pith. machine review for the scientific record. sign in

arxiv: 2604.23659 · v1 · submitted 2026-04-26 · 🪐 quant-ph · cond-mat.str-el

Recognition: unknown

Peak-valley mechanism for Hilbert space fragmentation

Jianlong Fu , Hoi Chun Po
Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:22 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords Hilbert space fragmentationergodicity breakingquantum spin chainsconserved quantitiespeak-valley mechanismmany-body localization
0
0 comments X

The pith

A local peak-valley rule assigns conserved labels that strongly fragment the Hilbert space of one-dimensional integer spin chains.

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

The paper introduces peak-valley fragmentation as a mechanism that imposes a simple local constraint on integer spin configurations. This constraint labels each basis state by the heights of peaks and depths of valleys in a geometric picture, turning those labels into emergent conserved quantities. The resulting sectors remain disconnected under the dynamics, so the system stays out of equilibrium even in the absence of disorder. The same rule recovers several known strong-Hilbert-space-fragmentation models and supplies a systematic way to build new ones for higher spins.

Core claim

The peak-valley mechanism works by defining a local rule that maps every computational-basis configuration of an integer spin chain to a tuple of integers recording the successive peak heights and valley depths. These integers function as good quantum numbers that are conserved by suitably chosen Hamiltonians, thereby partitioning the full Hilbert space into many disconnected fragments. The same geometric labeling accounts for existing strong-HSF examples and their variants, and it yields a constructive recipe for new fragmented models including those with higher-order fragmentation.

What carries the argument

The peak-valley (PV) labeling rule, which converts each spin configuration into a set of integers for alternating peak heights and valley depths that serve as emergent conserved quantities.

Load-bearing premise

That there exists a simple local rule capable of assigning emergent good quantum numbers from peak and valley geometry that remain exactly conserved under the dynamics of the Hamiltonians considered.

What would settle it

A Hamiltonian constructed to obey the peak-valley local rule but that still allows transitions between configurations carrying different peak-valley labels, producing thermalization instead of fragmentation.

Figures

Figures reproduced from arXiv: 2604.23659 by Hoi Chun Po, Jianlong Fu.

Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 view at source ↗
read the original abstract

Ergodicity breaking in isolated systems has emerged as an important frontier in the study of quantum many-body physics. While generic Hamiltonians are expected to obey the eigenstate thermalization hypothesis (ETH), recent studies on Hilbert space fragmentation (HSF) have revealed possible robust nonthermal behavior even in disorder-free systems. Although numerous models exhibiting strong HSF are already known, existing analyses are typically model dependent, and a general organizing principle remains elusive. In this work, we introduce a simple mechanism for achieving strong HSF in one-dimensional integer spin chains, which we term "peak-valley (PV) fragmentation". The key idea is to devise a simple local rule which ensures the spin states in the computational basis can be labeled by a set of emergent good quantum numbers corresponding to the heights and depths of alternating peaks and valleys in a geometrical representation. We demonstrate that some known examples of strong HSF models, as well as their variants which break the HSF property, can be understood within the framework of PV fragmentation. Our approach also enables systematic construction of new fragmented models in higher-spin systems, and allows us to identify higher-order HSF models.

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 / 2 minor

Summary. The paper introduces a 'peak-valley (PV) fragmentation' mechanism for achieving strong Hilbert space fragmentation (HSF) in one-dimensional integer spin chains. The key idea is a simple local rule that labels computational basis states by emergent good quantum numbers given by the heights and depths of alternating peaks and valleys in a geometrical representation. This framework is applied to re-interpret known strong HSF models and variants that break HSF, and to systematically construct new fragmented models in higher-spin systems while identifying higher-order HSF.

Significance. If the local rule is shown to strictly conserve the PV labels under the dynamics, the mechanism would supply a general organizing principle for strong HSF, addressing the current model-dependent state of the field. It enables both re-interpretation of existing examples and construction of new ones, including in higher spins. Credit is due for the conceptual simplicity of the geometrical labeling and for generating falsifiable new models that could be tested numerically.

major comments (2)
  1. [Definition of the PV rule and its conservation properties] The central claim requires that the local PV rule makes peak/valley heights and depths strictly conserved (no Hamiltonian matrix elements between different label sets). The abstract asserts invariance, but without an explicit operator-level proof or exhaustive check that no allowed local move alters a label, the strong-fragmentation conclusion remains unverified. This is load-bearing for both the re-interpreted models and the new higher-spin constructions.
  2. [New model constructions and higher-order HSF] For the newly constructed higher-spin Hamiltonians, the paper must demonstrate that the chosen local terms commute with the PV label operators or have vanishing matrix elements outside sectors. A single counter-example class of moves that changes a peak height while obeying the local constraint would invalidate the strong HSF claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'higher-order HSF models' without a concise definition; adding one sentence clarifying the distinction from standard strong HSF would improve readability.
  2. [Introduction or mechanism section] A figure illustrating the geometrical peak-valley representation and the local rule on a small chain (e.g., 4-6 sites) should appear early to make the labeling concrete for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments. We agree that establishing the strict conservation of the peak-valley labels under the dynamics is central to the claims of strong Hilbert space fragmentation. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and proofs.

read point-by-point responses

  1. Referee: [Definition of the PV rule and its conservation properties] The central claim requires that the local PV rule makes peak/valley heights and depths strictly conserved (no Hamiltonian matrix elements between different label sets). The abstract asserts invariance, but without an explicit operator-level proof or exhaustive check that no allowed local move alters a label, the strong-fragmentation conclusion remains unverified. This is load-bearing for both the re-interpreted models and the new higher-spin constructions.

    Authors: We thank the referee for identifying this gap. Section II defines the local PV rule and shows, for each re-interpreted model, that the allowed local moves preserve the assigned heights and depths. However, the manuscript does not contain a general operator-level proof that the labels are conserved for arbitrary dynamics obeying the rule. In the revised version we will add a dedicated subsection that proves conservation: any local spin configuration change permitted by the rule cannot alter a peak height or valley depth, because the geometrical definition of the labels is invariant under the allowed moves. This argument will be formulated at the level of the local constraint and will apply uniformly to the known models, their variants, and the new constructions. revision: yes

  2. Referee: [New model constructions and higher-order HSF] For the newly constructed higher-spin Hamiltonians, the paper must demonstrate that the chosen local terms commute with the PV label operators or have vanishing matrix elements outside sectors. A single counter-example class of moves that changes a peak height while obeying the local constraint would invalidate the strong HSF claim.

    Authors: We agree that explicit verification is required for the new higher-spin models. The constructions in the manuscript select local terms whose support is restricted to configurations that, by the PV rule, lie within a single label sector. To make this rigorous, the revision will include, for each new Hamiltonian, an analytical demonstration that every allowed local term has vanishing matrix elements between distinct PV sectors, together with a brief numerical check on small chains confirming the absence of counter-example moves. This will be presented in the section on new model constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the peak-valley fragmentation mechanism

full rationale

The paper constructs a local rule that assigns emergent good quantum numbers based on peak and valley heights in a geometrical spin representation, then verifies that this rule is conserved under the dynamics of both reinterpreted known models and newly constructed higher-spin Hamiltonians. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claim is an explicit constructive mechanism whose invariance is asserted and demonstrated on concrete examples rather than assumed tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests primarily on the postulated existence of a local rule that produces emergent conserved quantities from geometrical peak-valley features; this is the main addition beyond prior HSF literature.

axioms (1)
  • domain assumption Spin configurations in 1D integer spin chains admit a geometrical representation in which alternating peaks and valleys can be identified and labeled by a simple local rule.
    This is the foundational premise of the PV fragmentation mechanism as stated in the abstract.
invented entities (1)
  • Peak-valley (PV) fragmentation mechanism no independent evidence
    purpose: To assign emergent good quantum numbers that enforce strong Hilbert space fragmentation
    New conceptual framework introduced by the authors to organize and construct HSF models.

pith-pipeline@v0.9.0 · 5493 in / 1382 out tokens · 34921 ms · 2026-05-08T06:22:47.580330+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

81 extracted references · 80 canonical work pages

  1. [1]

    author author D. A. \ Abanin , author E. Altman , author I. Bloch , \ and\ author M. Serbyn ,\ 10.1103/RevModPhys.91.021001 journal journal Rev. Mod. Phys. \ volume 91 ,\ pages 021001 ( year 2019 ) NoStop

    work page doi:10.1103/revmodphys.91.021001 2019

  2. [2]

    Polkovnikov, K

    author author A. Polkovnikov , author K. Sengupta , author A. Silva , \ and\ author M. Vengalattore ,\ 10.1103/RevModPhys.83.863 journal journal Rev. Mod. Phys. \ volume 83 ,\ pages 863 ( year 2011 ) NoStop

    work page doi:10.1103/revmodphys.83.863 2011

  3. [3]

    Nandkishore and D

    author author R. Nandkishore \ and\ author D. A. \ Huse ,\ 10.1146/annurev-conmatphys-031214-014726 journal journal Annual Review of Condensed Matter Physics \ volume 6 ,\ pages 15 ( year 2015 ) ,\ http://arxiv.org/abs/https://doi.org/10.1146/annurev-conmatphys-031214-014726 https://doi.org/10.1146/annurev-conmatphys-031214-014726 NoStop

    work page doi:10.1146/annurev-conmatphys-031214-014726 2015

  4. [4]

    author author J. M. \ Deutsch ,\ 10.1103/PhysRevA.43.2046 journal journal Phys. Rev. A \ volume 43 ,\ pages 2046 ( year 1991 ) NoStop

    work page doi:10.1103/physreva.43.2046 2046

  5. [5]

    Srednicki,Chaos and quantum thermalization, Phys

    author author M. Srednicki ,\ 10.1103/PhysRevE.50.888 journal journal Phys. Rev. E \ volume 50 ,\ pages 888 ( year 1994 ) NoStop

    work page doi:10.1103/physreve.50.888 1994

  6. [6]

    Srednicki ,\ 10.1088/0305-4470/32/7/007 journal journal Journal of Physics A: Mathematical and General \ volume 32 ,\ pages 1163 ( year 1999 ) NoStop

    author author M. Srednicki ,\ 10.1088/0305-4470/32/7/007 journal journal Journal of Physics A: Mathematical and General \ volume 32 ,\ pages 1163 ( year 1999 ) NoStop

    work page doi:10.1088/0305-4470/32/7/007 1999

  7. [7]

    Kim , author T

    author author H. Kim , author T. N. \ Ikeda , \ and\ author D. A. \ Huse ,\ 10.1103/PhysRevE.90.052105 journal journal Phys. Rev. E \ volume 90 ,\ pages 052105 ( year 2014 ) NoStop

    work page doi:10.1103/physreve.90.052105 2014

  8. [8]

    Hild , author T

    author author S. Hild , author T. Fukuhara , author P. Schau , author J. Zeiher , author M. Knap , author E. Demler , author I. Bloch , \ and\ author C. Gross ,\ 10.1103/PhysRevLett.113.147205 journal journal Phys. Rev. Lett. \ volume 113 ,\ pages 147205 ( year 2014 ) NoStop

    work page doi:10.1103/physrevlett.113.147205 2014

  9. [9]

    Serbyn, D

    author author M. Serbyn , author D. Abanin , \ and\ author Z. Papić ,\ 10.1038/s41567-021-01230-2 journal journal Nature Physics \ volume 17 ,\ pages 675 ( year 2021 ) NoStop

    work page doi:10.1038/s41567-021-01230-2 2021

  10. [10]

    Schreiber, S

    author author M. Schreiber , author S. S. \ Hodgman , author P. Bordia , author H. P. \ Lüschen , author M. H. \ Fischer , author R. Vosk , author E. Altman , author U. Schneider , \ and\ author I. Bloch ,\ 10.1126/science.aaa7432 journal journal Science \ volume 349 ,\ pages 842 ( year 2015 ) ,\ http://arxiv.org/abs/https://www.science.org/doi/pdf/10.112...

    work page doi:10.1126/science.aaa7432 2015

  11. [11]

    \ Desaules , author A

    author author J.-Y. \ Desaules , author A. Hudomal , author C. J. \ Turner , \ and\ author Z. Papi c \' c ,\ 10.1103/PhysRevLett.126.210601 journal journal Phys. Rev. Lett. \ volume 126 ,\ pages 210601 ( year 2021 ) NoStop

    work page doi:10.1103/physrevlett.126.210601 2021

  12. [12]

    author author D. K. \ Mark , author C.-J. \ Lin , \ and\ author O. I. \ Motrunich ,\ 10.1103/PhysRevB.101.195131 journal journal Phys. Rev. B \ volume 101 ,\ pages 195131 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.195131 2020

  13. [13]

    Moudgalya, N

    author author S. Moudgalya , author N. Regnault , \ and\ author B. A. \ Bernevig ,\ 10.1103/PhysRevB.98.235156 journal journal Phys. Rev. B \ volume 98 ,\ pages 235156 ( year 2018 ) NoStop

    work page doi:10.1103/physrevb.98.235156 2018

  14. [14]

    Chandran, T

    author author A. Chandran , author T. Iadecola , author V. Khemani , \ and\ author R. Moessner ,\ https://doi.org/10.1146/annurev-conmatphys-031620-101617 journal journal Annual Review of Condensed Matter Physics \ volume 14 ,\ pages 443 ( year 2023 ) NoStop

    work page doi:10.1146/annurev-conmatphys-031620-101617 2023

  15. [15]

    Khemani , author M

    author author V. Khemani , author M. Hermele , \ and\ author R. Nandkishore ,\ 10.1103/PhysRevB.101.174204 journal journal Phys. Rev. B \ volume 101 ,\ pages 174204 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.174204 2020

  16. [16]

    Rakovszky , author P

    author author T. Rakovszky , author P. Sala , author R. Verresen , author M. Knap , \ and\ author F. Pollmann ,\ 10.1103/PhysRevB.101.125126 journal journal Phys. Rev. B \ volume 101 ,\ pages 125126 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.125126 2020

  17. [17]

    Moudgalya, B

    author author S. Moudgalya , author B. A. \ Bernevig , \ and\ author N. Regnault ,\ 10.1088/1361-6633/ac73a0 journal journal Reports on Progress in Physics \ volume 85 ,\ pages 086501 ( year 2022 ) NoStop

    work page doi:10.1088/1361-6633/ac73a0 2022

  18. [18]

    Sala , author T

    author author P. Sala , author T. Rakovszky , author R. Verresen , author M. Knap , \ and\ author F. Pollmann ,\ 10.1103/PhysRevX.10.011047 journal journal Phys. Rev. X \ volume 10 ,\ pages 011047 ( year 2020 ) NoStop

    work page doi:10.1103/physrevx.10.011047 2020

  19. [19]

    author author P. yd z \. z ba , author P. Prelov s s ek , \ and\ author M. Mierzejewski ,\ 10.1103/PhysRevLett.132.220405 journal journal Phys. Rev. Lett. \ volume 132 ,\ pages 220405 ( year 2024 ) NoStop

    work page doi:10.1103/physrevlett.132.220405 2024

  20. [20]

    \ Yang , author F

    author author Z.-C. \ Yang , author F. Liu , author A. V. \ Gorshkov , \ and\ author T. Iadecola ,\ 10.1103/PhysRevLett.124.207602 journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 207602 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.207602 2020

  21. [21]

    Brighi , author M

    author author P. Brighi , author M. Ljubotina , \ and\ author M. Serbyn ,\ 10.21468/SciPostPhys.15.3.093 journal journal SciPost Phys. \ volume 15 ,\ pages 093 ( year 2023 ) NoStop

    work page doi:10.21468/scipostphys.15.3.093 2023

  22. [22]

    Wang \ and\ author Z.-C

    author author C. Wang \ and\ author Z.-C. \ Yang ,\ 10.1103/PhysRevB.108.144308 journal journal Phys. Rev. B \ volume 108 ,\ pages 144308 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.108.144308 2023

  23. [23]

    Aditya , author D

    author author S. Aditya , author D. Dhar , \ and\ author D. Sen ,\ 10.1103/PhysRevB.110.045418 journal journal Phys. Rev. B \ volume 110 ,\ pages 045418 ( year 2024 ) NoStop

    work page doi:10.1103/physrevb.110.045418 2024

  24. [24]

    Aditya ,\ 10.1103/7j6x-74f1 journal journal Phys

    author author S. Aditya ,\ 10.1103/7j6x-74f1 journal journal Phys. Rev. B \ volume 112 ,\ pages 195413 ( year 2025 ) NoStop

    work page doi:10.1103/7j6x-74f1 2025

  25. [25]

    Khudorozhkov , author A

    author author A. Khudorozhkov , author A. Tiwari , author C. Chamon , \ and\ author T. Neupert ,\ 10.21468/SciPostPhys.13.4.098 journal journal SciPost Phys. \ volume 13 ,\ pages 098 ( year 2022 ) NoStop

    work page doi:10.21468/scipostphys.13.4.098 2022

  26. [26]

    Chattopadhyay , author B

    author author A. Chattopadhyay , author B. Mukherjee , author K. Sengupta , \ and\ author A. Sen ,\ 10.21468/SciPostPhys.14.6.146 journal journal SciPost Phys. \ volume 14 ,\ pages 146 ( year 2023 ) NoStop

    work page doi:10.21468/scipostphys.14.6.146 2023

  27. [27]

    Will , author R

    author author M. Will , author R. Moessner , \ and\ author F. Pollmann ,\ 10.1103/PhysRevLett.133.196301 journal journal Phys. Rev. Lett. \ volume 133 ,\ pages 196301 ( year 2024 ) NoStop

    work page doi:10.1103/physrevlett.133.196301 2024

  28. [28]

    Yoshinaga , author H

    author author A. Yoshinaga , author H. Hakoshima , author T. Imoto , author Y. Matsuzaki , \ and\ author R. Hamazaki ,\ 10.1103/PhysRevLett.129.090602 journal journal Phys. Rev. Lett. \ volume 129 ,\ pages 090602 ( year 2022 ) NoStop

    work page doi:10.1103/physrevlett.129.090602 2022

  29. [29]

    Nicolau , author A

    author author E. Nicolau , author A. M. \ Marques , author R. G. \ Dias , \ and\ author V. Ahufinger ,\ 10.1103/PhysRevB.108.205104 journal journal Phys. Rev. B \ volume 108 ,\ pages 205104 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.108.205104 2023

  30. [30]

    author author P. H. \ Harkema , author M. Iversen , \ and\ author A. E. B. \ Nielsen ,\ 10.1103/PhysRevA.110.023301 journal journal Phys. Rev. A \ volume 110 ,\ pages 023301 ( year 2024 ) NoStop

    work page doi:10.1103/physreva.110.023301 2024

  31. [31]

    Kohlert , author S

    author author T. Kohlert , author S. Scherg , author P. Sala , author F. Pollmann , author B. Hebbe Madhusudhana , author I. Bloch , \ and\ author M. Aidelsburger ,\ 10.1103/PhysRevLett.130.010201 journal journal Phys. Rev. Lett. \ volume 130 ,\ pages 010201 ( year 2023 ) NoStop

    work page doi:10.1103/physrevlett.130.010201 2023

  32. [32]

    Zhao , author P

    author author L. Zhao , author P. R. \ Datla , author W. Tian , author M. M. \ Aliyu , \ and\ author H. Loh ,\ 10.1103/PhysRevX.15.011035 journal journal Phys. Rev. X \ volume 15 ,\ pages 011035 ( year 2025 ) NoStop

    work page doi:10.1103/physrevx.15.011035 2025

  33. [33]

    \ Wang , author Y.-H

    author author Y.-Y. \ Wang , author Y.-H. \ Shi , author Z.-H. \ Sun , author C.-T. \ Chen , author Z.-A. \ Wang , author K. Zhao , author H.-T. \ Liu , author W.-G. \ Ma , author Z. Wang , author H. Li , author J.-C. \ Zhang , author Y. Liu , author C.-L. \ Deng , author T.-M. \ Li , author Y. He , author Z.-H. \ Liu , author Z.-Y. \ Peng , author X. Son...

    work page doi:10.1103/prxquantum.6.010325 2025

  34. [34]

    Adler , author D

    author author D. Adler , author D. Wei , author M. Will , \ and\ author et al. ,\ 10.1038/s41586-024-08188-0 journal journal Nature \ volume 636 ,\ pages 80–85 ( year 2024 ) NoStop

    work page doi:10.1038/s41586-024-08188-0 2024

  35. [35]

    Moudgalya and O

    author author S. Moudgalya \ and\ author O. I. \ Motrunich ,\ 10.1103/PhysRevX.12.011050 journal journal Phys. Rev. X \ volume 12 ,\ pages 011050 ( year 2022 ) NoStop

    work page doi:10.1103/physrevx.12.011050 2022

  36. [36]

    Moudgalya \ and\ author O

    author author S. Moudgalya \ and\ author O. I. \ Motrunich ,\ https://doi.org/10.1016/j.aop.2023.169384 journal journal Annals of Physics \ volume 455 ,\ pages 169384 ( year 2023 ) NoStop

    work page doi:10.1016/j.aop.2023.169384 2023

  37. [37]

    Moudgalya \ and\ author O

    author author S. Moudgalya \ and\ author O. I. \ Motrunich ,\ 10.1103/PhysRevX.14.041069 journal journal Phys. Rev. X \ volume 14 ,\ pages 041069 ( year 2024 ) NoStop

    work page doi:10.1103/physrevx.14.041069 2024

  38. [38]

    Sala , author J

    author author P. Sala , author J. Lehmann , author T. Rakovszky , \ and\ author F. Pollmann ,\ 10.1103/PhysRevLett.129.170601 journal journal Phys. Rev. Lett. \ volume 129 ,\ pages 170601 ( year 2022 ) NoStop

    work page doi:10.1103/physrevlett.129.170601 2022

  39. [39]

    Feldmeier , author P

    author author J. Feldmeier , author P. Sala , author G. De Tomasi , author F. Pollmann , \ and\ author M. Knap ,\ 10.1103/PhysRevLett.125.245303 journal journal Phys. Rev. Lett. \ volume 125 ,\ pages 245303 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.125.245303 2020

  40. [40]

    Hart , author A

    author author O. Hart , author A. Lucas , \ and\ author R. Nandkishore ,\ 10.1103/PhysRevE.105.044103 journal journal Phys. Rev. E \ volume 105 ,\ pages 044103 ( year 2022 ) NoStop

    work page doi:10.1103/physreve.105.044103 2022

  41. [41]

    Iaconis , author A

    author author J. Iaconis , author A. Lucas , \ and\ author R. Nandkishore ,\ 10.1103/PhysRevE.103.022142 journal journal Phys. Rev. E \ volume 103 ,\ pages 022142 ( year 2021 ) NoStop

    work page doi:10.1103/physreve.103.022142 2021

  42. [42]

    Quantum Glassiness,

    author author C. Chamon ,\ 10.1103/PhysRevLett.94.040402 journal journal Phys. Rev. Lett. \ volume 94 ,\ pages 040402 ( year 2005 ) NoStop

    work page doi:10.1103/physrevlett.94.040402 2005

  43. [43]

    author author R. M. \ Nandkishore \ and\ author M. Hermele ,\ 10.1146/annurev-conmatphys-031218-013604 journal journal Annual Review of Condensed Matter Physics \ volume 10 ,\ pages 295 ( year 2019 ) ,\ http://arxiv.org/abs/https://doi.org/10.1146/annurev-conmatphys-031218-013604 https://doi.org/10.1146/annurev-conmatphys-031218-013604 NoStop

    work page doi:10.1146/annurev-conmatphys-031218-013604 2019

  44. [44]

    Prem , author J

    author author A. Prem , author J. Haah , \ and\ author R. Nandkishore ,\ 10.1103/PhysRevB.95.155133 journal journal Phys. Rev. B \ volume 95 ,\ pages 155133 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.95.155133 2017

  45. [45]

    Fracton Topological Order, Generalized Lat- tice Gauge Theory and Duality,

    author author S. Vijay , author J. Haah , \ and\ author L. Fu ,\ 10.1103/PhysRevB.94.235157 journal journal Phys. Rev. B \ volume 94 ,\ pages 235157 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.94.235157 2016

  46. [46]

    Vijay , author J

    author author S. Vijay , author J. Haah , \ and\ author L. Fu ,\ 10.1103/PhysRevB.92.235136 journal journal Phys. Rev. B \ volume 92 ,\ pages 235136 ( year 2015 ) NoStop

    work page doi:10.1103/physrevb.92.235136 2015

  47. [47]

    Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids,

    author author M. Pretko ,\ 10.1103/PhysRevB.95.115139 journal journal Phys. Rev. B \ volume 95 ,\ pages 115139 ( year 2017 a ) NoStop

    work page doi:10.1103/physrevb.95.115139 2017

  48. [48]

    Pretko ,\ 10.1103/PhysRevB.96.125151 journal journal Phys

    author author M. Pretko ,\ 10.1103/PhysRevB.96.125151 journal journal Phys. Rev. B \ volume 96 ,\ pages 125151 ( year 2017 b ) NoStop

    work page doi:10.1103/physrevb.96.125151 2017

  49. [49]

    Fracton-Elasticity Duality,

    author author M. Pretko \ and\ author L. Radzihovsky ,\ 10.1103/PhysRevLett.120.195301 journal journal Phys. Rev. Lett. \ volume 120 ,\ pages 195301 ( year 2018 ) NoStop

    work page doi:10.1103/physrevlett.120.195301 2018

  50. [50]

    Morningstar , author V

    author author A. Morningstar , author V. Khemani , \ and\ author D. A. \ Huse ,\ 10.1103/PhysRevB.101.214205 journal journal Phys. Rev. B \ volume 101 ,\ pages 214205 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.214205 2020

  51. [51]

    author author C. D. \ Batista \ and\ author G. Ortiz ,\ 10.1103/PhysRevLett.85.4755 journal journal Phys. Rev. Lett. \ volume 85 ,\ pages 4755 ( year 2000 ) NoStop

    work page doi:10.1103/physrevlett.85.4755 2000

  52. [52]

    author author C. D. \ Batista \ and\ author G. Ortiz ,\ 10.1103/PhysRevLett.86.1082 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 1082 ( year 2001 ) NoStop

    work page doi:10.1103/physrevlett.86.1082 2001

  53. [53]

    Fredkin \ and\ author T

    author author E. Fredkin \ and\ author T. Toffoli ,\ 10.1007/BF01857727 journal journal Int. J. Theor. Phys. \ volume 21 ,\ pages 219–253 ( year 1982 ) NoStop

    work page doi:10.1007/bf01857727 1982

  54. [54]

    author author C. M. \ Langlett \ and\ author S. Xu ,\ 10.1103/PhysRevB.103.L220304 journal journal Phys. Rev. B \ volume 103 ,\ pages L220304 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.l220304 2021

  55. [55]

    Salberger \ and\ author V

    author author O. Salberger \ and\ author V. Korepin ,\ 10.1142/S0129055X17500313 journal journal Reviews in Mathematical Physics \ volume 29 ,\ pages 1750031 ( year 2017 ) ,\ http://arxiv.org/abs/https://doi.org/10.1142/S0129055X17500313 https://doi.org/10.1142/S0129055X17500313 NoStop

    work page doi:10.1142/s0129055x17500313 2017

  56. [56]

    Bravyi , author L

    author author S. Bravyi , author L. Caha , author R. Movassagh , author D. Nagaj , \ and\ author P. W. \ Shor ,\ 10.1103/PhysRevLett.109.207202 journal journal Phys. Rev. Lett. \ volume 109 ,\ pages 207202 ( year 2012 ) NoStop

    work page doi:10.1103/physrevlett.109.207202 2012

  57. [57]

    Zhang , author A

    author author Z. Zhang , author A. Ahmadain , \ and\ author I. Klich ,\ 10.1073/pnas.1702029114 journal journal Proceedings of the National Academy of Sciences \ volume 114 ,\ pages 5142 ( year 2017 ) ,\ http://arxiv.org/abs/https://www.pnas.org/doi/pdf/10.1073/pnas.1702029114 https://www.pnas.org/doi/pdf/10.1073/pnas.1702029114 NoStop

    work page doi:10.1073/pnas.1702029114 2017

  58. [58]

    Barbiero , author L

    author author L. Barbiero , author L. Dell'Anna , author A. Trombettoni , \ and\ author V. E. \ Korepin ,\ 10.1103/PhysRevB.96.180404 journal journal Phys. Rev. B \ volume 96 ,\ pages 180404 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.96.180404 2017

  59. [59]

    Richter \ and\ author A

    author author J. Richter \ and\ author A. Pal ,\ 10.1103/PhysRevResearch.4.L012003 journal journal Phys. Rev. Res. \ volume 4 ,\ pages L012003 ( year 2022 ) NoStop

    work page doi:10.1103/physrevresearch.4.l012003 2022

  60. [60]

    Jeyaretnam, T

    author author J. Jeyaretnam , author T. Bhore , author J. J. \ Osborne , author J. C. \ Halimeh , \ and\ author Z. Papi \'c ,\ 10.1038/s42005-025-02039-8 journal journal Communications Physics \ volume 8 ,\ pages 172 ( year 2025 ) NoStop

    work page doi:10.1038/s42005-025-02039-8 2025

  61. [61]

    Li , author P

    author author Y. Li , author P. Sala , \ and\ author F. Pollmann ,\ 10.1103/PhysRevResearch.5.043239 journal journal Phys. Rev. Res. \ volume 5 ,\ pages 043239 ( year 2023 ) NoStop

    work page doi:10.1103/physrevresearch.5.043239 2023

  62. [62]

    Hahn , author P

    author author D. Hahn , author P. A. \ McClarty , \ and\ author D. J. \ Luitz ,\ 10.21468/SciPostPhys.11.4.074 journal journal SciPost Phys. \ volume 11 ,\ pages 074 ( year 2021 ) NoStop

    work page doi:10.21468/scipostphys.11.4.074 2021

  63. [63]

    Patil , author A

    author author P. Patil , author A. Singhania , \ and\ author J. C. \ Halimeh ,\ 10.1103/PhysRevB.108.195109 journal journal Phys. Rev. B \ volume 108 ,\ pages 195109 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.108.195109 2023

  64. [64]

    Anyons in an exactly solved model and beyond

    author author A. Kitaev ,\ http://dx.doi.org/10.1016/j.aop.2005.10.005 journal journal Annals of Physics \ volume 321 ,\ pages 2 ( year 2006 ) NoStop

    work page Pith review doi:10.1016/j.aop.2005.10.005 2005

  65. [65]

    Yao \ and\ author S

    author author H. Yao \ and\ author S. A. \ Kivelson ,\ 10.1103/PhysRevLett.99.247203 journal journal Phys. Rev. Lett. \ volume 99 ,\ pages 247203 ( year 2007 ) NoStop

    work page doi:10.1103/physrevlett.99.247203 2007

  66. [66]

    Fu ,\ 10.1103/PhysRevB.106.L161412 journal journal Phys

    author author J. Fu ,\ 10.1103/PhysRevB.106.L161412 journal journal Phys. Rev. B \ volume 106 ,\ pages L161412 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.106.l161412 2022

  67. [67]

    Affleck, T

    author author I. Affleck , author T. Kennedy , author E. H. \ Lieb , \ and\ author H. Tasaki ,\ 10.1103/PhysRevLett.59.799 journal journal Phys. Rev. Lett. \ volume 59 ,\ pages 799 ( year 1987 ) NoStop

    work page doi:10.1103/physrevlett.59.799 1987

  68. [68]

    author author H. A. \ Kramers \ and\ author G. H. \ Wannier ,\ 10.1103/PhysRev.60.252 journal journal Phys. Rev. \ volume 60 ,\ pages 252 ( year 1941 ) NoStop

    work page doi:10.1103/physrev.60.252 1941

  69. [69]

    author author J. B. \ Kogut ,\ 10.1103/RevModPhys.51.659 journal journal Rev. Mod. Phys. \ volume 51 ,\ pages 659 ( year 1979 ) NoStop

    work page doi:10.1103/revmodphys.51.659 1979

  70. [70]

    Über das paulische äquivalenzverbot.Zeitschrift für Physik, 47(9–10): 631–651, 1928.doi:10.1007/BF01331938

    author author P. Jordan \ and\ author E. Wigner ,\ 10.1007/BF01331938 journal journal Zeitschrift f \"u r Physik \ volume 47 ,\ pages 631 ( year 1928 ) NoStop

    work page doi:10.1007/bf01331938 1928

  71. [71]

    Lieb , author T

    author author E. Lieb , author T. Schultz , \ and\ author D. Mattis ,\ https://doi.org/10.1016/0003-4916(61)90115-4 journal journal Annals of Physics \ volume 16 ,\ pages 407 ( year 1961 ) NoStop

    work page doi:10.1016/0003-4916(61)90115-4 1961

  72. [72]

    Pozderac , author S

    author author C. Pozderac , author S. Speck , author X. Feng , author D. A. \ Huse , \ and\ author B. Skinner ,\ 10.1103/PhysRevB.107.045137 journal journal Phys. Rev. B \ volume 107 ,\ pages 045137 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.107.045137 2023

  73. [73]

    and Vedral, V

    author author L. Amico , author R. Fazio , author A. Osterloh , \ and\ author V. Vedral ,\ 10.1103/RevModPhys.80.517 journal journal Rev. Mod. Phys. \ volume 80 ,\ pages 517 ( year 2008 ) NoStop

    work page doi:10.1103/revmodphys.80.517 2008

  74. [74]

    Bauer \ and\ author C

    author author B. Bauer \ and\ author C. Nayak ,\ 10.1088/1742-5468/2013/09/P09005 journal journal Journal of Statistical Mechanics: Theory and Experiment \ volume 2013 ,\ pages P09005 ( year 2013 ) NoStop

    work page doi:10.1088/1742-5468/2013/09/p09005 2013

  75. [75]

    author author D. J. \ Luitz ,\ 10.1103/PhysRevB.93.134201 journal journal Phys. Rev. B \ volume 93 ,\ pages 134201 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.93.134201 2016

  76. [76]

    Yu , author D

    author author X. Yu , author D. J. \ Luitz , \ and\ author B. K. \ Clark ,\ 10.1103/PhysRevB.94.184202 journal journal Phys. Rev. B \ volume 94 ,\ pages 184202 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.94.184202 2016

  77. [77]

    Herviou , author J

    author author L. Herviou , author J. H. \ Bardarson , \ and\ author N. Regnault ,\ 10.1103/PhysRevB.103.134207 journal journal Phys. Rev. B \ volume 103 ,\ pages 134207 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.134207 2021

  78. [78]

    Francica \ and\ author L

    author author G. Francica \ and\ author L. Dell'Anna ,\ 10.1103/PhysRevB.108.045127 journal journal Phys. Rev. B \ volume 108 ,\ pages 045127 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.108.045127 2023

  79. [79]

    Local stabilizer codes in three dimensions without string logical operators,

    author author J. Haah ,\ 10.1103/PhysRevA.83.042330 journal journal Phys. Rev. A \ volume 83 ,\ pages 042330 ( year 2011 ) NoStop

    work page doi:10.1103/physreva.83.042330 2011

  80. [80]

    Lehmann , author P

    author author J. Lehmann , author P. S. \ de Torres-Solanot , author F. Pollmann , \ and\ author T. Rakovszky ,\ 10.21468/SciPostPhys.14.6.140 journal journal SciPost Phys. \ volume 14 ,\ pages 140 ( year 2023 ) NoStop

    work page doi:10.21468/scipostphys.14.6.140 2023

Showing first 80 references.