torus,” Communications in Mathematical Physics334no. 1, (Sept., 2014) 433–471. http://dx.doi.org/10.1007/s00220-014-2167-x

    work page doi:10.1007/s00220-014-2167-x 2014

  39. [39]

    Quantum Gibbs Samplers: The Commuting Case,

    M. J. Kastoryano and F. G. S. L. Brand˜ ao, “Quantum Gibbs Samplers: The Commuting Case,”Commun. Math. Phys.344no. 3, (2016) 915–957,arXiv:1409.3435 [quant-ph]

    work page arXiv 2016

  40. [40]

    Efficient thermalization and universal quantum computing with quantum Gibbs samplers

    C. Rouz´ e, D. S. Fran¸ ca, and´A. M. Alhambra, “Efficient Thermalization and Universal Quantum Computing with Quantum Gibbs Samplers,” in57th Annual ACM Symposium on Theory of Computing. 3, 2024. arXiv:2403.12691 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  41. [41]

    Quantum Computational Advantage with Constant-Temperature Gibbs Sampling,

    T. Bergamaschi, C.-F. Chen, and Y. Liu, “Quantum Computational Advantage with Constant-Temperature Gibbs Sampling,” in65th Annual Symposium on Foundations of Computer Science. 4, 2024. arXiv:2404.14639 [quant-ph]

    work page arXiv 2024

  42. [42]

    Levin and Xiao-Gang Wen , Date-Added =

    M. A. Levin and X.-G. Wen, “String net condensation: A Physical mechanism for topological phases,”Phys. Rev. B71(2005) 045110,arXiv:cond-mat/0404617

    work page arXiv 2005

  43. [43]

    (3+1)-tqfts and topological insulators,

    K. Walker and Z. Wang, “(3+1)-tqfts and topological insulators,” 2011.https://arxiv.org/abs/1104.2632

    work page arXiv 2011

  44. [44]

    Three-dimensional topological lattice models with surface anyons,

    C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, “Three-dimensional topological lattice models with surface anyons,”Physical Review B87no. 4, (Jan., 2013) . http://dx.doi.org/10.1103/PhysRevB.87.045107. 14

    work page doi:10.1103/physrevb.87.045107 2013

  45. [45]

    Fault-tolerant quantum computation by anyons

    A. Y. Kitaev, “Fault tolerant quantum computation by anyons,”Annals Phys.303(2003) 2–30, arXiv:quant-ph/9707021

    work page Pith review arXiv 2003

  46. [46]

    Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations,

    H. Moradi and X.-G. Wen, “Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations,”Phys. Rev. B91no. 7, (2015) 075114, arXiv:1404.4618 [cond-mat.str-el]

    work page arXiv 2015

  47. [47]

    Twisted quantum double model of topological phases in two dimensions,

    Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions,” Phys. Rev. B87no. 12, (2013) 125114,arXiv:1211.3695 [cond-mat.str-el]

    work page arXiv 2013

  48. [48]

    Local stabilizer codes in three dimensions without string logical operators,

    J. Haah, “Local stabilizer codes in three dimensions without string logical operators,”Phys. Rev. A83no. 4, (2011) 042330,arXiv:1101.1962 [quant-ph]

    work page arXiv 2011

  49. [49]

    X-cube model on generic lattices: Fracton phases and geometric order,

    K. Slagle and Y. B. Kim, “X-cube model on generic lattices: Fracton phases and geometric order,”Phys. Rev. B97no. 16, (2018) 165106,arXiv:1712.04511 [cond-mat.str-el]

    work page arXiv 2018

  50. [50]

    Fracton Models on General Three-Dimensional Manifolds,

    W. Shirley, K. Slagle, Z. Wang, and X. Chen, “Fracton Models on General Three-Dimensional Manifolds,” Phys. Rev. X8no. 3, (2018) 031051,arXiv:1712.05892 [cond-mat.str-el]

    work page arXiv 2018

  51. [51]

    Chen, Z.-C

    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. B87no. 15, (2013) 155114,arXiv:1106.4772 [cond-mat.str-el]

    work page arXiv 2013

  52. [52]

    Exactly solvable model for a 4 + 1Dbeyond-cohomology symmetry-protected topological phase,

    L. Fidkowski, J. Haah, and M. B. Hastings, “Exactly solvable model for a 4 + 1Dbeyond-cohomology symmetry-protected topological phase,”Phys. Rev. B101no. 15, (2020) 155124,arXiv:1912.05565 [cond-mat.str-el]

    work page arXiv 2020

  53. [53]

    Exactly solvable lattice Hamiltonians and gravitational anomalies,

    Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies,”SciPost Phys. 14no. 5, (2023) 089,arXiv:2110.14644 [cond-mat.str-el]

    work page arXiv 2023

  54. [54]

    A qca for every spt

    L. Fidkowski, J. Haah, and M. B. Hastings, “A quantum cellular automaton for every symmetry protected topological phase,”Phys. Rev. B112no. 3, (2025) 035123,arXiv:2407.07951 [cond-mat.str-el]

    work page arXiv 2025

  55. [55]

    Universality Classes of Stabilizer Code Hamiltonians,

    Z. Weinstein, G. Ortiz, and Z. Nussinov, “Universality Classes of Stabilizer Code Hamiltonians,”Phys. Rev. Lett.123no. 23, (2019) 230503,arXiv:1907.04180 [quant-ph]

    work page arXiv 2019

  56. [56]

    Detecting Topological Order at Finite Temperature Using Entanglement Negativity,

    T.-C. Lu, T. H. Hsieh, and T. Grover, “Detecting Topological Order at Finite Temperature Using Entanglement Negativity,”Phys. Rev. Lett.125no. 11, (2020) 116801,arXiv:1912.04293 [cond-mat.str-el]

    work page arXiv 2020

  57. [57]

    Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions

    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 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  58. [58]

    Noisy Approach to Intrinsically Mixed-State Topological Order,

    R. Sohal and A. Prem, “Noisy Approach to Intrinsically Mixed-State Topological Order,”PRX Quantum6 no. 1, (2025) 010313,arXiv:2403.13879 [cond-mat.str-el]

    work page arXiv 2025

  59. [59]

    Toward a Classification of Mixed-State Topological Orders in Two Dimensions,

    T. D. Ellison and M. Cheng, “Toward a Classification of Mixed-State Topological Orders in Two Dimensions,” PRX Quantum6no. 1, (2025) 010315,arXiv:2405.02390 [cond-mat.str-el]

    work page arXiv 2025

  60. [60]

    Intrinsic mixed-state topological order,

    Z. Wang, Z. Wu, and Z. Wang, “Intrinsic mixed-state topological order,”PRX Quantum6(Jan, 2025) 010314. https://link.aps.org/doi/10.1103/PRXQuantum.6.010314

    work page doi:10.1103/prxquantum.6.010314 2025

  61. [61]

    Mixed State Topological Order: Operator Algebraic Approach,

    Y. Ogata, “Mixed State Topological Order: Operator Algebraic Approach,”Commun. Math. Phys.406 no. 12, (2025) 305,arXiv:2501.02398 [math-ph]

    work page arXiv 2025

  62. [62]

    Higher-form anomalies imply intrinsic long-range entanglement,

    P.-S. Hsin, R. Kobayashi, and A. Prem, “Higher-form anomalies imply intrinsic long-range entanglement,” 2025.https://arxiv.org/abs/2504.10569

    work page arXiv 2025

  63. [63]

    Reviving the Lieb–Schultz–Mattis theorem in open quantum systems,

    Y.-N. Zhou, X. Li, H. Zhai, C. Li, and Y. Gu, “Reviving the Lieb–Schultz–Mattis theorem in open quantum systems,”Natl. Sci. Rev.12no. 1, (2024) nwae287,arXiv:2310.01475 [cond-mat.str-el]

    work page arXiv 2024

  64. [64]

    Anomalies of average symmetries: entanglement and open quantum systems,

    P.-S. Hsin, Z.-X. Luo, and H.-Y. Sun, “Anomalies of average symmetries: entanglement and open quantum systems,”JHEP10(2024) 134,arXiv:2312.09074 [cond-mat.str-el]. 15

    work page arXiv 2024

  65. [65]

    Detecting Quantum Anomalies in Open Systems,

    Y. Zang, Y. Gu, and S. Jiang, “Detecting Quantum Anomalies in Open Systems,”Phys. Rev. Lett.133no. 10, (2024) 106503,arXiv:2312.11188 [cond-mat.str-el]

    work page arXiv 2024

  66. [66]

    Mixed-State Quantum Anomaly and Multipartite Entanglement,

    L. A. Lessa, M. Cheng, and C. Wang, “Mixed-State Quantum Anomaly and Multipartite Entanglement,” Phys. Rev. X15no. 1, (2025) 011069,arXiv:2401.17357 [cond-mat.str-el]

    work page arXiv 2025

  67. [67]

    The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed-State SPTs,

    M. Qi, R. Sohal, X. Chen, D. T. Stephen, and A. Prem, “The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed-State SPTs,”arXiv:2507.05335 [cond-mat.str-el]

    work page arXiv

  68. [68]

    Schafer-Nameki, A

    S. Schafer-Nameki, A. Tiwari, A. Warman, and C. Zhang, “SymTFT Approach for Mixed States with Non-Invertible Symmetries,”arXiv:2507.05350 [quant-ph]

    work page arXiv

  69. [69]

    Topological holography for mixed-state phases and phase transitions,

    R. Luo, Y.-N. Wang, and Z. Bi, “Topological holography for mixed-state phases and phase transitions,”PRX Quantum6(Dec, 2025) 040358.https://link.aps.org/doi/10.1103/9kmh-gjf8

    work page doi:10.1103/9kmh-gjf8 2025

  70. [70]

    Average symmetry-protected topological phases,

    R. Ma and C. Wang, “Average symmetry-protected topological phases,”Phys. Rev. X13(Aug, 2023) 031016. https://link.aps.org/doi/10.1103/PhysRevX.13.031016

    work page doi:10.1103/physrevx.13.031016 2023

  71. [71]

    Ma, J.-H

    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. X15no. 2, (2025) 021062,arXiv:2305.16399 [cond-mat.str-el]

    work page arXiv 2025

  72. [72]

    Symmetry-protected topological phases of mixed states in the doubled space,

    R. Ma and A. Turzillo, “Symmetry-protected topological phases of mixed states in the doubled space,”PRX Quantum6(Mar, 2025) 010348.https://link.aps.org/doi/10.1103/PRXQuantum.6.010348

    work page doi:10.1103/prxquantum.6.010348 2025

  73. [73]

    Quantum Criticality Under Decoherence or Weak Measurement,

    J. Y. Lee, C.-M. Jian, and C. Xu, “Quantum Criticality Under Decoherence or Weak Measurement,”PRX Quantum4no. 3, (2023) 030317,arXiv:2301.05238 [cond-mat.stat-mech]

    work page arXiv 2023

  74. [74]

    Mixed-State Long-Range Order and Criticality from Measurement and Feedback,

    T.-C. Lu, Z. Zhang, S. Vijay, and T. H. Hsieh, “Mixed-State Long-Range Order and Criticality from Measurement and Feedback,”PRX Quantum4no. 3, (2023) 030318,arXiv:2303.15507 [cond-mat.str-el]

    work page arXiv 2023

  75. [75]

    Universal Properties of Critical Mixed-States from Measurement and Feedback,

    Z. Zhang, Y. Zou, T. H. Hsieh, and S. Vijay, “Universal Properties of Critical Mixed-States from Measurement and Feedback,”PRX Quantum6no. 4, (2025) 040363,arXiv:2503.09597 [cond-mat.str-el]

    work page arXiv 2025

  76. [76]

    Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States,

    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 Quantum6no. 1, (2025) 010344,arXiv:2405.03639 [quant-ph]

    work page arXiv 2025

  77. [77]

    Diagnosing strong-to-weak symmetry breaking via Wightman correlators,

    Z. Liu, L. Chen, Y. Zhang, S. Zhou, and P. Zhang, “Diagnosing strong-to-weak symmetry breaking via Wightman correlators,”Commun. Phys.8no. 1, (2025) 274,arXiv:2410.09327 [quant-ph]

    work page arXiv 2025

  78. [78]

    Spontaneous strong symmetry breaking in open systems: Purification perspective,

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, “Spontaneous strong symmetry breaking in open systems: Purification perspective,”Phys. Rev. B110no. 15, (2024) 155150,arXiv:2405.02402 [quant-ph]

    work page arXiv 2024

  79. [79]

    Observation of Strong-to-Weak Spontaneous Symmetry Breaking in a Dephased Fermi Gas

    S. Wanget al., “Observation of Strong-to-Weak Spontaneous Symmetry Breaking in a Dephased Fermi Gas,” arXiv:2604.16137 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv

  80. [80]

    Thermal hall conductance and a relative topological invariant of gapped two-dimensional systems,

    A. Kapustin and L. Spodyneiko, “Thermal hall conductance and a relative topological invariant of gapped two-dimensional systems,”Physical Review B101no. 4, (Jan., 2020) . http://dx.doi.org/10.1103/PhysRevB.101.045137

    work page doi:10.1103/physrevb.101.045137 2020

Showing first 80 references.