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arxiv: 2510.23701 · v1 · submitted 2025-10-27 · ❄️ cond-mat.str-el · hep-th· quant-ph

Onsiteability of Higher-Form Symmetries

Yitao Feng , Yu-An Chen , Po-Shen Hsin , Ryohei Kobayashi This is my paper

Pith reviewed 2026-05-18 03:14 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thquant-ph
keywords higher-form symmetryonsiteabilityt Hooft anomalyhigher gauginglattice modelsquantum many-body systemsanomalies
0
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The pith

A higher-form symmetry is onsiteable exactly when its lattice 't Hooft anomaly allows higher gauging.

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

The paper asks when an internal higher-form symmetry on a lattice can be made to act entirely onsite by adding ancilla sites and conjugating with a finite-depth circuit. For ordinary zero-form symmetries the answer coincides with anomaly freedom, but the two notions diverge for higher-form symmetries. The authors prove that onsiteability is equivalent to the possibility of performing higher gauging. For a finite one-form symmetry in two spatial dimensions this equivalence reduces to a concrete algebraic condition on the symmetry's lattice 't Hooft anomaly. They further show that any symmetry satisfying the condition can be rewritten as a set of transversal Pauli operators.

Core claim

For a finite 1-form symmetry in (2+1)D, the symmetry is onsiteable if and only if its 't Hooft anomaly satisfies a specific algebraic condition that ensures the symmetry can be 1-gauged. Onsiteable 1-form symmetries can always be brought into transversal Pauli operators by ancillas and circuit conjugation. In generic dimensions, lattice 't Hooft anomalies of higher-form symmetries supply necessary conditions for onsiteability, supporting the conjecture that onsiteability is equivalent to the possibility of higher gauging on the lattice.

What carries the argument

the algebraic condition on the lattice 't Hooft anomaly that permits 1-gauging of a 1-form symmetry

If this is right

  • Any onsiteable 1-form symmetry in (2+1)D admits a representation as transversal Pauli operators after ancilla addition.
  • The obstruction to onsiteability is completely determined by whether higher gauging is possible.
  • Necessary conditions for onsiteability in higher dimensions are read off directly from the lattice anomaly.
  • The equivalence supplies a practical test for whether a given higher-form symmetry can be realized locally on a lattice.

Where Pith is reading between the lines

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

  • Lattice models realizing anomalous yet onsiteable higher symmetries may now be constructed systematically by enforcing the algebraic gauging condition.
  • The same distinction between onsiteability and anomaly freedom could reappear in the classification of symmetry-enriched topological phases.
  • Numerical checks on small lattices with known anomalies would provide direct tests of the algebraic condition.

Load-bearing premise

The lattice 't Hooft anomaly is assumed to capture every obstruction to onsiteability, with no extra lattice-specific effects that survive the continuum limit.

What would settle it

A (2+1)D lattice model whose 1-form symmetry obeys the algebraic anomaly condition for 1-gauging yet cannot be made onsite by any choice of ancillas and finite-depth circuit would disprove the claimed equivalence.

Figures

Figures reproduced from arXiv: 2510.23701 by Po-Shen Hsin, Ryohei Kobayashi, Yitao Feng, Yu-An Chen.

Figure 1
Figure 1. Figure 1: (a): Symmetry operators are defined on a mesoscopic dual lattice [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A cycle ˆγ of the dual lattice separates the disk region R into half, up (u) and down (d) region. The remaining task is to show that the above [ω3] ∈ H3 (BG,U(1)) is a transgression of the anomaly index [ω4] ∈ H4 (B2G,U(1)). Let us first consider the 1-cycle of the dual lattice ˆγ which cuts R into a bipartition. See [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) By introducing a 1d ancilla and disentangler on each mesoscopic edge, the Gauss law operator [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For (3+1)D G 1-form symmetry, the operator Ω(ϵ01, ϵ12, g012) is a network of 1d QCAs supported at the 2d dual lattice Λˆ ∂R. Each red edge in the figure carries a 1d QCA, therefore assigns a QCA index valued in Q+. This assignment defines a 1-cocycle F ∈ Z 1 (Λ∂R, Q+). Each line operator of Ω(ϵ01, ϵ12, g012) on an edge of Λˆ ∂R carries a QCA index valued in Q+. This assignment of the QCA index on each dual… view at source ↗
read the original abstract

An internal symmetry in a lattice model is said to be onsiteable if it can be disentangled into an onsite action by introducing ancillas and conjugating with a finite-depth circuit. A standard lore holds that onsiteability is equivalent to being anomaly-free, which is indeed valid for finite 0-form symmetries in (1+1)D. However, for higher-form symmetries, these notions become inequivalent: a symmetry may be onsite while still anomalous. In this work, we clarify the conditions for onsiteability of higher-form symmetries by proposing an equivalence between onsiteability and the possibility of $higher$ gauging. For a finite 1-form symmetry in (2+1)D, we show that the symmetry is onsiteable if and only if its 't Hooft anomaly satisfies a specific algebraic condition that ensures the symmetry can be 1-gauged. We further demonstrate that onsiteable 1-form symmetry in (2+1)D can always be brought into transversal Pauli operators by ancillas and circuit conjugation. In generic dimensions, we derive necessary conditions for onsiteability using lattice 't Hooft anomaly of higher-form symmetry, and conjecture a general equivalence between onsiteability and possibility of higher gauging on lattices.

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 defines onsiteability of a lattice symmetry as the existence of ancillas and a finite-depth circuit that renders the symmetry action strictly onsite. It argues that for higher-form symmetries this is inequivalent to anomaly-freeness and instead proposes an equivalence between onsiteability and the possibility of higher gauging. For finite 1-form symmetries in (2+1)D the central claim is an if-and-only-if statement: the symmetry is onsiteable precisely when its lattice 't Hooft anomaly satisfies an algebraic condition permitting 1-gauging. The authors further show that any such onsiteable 1-form symmetry can be realized by transversal Pauli operators after circuit conjugation, derive necessary anomaly conditions in higher dimensions, and conjecture the general equivalence on lattices.

Significance. If the claimed equivalence is established, the work supplies a concrete lattice criterion linking higher-form anomalies to the possibility of gauging and onsite realization, which is directly relevant to the classification of symmetry-enriched topological phases and to the design of fault-tolerant gates in quantum error-correcting codes. The explicit transversal-Pauli construction for (2+1)D 1-form symmetries is a useful technical result; the conjecture for generic dimensions identifies a clear direction for subsequent work.

major comments (2)
  1. [§4] §4 (proof of the 'if' direction): the construction that converts an anomaly satisfying the 1-gauging condition into a finite-depth circuit plus ancillas is presented, but the argument assumes without independent verification that no additional lattice obstructions (e.g., support constraints on the symmetry operators or entanglement patterns outside the cocycle data) survive. An explicit check or counter-example family would be needed to confirm that the anomaly condition is sufficient.
  2. [§5] §5 (converse direction and general dimensions): the derivation of necessary conditions from the lattice 't Hooft anomaly is given, yet the manuscript does not exhibit a separate argument or numerical test showing that onsiteability cannot be obstructed by effects not captured by the anomaly. This assumption is load-bearing for the conjectured general equivalence.
minor comments (2)
  1. [§3] Notation for the lattice cocycle and the precise algebraic condition for 1-gauging should be stated explicitly in a single displayed equation early in §3 to improve readability.
  2. [Figure 2] Figure 2 (circuit diagram) would benefit from an accompanying caption that lists the depth and ancilla count used in the transversal-Pauli construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below. Revisions have been made to clarify the arguments and strengthen the presentation where appropriate.

read point-by-point responses

  1. Referee: [§4] §4 (proof of the 'if' direction): the construction that converts an anomaly satisfying the 1-gauging condition into a finite-depth circuit plus ancillas is presented, but the argument assumes without independent verification that no additional lattice obstructions (e.g., support constraints on the symmetry operators or entanglement patterns outside the cocycle data) survive. An explicit check or counter-example family would be needed to confirm that the anomaly condition is sufficient.

    Authors: We thank the referee for this observation. The construction in §4 is formulated directly from the algebraic 1-gauging condition on the anomaly cocycle and defines the ancilla degrees of freedom and the finite-depth circuit in a manner that is independent of further lattice-specific details. Because onsiteability is defined precisely as the existence of such a circuit that renders the symmetry action strictly onsite, and the gauging condition guarantees that the symmetry operators remain consistent after conjugation, the construction accounts for support and entanglement requirements through the cocycle data alone. To provide additional explicit verification as suggested, we have added a concrete illustrative example in the revised manuscript for a finite 1-form symmetry satisfying the condition, walking through the circuit construction and confirming that no extraneous obstructions appear. revision: yes

  2. Referee: [§5] §5 (converse direction and general dimensions): the derivation of necessary conditions from the lattice 't Hooft anomaly is given, yet the manuscript does not exhibit a separate argument or numerical test showing that onsiteability cannot be obstructed by effects not captured by the anomaly. This assumption is load-bearing for the conjectured general equivalence.

    Authors: In the converse direction for (2+1)D, necessity follows because any finite-depth circuit together with ancillas preserves the lattice 't Hooft anomaly; hence onsiteability can hold only when the anomaly admits 1-gauging. In higher dimensions we likewise extract the necessary anomaly conditions that must be satisfied for onsiteability. We agree that a fully general proof that these conditions are also sufficient (i.e., that the anomaly captures all possible obstructions) would complete the conjecture, and we have expanded the discussion in the revised §5 to make this scope and the underlying assumptions explicit. A complete sufficiency argument or numerical test in generic dimensions lies beyond the present work and is left as an open direction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; equivalence derived from anomaly data as independent input

full rationale

The paper defines onsiteability via finite-depth circuit plus ancillas and defines the lattice 't Hooft anomaly separately via cocycle/projective representation data. It then proves an iff equivalence to an algebraic condition permitting 1-gauging for (2+1)D 1-form symmetries, with explicit demonstration that onsiteable symmetries become transversal Pauli operators. This is distinguished from the 0-form lore and labeled as a new result. Necessary conditions are derived in higher dimensions with an explicit conjecture for the general case. No step reduces by construction to its own inputs, no fitted parameter is renamed as prediction, and no load-bearing self-citation chain is invoked; the anomaly supplies external data against which the equivalence is checked.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions of higher-form symmetries, 't Hooft anomalies on lattices, and finite-depth circuits; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Finite higher-form symmetries on lattices admit a well-defined 't Hooft anomaly that can be expressed algebraically.
    Invoked when stating the if-and-only-if condition for 1-form symmetries in (2+1)D.
  • ad hoc to paper Onsiteability is equivalent to the possibility of higher gauging.
    Proposed as the clarifying equivalence in the abstract.

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 3 Pith papers · 2 internal anchors

  1. [1]

    Physical Review Letters 102(11), doi:10.1103/physrevlett.102.110502

    B. Eastin and E. Knill, “Restrictions on transversal encoded quantum gate sets,” Physical Review Letters102no. 11, (Mar., 2009) .http://dx.doi.org/10.1103/PhysRevLett.102.110502

    work page doi:10.1103/physrevlett.102.110502 2009

  2. [2]

    Disentangling anomaly-free symmetries of quantum spin chains,

    S. Seifnashri and W. Shirley, “Disentangling anomaly-free symmetries of quantum spin chains,” arXiv:2503.09717 [cond-mat.str-el].https://arxiv.org/abs/2503.09717

    work page arXiv

  3. [3]

    Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems,

    K. Kawagoe and W. Shirley, “Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems,”arXiv:2507.07430 [cond-mat.str-el].https://arxiv.org/abs/2507.07430

    work page arXiv

  4. [4]

    Higher symmetries and anomalies in quantum lattice systems,

    A. Kapustin and S. Xu, “Higher symmetries and anomalies in quantum lattice systems,” arXiv:2505.04719 [math-ph].https://arxiv.org/abs/2505.04719

    work page arXiv

  5. [5]

    Higher symmetries, anomalies, and crossed squares in lattice gauge theory,

    A. Kapustin and L. Spodyneiko, “Higher symmetries, anomalies, and crossed squares in lattice gauge theory,”arXiv:2507.16966 [hep-th].https://arxiv.org/abs/2507.16966. 20

    work page arXiv

  6. [6]

    Anomaly-free symmetries with obstructions to gauging and onsiteability,

    W. Shirley, C. Zhang, W. Ji, and M. Levin, “Anomaly-free symmetries with obstructions to gauging and onsiteability,”arXiv:2507.21267 [cond-mat.str-el].https://arxiv.org/abs/2507.21267

    work page arXiv

  7. [7]

    Anomalies of Global Symmetries on the Lattice,

    Y.-T. Tu, D. M. Long, and D. V. Else, “Anomalies of global symmetries on the lattice,” arXiv:2507.21209 [cond-mat.str-el].https://arxiv.org/abs/2507.21209

    work page arXiv

  8. [8]

    doi: 10.1103/physrevb.82.155138

    X. Chen, Z.-C. Gu, and X.-G. Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys. Rev. B82(Oct, 2010) 155138. https://link.aps.org/doi/10.1103/PhysRevB.82.155138

    work page doi:10.1103/physrevb.82.155138 2010

  9. [9]

    Partition function of the Eight-Vertex lattice model

    E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics16no. 3, (1961) 407 – 466. http://www.sciencedirect.com/science/article/pii/0003491661901154

    work page arXiv 1961

  10. [10]

    Commensurability, excitation gap, and t opology in quantum many-particle systems on a periodic lattice,

    M. Oshikawa, “Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice,” Phys. Rev. Lett.84(Feb, 2000) 1535–1538. https://link.aps.org/doi/10.1103/PhysRevLett.84.1535

    work page doi:10.1103/physrevlett.84.1535 2000

  11. [11]

    Lieb-schultz-mattis in higher dimens ions,

    M. B. Hastings, “Lieb-schultz-mattis in higher dimensions,” Phys. Rev. B69(Mar, 2004) 104431. https://link.aps.org/doi/10.1103/PhysRevB.69.104431

    work page doi:10.1103/physrevb.69.104431 2004

  12. [12]

    Anomalous symmetries of quantum spin chains and a generalization of the lieb–schultz–mattis theorem,

    A. Kapustin and N. Sopenko, “Anomalous symmetries of quantum spin chains and a generalization of the lieb–schultz–mattis theorem,” Communications in Mathematical Physics406no. 10, (2025) 238. https://doi.org/10.1007/s00220-025-05422-2

    work page doi:10.1007/s00220-025-05422-2 2025

  13. [13]

    Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface,

    M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, “Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface,” Phys. Rev. X6(Dec, 2016) 041068.https://link.aps.org/doi/10.1103/PhysRevX.6.041068

    work page doi:10.1103/physrevx.6.041068 2016

  14. [14]

    Anomaly manifestation of lieb-schultz-mattis theorem and topological phases,

    G. Y. Cho, C.-T. Hsieh, and S. Ryu, “Anomaly manifestation of lieb-schultz-mattis theorem and topological phases,” Phys. Rev. B96(Nov, 2017) 195105. https://link.aps.org/doi/10.1103/PhysRevB.96.195105

    work page doi:10.1103/physrevb.96.195105 2017

  15. [15]

    Lieb-schultz-mattis type theorem with higher-form symmetry and the quantum dimer models,

    R. Kobayashi, K. Shiozaki, Y. Kikuchi, and S. Ryu, “Lieb-schultz-mattis type theorem with higher-form symmetry and the quantum dimer models,” Physical Review B99no. 1, (Jan., 2019) . http://dx.doi.org/10.1103/PhysRevB.99.014402

    work page doi:10.1103/physrevb.99.014402 2019

  16. [16]

    Topological theory of lieb-schultz-mattis theorems in quantum spin systems,

    D. V. Else and R. Thorngren, “Topological theory of lieb-schultz-mattis theorems in quantum spin systems,” Phys. Rev. B101(Jun, 2020) 224437. https://link.aps.org/doi/10.1103/PhysRevB.101.224437

    work page doi:10.1103/physrevb.101.224437 2020

  17. [17]

    Lieb-schultz-mattis–type constraints on fractonic matter,

    H. He, Y. You, and A. Prem, “Lieb-schultz-mattis–type constraints on fractonic matter,” Phys. Rev. B101(Apr, 2020) 165145.https://link.aps.org/doi/10.1103/PhysRevB.101.165145

    work page doi:10.1103/physrevb.101.165145 2020

  18. [18]

    Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,

    M. Cheng and N. Seiberg, “Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,” SciPost Phys.15(2023) 051.https://scipost.org/10.21468/SciPostPhys.15.2.051

    work page doi:10.21468/scipostphys.15.2.051 2023

  19. [20]

    Anomaly constraints on deconfinement and chiral phase transition,

    H. Shimizu and K. Yonekura, “Anomaly constraints on deconfinement and chiral phase transition,” Phys. Rev. D97(May, 2018) 105011.https://link.aps.org/doi/10.1103/PhysRevD.97.105011

    work page doi:10.1103/physrevd.97.105011 2018

  20. [21]

    Symmetry-enriched quantum spin liquids in (3 + 1)d,

    P.-S. Hsin and A. Turzillo, “Symmetry-enriched quantum spin liquids in (3 + 1)d,” Journal of High Energy Physics2020no. 9, (Sept., 2020) 22

    work page 2020

  21. [22]

    Symmetries and strings of adjoint QCD2,

    Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, “Symmetries and strings of adjoint QCD2,” Journal of High Energy Physics2021no. 3, (Mar., 2021) 103. 21

    work page 2021

  22. [23]

    Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations,

    X. Chen, Z.-X. Liu, and X.-G. Wen, “Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations,” Phys. Rev. B84(Dec, 2011) 235141. https://link.aps.org/doi/10.1103/PhysRevB.84.235141

    work page doi:10.1103/physrevb.84.235141 2011

  23. [24]

    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. B87(Apr, 2013) 155114. https://link.aps.org/doi/10.1103/PhysRevB.87.155114

    work page doi:10.1103/physrevb.87.155114 2013

  24. [26]

    Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging,

    A. Tiwari, X. Chen, K. Shiozaki, and S. Ryu, “Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging,” Phys. Rev. B97(Jun,

    work page

  25. [27]

    245133.https://link.aps.org/doi/10.1103/PhysRevB.97.245133

    work page doi:10.1103/physrevb.97.245133

  26. [28]

    Mixed-state quantum anomaly and multipartite entan- glement,

    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

  27. [29]

    How much entanglement is needed for topological codes and mixed states with anomalous symmetry?,

    Z. Li, D. Lee, and B. Yoshida, “How much entanglement is needed for topological codes and mixed states with anomalous symmetry?,” Phys. Rev. X15(Jun, 2025) 021090. https://link.aps.org/doi/10.1103/pw12-kdjx

    work page doi:10.1103/pw12-kdjx 2025

  28. [30]

    Higher-form anomaly and long-range entanglement of mixed states,

    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 e-prints (Mar., 2025) arXiv:2503.12792,arXiv:2503.12792 [quant-ph]

    work page arXiv 2025

  29. [31]

    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,” arXiv e-prints (Mar., 2025) arXiv:2503.02928, arXiv:2503.02928 [cond-mat.str-el]

    work page arXiv 2025

  30. [32]

    Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement,

    P.-S. Hsin, R. Kobayashi, and A. Prem, “Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement,”arXiv:2504.10569 [quant-ph]

    work page arXiv

  31. [33]

    Higher-Form Anomalies on Lattices,

    Y. Feng, R. Kobayashi, Y.-A. Chen, and S. Ryu, “Higher-form anomalies on lattices,” arXiv:2509.12304 [cond-mat.str-el].https://arxiv.org/abs/2509.12304

    work page arXiv

  32. [34]

    Generalized statistics on lattices,

    R. Kobayashi, Y. Li, H. Xue, P.-S. Hsin, and Y.-A. Chen, “Generalized statistics on lattices,” arXiv:2412.01886 [quant-ph].https://arxiv.org/abs/2412.01886

    work page arXiv

  33. [35]

    Statistics of invertible topological excitations,

    H. Xue, “Statistics of invertible topological excitations,”arXiv:2412.07653 [quant-ph]. https://arxiv.org/abs/2412.07653

    work page arXiv

  34. [36]

    Anyonic membranes and pontryagin statistics,

    Y. Feng, H. Xue, Y. Li, M. Cheng, R. Kobayashi, P.-S. Hsin, and Y.-A. Chen, “Anyonic membranes and pontryagin statistics,”arXiv:2509.14314 [quant-ph].https://arxiv.org/abs/2509.14314

    work page arXiv

  35. [37]

    Higher Gauging and Non-invertible Condensation Defects

    K. Roumpedakis, S. Seifnashri, and S.-H. Shao, “Higher gauging and non-invertible condensation defects,” Communications in Mathematical Physics401no. 3, (May, 2023) 3043–3107. http://dx.doi.org/10.1007/s00220-023-04706-9

    work page doi:10.1007/s00220-023-04706-9 2023

  36. [38]

    Topological Gauge Theories and Group Cohomology,

    R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,” Commun. Math. Phys.129(1990) 393

    work page 1990

  37. [39]

    Lattice models that realizeZ n-1 symmetry-protected topological states for evenn,

    L. Tsui and X.-G. Wen, “Lattice models that realizeZ n-1 symmetry-protected topological states for evenn,” Phys. Rev. B101no. 3, (2020) 035101,arXiv:1908.02613 [cond-mat.str-el]

    work page arXiv 2020

  38. [40]

    Index theory of one dimensional quantum walks and cellular automata,

    D. Gross, V. Nesme, H. Vogts, and R. F. Werner, “Index theory of one dimensional quantum walks and cellular automata,” Communications in Mathematical Physics310no. 2, (Jan, 2012) 419–454. http://dx.doi.org/10.1007/s00220-012-1423-1. 22

    work page doi:10.1007/s00220-012-1423-1 2012

  39. [41]

    Nontrivial quantum cellular automata in higher dimensions,

    J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial quantum cellular automata in higher dimensions,” Communications in Mathematical Physics398no. 1, (2023) 469–540. https://doi.org/10.1007/s00220-022-04528-1

    work page doi:10.1007/s00220-022-04528-1 2023

  40. [42]

    Clifford quantum cellular automata: Trivial group in 2d and witt group in 3d,

    J. Haah, “Clifford quantum cellular automata: Trivial group in 2d and witt group in 3d,” Journal of Mathematical Physics62no. 9, (09, 2021) 092202.https://doi.org/10.1063/5.0022185

    work page doi:10.1063/5.0022185 2021

  41. [43]

    PRX Quantum3, 030326 (2022) https://doi.org/10.1103/PRXQuantum.3.030326

    W. Shirley, Y.-A. Chen, A. Dua, T. D. Ellison, N. Tantivasadakarn, and D. J. Williamson, “Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond,” PRX Quantum3(Aug, 2022) 030326. https://link.aps.org/doi/10.1103/PRXQuantum.3.030326

    work page doi:10.1103/prxquantum.3.030326 2022

  42. [44]

    A quantum cellular automaton for every symmetry protected topological phase,

    L. Fidkowski, J. Haah, and M. B. Hastings, “A quantum cellular automaton for every symmetry protected topological phase,” Phys. Rev. B112(Jul, 2025) 035123. https://link.aps.org/doi/10.1103/kw68-mkkd

    work page doi:10.1103/kw68-mkkd 2025

  43. [45]

    Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras

    M. Sun, B. Yang, Z. Wang, N. Tantivasadakarn, and Y.-A. Chen, “Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras,”arXiv:2509.07099 [math.QA]. https://arxiv.org/abs/2509.07099

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [46]

    Kramers-wannier-like duality defects in (3+1)d gauge theories,

    J. Kaidi, K. Ohmori, and Y. Zheng, “Kramers-wannier-like duality defects in (3+1)d gauge theories,” Physical Review Letters128no. 11, (Mar., 2022) . http://dx.doi.org/10.1103/PhysRevLett.128.111601

    work page doi:10.1103/physrevlett.128.111601 2022

  45. [47]

    Non-invertible condensation, duality, and triality defects in 3+1 dimensions,

    Y. Choi, C. C´ ordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, “Non-invertible condensation, duality, and triality defects in 3+1 dimensions,” Communications in Mathematical Physics402no. 1, (May,

    work page

  46. [48]

    489–542.http://dx.doi.org/10.1007/s00220-023-04727-4

    work page doi:10.1007/s00220-023-04727-4

  47. [49]

    Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,

    D. V. Else and C. Nayak, “Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,” Phys. Rev. B90(Dec, 2014) 235137. https://link.aps.org/doi/10.1103/PhysRevB.90.235137

    work page doi:10.1103/physrevb.90.235137 2014

  48. [50]

    Fermions, strings, and gauge fields in lattice spin models,

    M. Levin and X.-G. Wen, “Fermions, strings, and gauge fields in lattice spin models,” Phys. Rev. B67 (Jun, 2003) 245316.https://link.aps.org/doi/10.1103/PhysRevB.67.245316

    work page doi:10.1103/physrevb.67.245316 2003

  49. [51]

    Symmetries and anomalies of Kitaev spin-S models: Identifying symmetry-enforced exotic quantum matter,

    R. Liu, H. T. Lam, H. Ma, and L. Zou, “Symmetries and anomalies of Kitaev spin-S models: Identifying symmetry-enforced exotic quantum matter,” SciPost Phys.16(2024) 100. https://scipost.org/10.21468/SciPostPhys.16.4.100

    work page doi:10.21468/scipostphys.16.4.100 2024

  50. [52]

    Pauli stabilizer models of twisted quantum doubles,

    T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli stabilizer models of twisted quantum doubles,” PRX Quantum3(Mar, 2022) 010353. https://link.aps.org/doi/10.1103/PRXQuantum.3.010353

    work page doi:10.1103/prxquantum.3.010353 2022

  51. [53]

    SciPost Phys.14, 065 (2023) https: //doi.org/10.21468/SciPostPhys.14.4.065

    M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, “Codimension-2 defects and higher symmetries in (3+1)D topological phases,” SciPost Phys.14 (2023) 065.https://scipost.org/10.21468/SciPostPhys.14.4.065

    work page doi:10.21468/scipostphys.14.4.065 2023

  52. [54]

    Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups

    C. C´ ordova, P.-S. Hsin, and N. Seiberg, “Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups,” SciPost Phys.4no. 4, (2018) 021, arXiv:1711.10008 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  53. [55]

    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,”arXiv:2312.09074 [cond-mat.str-el]. 23

    work page arXiv