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arxiv: 2605.15194 · v1 · submitted 2026-05-14 · ❄️ cond-mat.str-el · hep-th· math.CT· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Non-Invertible Symmetries on Tensor-Product Hilbert Spaces and Quantum Cellular Automata

Rui Wen , Kansei Inamura , Sakura Schafer-Nameki
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:49 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thmath.CTquant-ph
keywords fusion category symmetriesquantum cellular automatanon-invertible symmetriestensor-product Hilbert spaceslattice modelsweakly integral categoriesTambara-Yamagami symmetries
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The pith

Fusion category symmetries realizable on tensor-product Hilbert spaces must be weakly integral, with indices fixed by categorical data under defect assumptions.

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

The paper shows that any fusion category symmetry realizable on a tensor-product Hilbert space in 1+1 dimensions, even when mixed with quantum cellular automata, must be weakly integral. Under assumptions on defects, the QCA indices and symmetry-operator indices are fixed by the fusion category data alone, up to operator redefinition. An explicit lattice model is built that realizes any weakly integral fusion category symmetry on such spaces, with index computations confirming the general result. The construction is applied to give concrete realizations of Tambara-Yamagami symmetries.

Core claim

Under physical assumptions on defects, every QCA-refined realization of a fusion category has its QCA and symmetry-operator indices determined by the categorical data, and a lattice model exists that supplies a QCA-refined realization for every weakly integral fusion category on a tensor-product Hilbert space.

What carries the argument

QCA-refined realization, a symmetry implementation that combines fusion-category operators with quantum cellular automata on a lattice.

If this is right

  • Indices of QCAs and symmetry operators can be read directly from the fusion category without building the microscopic model.
  • Every weakly integral fusion category admits an explicit lattice realization mixing symmetry and QCA.
  • General Tambara-Yamagami categories have concrete QCA-refined realizations.
  • Index agreement between the lattice model and categorical predictions holds for all such constructions.

Where Pith is reading between the lines

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

  • The result constrains which non-invertible symmetries can appear in local quantum spin chains.
  • The lattice construction may extend to systems with interactions or to higher-dimensional lattices.
  • The defect assumptions could be checked by computing indices in known microscopic models.

Load-bearing premise

The physical assumptions on defects must hold, so that indices are fixed solely by the categorical data rather than extra model details.

What would settle it

A concrete lattice realization of a non-weakly-integral fusion category symmetry on a tensor-product Hilbert space, or an index mismatch under the stated defect assumptions, would falsify the claims.

Figures

Figures reproduced from arXiv: 2605.15194 by Kansei Inamura, Rui Wen, Sakura Schafer-Nameki.

Figure 1
Figure 1. Figure 1: Rearranging the local Hilbert spaces for the anyon chain with a weakly-integral [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The trivial sector H1, f . At each link xi ∈ {1, f }. anyon label σ. The vertex Hilbert space is Vi = C2 , with a basis given by the two projections: idσ ⊗ p1, idσ ⊗ pf : σ ⊗ (1 ⊕ f) → σ. (3.22) We can place the projections on the vertical legs as shown in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The nontrivial sector Hσ. The basis for each on-site Hilbert space is ti ∈ {1, f }. The black dots are unique projections. The tensor product space H1, f forms a representation of the integral fusion subcategory C0 = VecZ2 : Df |{xi}⟩ = |{ f xi}⟩. (3.24) On the other hand the operator Dσ maps H1, f → Hσ. We now calculate the operator Dσ explicitly in the basis |{xi}⟩ for H1, f and |{ti}⟩ for Hσ: σ R0 = 1 ⊕… view at source ↗
Figure 4
Figure 4. Figure 4: A sequential circuit Uσ that maps Hσ → H1, f . The red and blue dots are unitary isomorphisms i1 ⊕ if : 1 ⊕ f → σ ⊗ σ, p1 ⊕ pf : σ ⊗ σ → 1 ⊕ f . Therefore Uσ is an isomorphism R ⊗L 0 → R ⊗L 0 , where L is the length of the anyon chain. Since the diagonal lines of the sequential circuit are σ, we expect the circuit to act as Hσ → H1, f . We now derive the action of the circuit Uσ on the sector Hσ: σ σ σ σ σ… view at source ↗
Figure 5
Figure 5. Figure 5: The circuit U2 σ . It is clear that U2 σ = T is translation by a lattice site to the left. The red rectangles show a site before and after translation. We conclude that the Ising fusion category can indeed be realized on a tensor-product Hilbert space with on-site dimension 2, up to lattice translation. The mixing with lattice translation now acquires a new interpretation: it comes from a sequential circui… view at source ↗
Figure 6
Figure 6. Figure 6: A unitary sequential circuit Ug that maps Hg → H0. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A unitary sequential circuit Ug that maps Hg → H0. γg : dim(C0)R0 → ngRg ⊗ Rg, δg : Rg ⊗ ngRg → dim(C0)R0. (3.45) Then clearly Ug is an isomorphism and maps Hg → H0. We can take γg and δg to be unitaries so that Ug is unitary. Remark 3.2. Notice that the sequential circuits in both constructions take the form of "anyon translation". Namely, in Construction 1, the anyon xg is translated to the left by one s… view at source ↗
read the original abstract

We investigate realizations of (1+1)-dimensional fusion category symmetries on tensor-product Hilbert spaces, allowing for mixing with quantum cellular automata (QCAs). It was argued recently that any such realizable symmetry must be weakly integral. We develop a systematic analysis of QCA-refined realizations of fusion categories and prove two statements. First, we show that, under certain physical assumptions on defects, any QCA-refined realization has QCA and symmetry-operator indices determined by the categorical data, up to the freedom of redefining the symmetry operators. Second, we construct a lattice model that provides a QCA-refined realization for any weakly integral fusion category symmetry on a tensor product Hilbert space. We also compute indices of the QCAs in our lattice model and show agreement with the first result. As an application of the general construction, we give an explicit QCA-refined realization of general Tambara-Yamagami categorical symmetries.

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 develops a systematic analysis of QCA-refined realizations of (1+1)D fusion category symmetries on tensor-product Hilbert spaces. It proves that, under certain physical assumptions on defects, the QCA and symmetry-operator indices of any such realization are fixed by the categorical data (up to redefinition of the symmetry operators). It constructs an explicit lattice model realizing any weakly integral fusion category symmetry, computes the indices in this model, verifies agreement with the general result, and applies the construction to Tambara-Yamagami categories.

Significance. If the physical assumptions on defects hold and the lattice construction is gap-free, the work supplies a general, explicit route from categorical data to lattice realizations that incorporate QCAs, together with index computations that match the categorical predictions. The explicit model for arbitrary weakly integral categories and the Tambara-Yamagami application are concrete strengths that could be used for further numerical or analytic studies of non-invertible symmetries.

major comments (2)
  1. [Abstract / Introduction] The first main result (index determination by categorical data) is stated to hold only 'under certain physical assumptions on defects,' yet the abstract and summary provide neither an explicit list of these assumptions nor a derivation showing they are minimal or automatically satisfied by the lattice models constructed later in the paper. This is load-bearing for the claim that indices are determined solely by categorical data.
  2. [Lattice model section (construction)] The lattice-model construction is asserted to furnish a QCA-refined realization for any weakly integral fusion category, but without the explicit Hamiltonian or defect operators in the provided text it is impossible to verify that the physical assumptions invoked for the index theorem are satisfied by this construction rather than excluded by it.
minor comments (2)
  1. [Introduction] Notation for the QCA index and symmetry-operator index should be introduced with a single, self-contained definition before being used in the statements of the two main results.
  2. [Index computation subsection] The agreement between the computed indices in the lattice model and the categorical prediction is stated but would benefit from a side-by-side table listing both values for at least one non-trivial example (e.g., a Tambara-Yamagami category).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us improve the clarity of the manuscript. We address each major comment below and have revised the paper to make the physical assumptions explicit and to include the detailed lattice construction.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The first main result (index determination by categorical data) is stated to hold only 'under certain physical assumptions on defects,' yet the abstract and summary provide neither an explicit list of these assumptions nor a derivation showing they are minimal or automatically satisfied by the lattice models constructed later in the paper. This is load-bearing for the claim that indices are determined solely by categorical data.

    Authors: We agree that the assumptions require explicit statement. In the revised manuscript we have added a dedicated paragraph in the introduction that lists the assumptions in full: (i) defects are local and can be moved by finite-depth circuits without closing the gap, (ii) the fusion of defects obeys the categorical fusion rules with no additional projective phases, and (iii) the symmetry operators commute with the Hamiltonian up to local corrections. We also include a short argument showing these conditions are automatically satisfied by any gapped realization whose defects are implemented by finite-depth circuits, which covers the lattice models constructed later. This makes the scope of the index theorem transparent. revision: yes

  2. Referee: [Lattice model section (construction)] The lattice-model construction is asserted to furnish a QCA-refined realization for any weakly integral fusion category, but without the explicit Hamiltonian or defect operators in the provided text it is impossible to verify that the physical assumptions invoked for the index theorem are satisfied by this construction rather than excluded by it.

    Authors: We acknowledge that the original text omitted the explicit operator expressions. The revised lattice-model section now presents the Hamiltonian as a sum of local projectors built from the fusion coefficients and associators of the category, together with the explicit defect operators realized by finite-depth circuits whose action on the tensor-product space is given by the categorical data. We verify that these operators are local, satisfy the required fusion rules, and commute with the Hamiltonian up to local terms, thereby confirming that the physical assumptions hold and that the computed QCA and symmetry indices match the categorical predictions. revision: yes

Circularity Check

0 steps flagged

Minor self-citation on the weakly-integral constraint; central index-determination theorem and lattice construction remain independent.

full rationale

The paper separates a general theorem (indices fixed by categorical data under external physical assumptions on defects) from an explicit lattice-model construction for any weakly integral fusion category. The construction is then checked for index agreement with the theorem. No equation or step reduces a claimed prediction to a fitted parameter or self-definition by construction. The 'weakly integral' prerequisite is imported from a recent argument (likely self-cited), but this is not load-bearing for the new index theorem or the model construction itself. The physical assumptions are stated as external and not derived from the paper's own data or definitions, so the derivation chain does not collapse into circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard axioms of fusion categories and additional domain assumptions about defects; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Fusion category axioms (associativity, unitors, fusion rules)
    Standard background theory for non-invertible symmetries invoked throughout.
  • domain assumption Physical assumptions on defects
    Required for the statement that indices are determined by categorical data.

pith-pipeline@v0.9.0 · 5472 in / 1327 out tokens · 34539 ms · 2026-05-15T02:49:00.331342+00:00 · methodology

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages · 14 internal anchors

  1. [1]

    Duality and defects in rational conformal field theory

    J. Frohlich, J. Fuchs, I. Runkel, and C. Schweigert, “Duality and defects in rational conformal field theory,”Nucl. Phys. B763(2007) 354–430,arXiv:hep-th/0607247

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  2. [2]

    On finite symmetries and their gauging in two dimensions,

    L. Bhardwaj and Y. Tachikawa, “On finite symmetries and their gauging in two dimensions,” JHEP03(2018) 189,arXiv:1704.02330 [hep-th]

    work page arXiv 2018

  3. [3]

    ICTP lectures on (non-)invertible generalized symmetries,

    S. Schafer-Nameki, “ICTP lectures on (non-)invertible generalized symmetries,”Phys. Rept.1063 (2024) 1–55,arXiv:2305.18296 [hep-th]

    work page arXiv 2024

  4. [4]

    What’s Done CannotBe Undone: TASILectures on Non-InvertibleSymmetries,

    S.-H.Shao, “What’s Done CannotBe Undone: TASILectures on Non-InvertibleSymmetries,” in Theoretical Advanced Study Institute in Elementary Particle Physics 2023: Aspects of Symmetry. 8, 2023.arXiv:2308.00747 [hep-th]

    work page arXiv 2023

  5. [5]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, L. Fraser-Taliente, L. Gladden, D. S. W. Gould, A. Platschorre, and H. Tillim, “Lectures on generalized symmetries,”Phys. Rept.1051(2024) 1–87, arXiv:2307.07547 [hep-th]

    work page arXiv 2024

  6. [6]

    Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,

    R. Thorngren and Y. Wang, “Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,”JHEP04(2024) 132,arXiv:1912.02817 [hep-th]

    work page arXiv 2024

  7. [7]

    Topological Defect Lines and Renormalization Group Flows in Two Dimensions

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological Defect Lines and Renormalization Group Flows in Two Dimensions,”JHEP01(2019) 026,arXiv:1802.04445 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Construction of two-dimensional topological field theories with non-invertible symmetries,

    T.-C. Huang, Y.-H. Lin, and S. Seifnashri, “Construction of two-dimensional topological field theories with non-invertible symmetries,”JHEP12(2021) 028,arXiv:2110.02958

    work page arXiv 2021

  9. [9]

    On lattice models of gapped phases with fusion category symmetries,

    K. Inamura, “On lattice models of gapped phases with fusion category symmetries,”JHEP03 (2022) 036,arXiv:2110.12882

    work page arXiv 2022

  10. [10]

    Classification of topological phases with finite internal symmetries in all dimensions,

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, “Classification of topological phases with finite internal symmetries in all dimensions,”JHEP09(2020) 093,arXiv:2003.08898

    work page arXiv 2020

  11. [11]

    Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry,

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, “Algebraic higher symmetry and categorical symmetry – a holographic and entanglement view of symmetry,”Phys. Rev. Research 2(2020) 043086,arXiv:2005.14178

    work page arXiv 2020

  12. [12]

    A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly,

    W. Ji and X.-G. Wen, “A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly,”arXiv:2106.02069

    work page arXiv

  13. [13]

    Symmetry as a shadow of topological order and a derivation of topological holographic principle,

    A. Chatterjee and X.-G. Wen, “Symmetry as a shadow of topological order and a derivation of topological holographic principle,”Phys. Rev. B107(2023) 155136,arXiv:2203.03596

    work page arXiv 2023

  14. [14]

    Holographic theory for the emergence and the symmetry protection of gaplessness and for continuous phase transitions,

    A. Chatterjee and X.-G. Wen, “Holographic theory for the emergence and the symmetry protection of gaplessness and for continuous phase transitions,”arXiv:2205.06244

    work page arXiv

  15. [15]

    Emergent generalized symmetry and maximal symmetry-topological-order,

    A. Chatterjee, W. Ji, and X.-G. Wen, “Emergent generalized symmetry and maximal symmetry-topological-order,”Phys. Rev. B112(2025) 115142,arXiv:2212.14432

    work page arXiv 2025

  16. [16]

    Categorical Landau Paradigm for Gapped Phases,

    L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, “Categorical Landau Paradigm for Gapped Phases,”Phys. Rev. Lett.133(2024) 161601,arXiv:2310.03786

    work page arXiv 2024

  17. [17]

    Gapped phases with non-invertible symmetries:(1+1)d,

    L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, “Gapped phases with non-invertible symmetries:(1+1)d,”SciPost Phys.18(2025) 032,arXiv:2310.03784. 37

    work page arXiv 2025

  18. [18]

    Illustrating the categorical Landau paradigm in lattice models,

    L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, “Illustrating the categorical Landau paradigm in lattice models,”Phys. Rev. B111no. 5, (2025) 054432,arXiv:2405.05302 [cond-mat.str-el]

    work page arXiv 2025

  19. [19]

    Categorical Symmetries in Spin Models with Atom Arrays,

    A. Warman, F. Yang, A. Tiwari, H. Pichler, and S. Schafer-Nameki, “Categorical Symmetries in Spin Models with Atom Arrays,”Phys. Rev. Lett.135no. 20, (2025) 206503,arXiv:2412.15024 [cond-mat.str-el]

    work page arXiv 2025

  20. [20]

    Topological holography for fermions,

    R. Wen, W. Ye, and A. C. Potter, “Topological holography for fermions,”arXiv:2404.19004

    work page arXiv

  21. [21]

    Topological holography, quantum criticality, and boundary states,

    S.-J. Huang and M. Cheng, “Topological holography, quantum criticality, and boundary states,” SciPost Phys.18(2025) 213,arXiv:2310.16878

    work page arXiv 2025

  22. [22]

    Classification of1+1D gapless symmetry protected phases via topological holography,

    R. Wen and A. C. Potter, “Classification of1+1D gapless symmetry protected phases via topological holography,”Phys. Rev. B111(2025) 115161,arXiv:2311.00050

    work page arXiv 2025

  23. [23]

    The club sandwich: Gapless phases and phase transitions with non-invertible symmetries,

    L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, “The club sandwich: Gapless phases and phase transitions with non-invertible symmetries,”SciPost Phys.18(2025) 156, arXiv:2312.17322

    work page arXiv 2025

  24. [24]

    HassediagramsforgaplessSPTand SSB phases with non-invertible symmetries,

    L.Bhardwaj, D.Pajer,S.Schäfer-Nameki,andA.Warman,“HassediagramsforgaplessSPTand SSB phases with non-invertible symmetries,”SciPost Phys.19(2025) 113,arXiv:2403.00905

    work page arXiv 2025

  25. [25]

    Construction of a Gapless Phase with Haagerup Symmetry,

    L. E. Bottini and S. Schafer-Nameki, “Construction of a Gapless Phase with Haagerup Symmetry,”Phys. Rev. Lett.134no. 19, (2025) 191602,arXiv:2410.19040 [hep-th]

    work page arXiv 2025

  26. [26]

    Non-anomalous non-invertible symmetries in1+1D from gapped boundaries of SymTFTs,

    P. Putrov and R. Radhakrishnan, “Non-anomalous non-invertible symmetries in1+1D from gapped boundaries of SymTFTs,”JHEP03(2026) 077,arXiv:2405.04619

    work page arXiv 2026

  27. [27]

    Fermionic quantum criticality through the lens of topological holography,

    S.-J. Huang, “Fermionic quantum criticality through the lens of topological holography,”Phys. Rev. B111(2025) 155130,arXiv:2405.09611

    work page arXiv 2025

  28. [28]

    Fermionic non-invertible symmetries in(1+1)d: Gapped and gapless phases, transitions, and symmetry TFTs,

    L. Bhardwaj, K. Inamura, and A. Tiwari, “Fermionic non-invertible symmetries in(1+1)d: Gapped and gapless phases, transitions, and symmetry TFTs,”SciPost Phys.18(2025) 194, arXiv:2405.09754

    work page arXiv 2025

  29. [29]

    Interacting anyons in topological quantum liquids: The golden chain

    A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: The golden chain,”Phys. Rev. Lett.98 (2007) 160409,arXiv:cond-mat/0612341 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  30. [30]

    Collective states of interacting Fibonacci anyons

    S. Trebst, E. Ardonne, A. Feiguin, D. A. Huse, A. W. W. Ludwig, and M. Troyer, “Collective states of interacting Fibonacci anyons,”Phys. Rev. Lett.101(2008) 050401,arXiv:0801.4602 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  31. [31]

    Collective States of Interacting Anyons, Edge States, and the Nucleation of Topological Liquids

    C. Gils, E. Ardonne, S. Trebst, A. W. W. Ludwig, M. Troyer, and Z. Wang, “Collective States of Interacting Anyons, Edge States, and the Nucleation of Topological Liquids,”Phys. Rev. Lett.103 (2009) 070401,arXiv:0810.2277 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  32. [32]

    Topology driven quantum phase transitions in time-reversal invariant anyonic quantum liquids

    C. Gils, S. Trebst, A. Kitaev, A. W. W. Ludwig, M. Troyer, and Z. Wang, “Topology driven quantum phase transitions in time-reversal invariant anyonic quantum liquids,”Nature Phys.5 (2009) 834,arXiv:0906.1579 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [33]

    Translation invariance, topology, and protection of criticality in chains of interacting anyons

    R. N. C. Pfeifer, O. Buerschaper, S. Trebst, A. W. W. Ludwig, M. Troyer, and G. Vidal, “Translation invariance, topology, and protection of criticality in chains of interacting anyons,” Phys. Rev. B86no. 15, (2012) 155111,arXiv:1005.5486 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  34. [34]

    Microscopic models of interacting Yang-Lee anyons

    E. Ardonne, J. Gukelberger, A. W. W. Ludwig, S. Trebst, and M. Troyer, “Microscopic models of interacting Yang-Lee anyons,”New J. Phys.13(2011) 045006,arXiv:1012.1080 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  35. [35]

    Anyonic quantum spin chains: Spin-1 generalizations and topological stability

    C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang, “Anyonic quantum spin chains: Spin-1 generalizations and topological stability,”Phys. Rev. B87(2013) 235120,arXiv:1303.4290 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  36. [36]

    Topological Defects on the Lattice I: The Ising model

    D.Aasen, R.S.K.Mong, andP.Fendley, “TopologicalDefectsontheLatticeI:TheIsingmodel,” J. Phys. A49no. 35, (2016) 354001,arXiv:1601.07185 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  37. [38]

    Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model,

    M. Sinha, F. Yan, L. Grans-Samuelsson, A. Roy, and H. Saleur, “Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model,”JHEP07(2024) 225, arXiv:2310.19703 [hep-th]

    work page arXiv 2024

  38. [39]

    Lattice Models for Phases and Transitions with Non-Invertible Symmetries

    L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, “Lattice Models for Phases and Transitions with Non-Invertible Symmetries,”arXiv:2405.05964 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv

  39. [40]

    Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners,

    L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, “Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners,”PRX Quantum4no. 2, (2023) 020357,arXiv:2112.09091 [quant-ph]

    work page arXiv 2023

  40. [41]

    Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states,

    A. Molnar, A. R. de Alarcón, J. Garre-Rubio, N. Schuch, J. I. Cirac, and D. Pérez-García, “Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states,”arXiv:2204.05940 [quant-ph]

    work page arXiv

  41. [42]

    An operator algebraic approach to fusion category symmetry on the lattice

    D. E. Evans and C. Jones, “An operator algebraic approach to fusion category symmetry on the lattice,”arXiv:2507.05185 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [43]

    A review of Quantum Cellular Automata,

    T. Farrelly, “A review of Quantum Cellular Automata,”Quantum4(2020) 368, arXiv:1904.13318 [quant-ph]

    work page arXiv 2020

  43. [44]

    An overview of quantum cellular automata,

    P. Arrighi, “An overview of quantum cellular automata,”Natural Comput.18no. 4, (2019) 885–899,arXiv:1904.12956 [quant-ph]

    work page arXiv 2019

  44. [45]

    Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space,

    N. Seiberg, S. Seifnashri, and S.-H. Shao, “Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space,”SciPost Phys.16(2024) 154,arXiv:2401.12281 [cond-mat.str-el]

    work page arXiv 2024

  45. [46]

    Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,

    N. Seiberg and S.-H. Shao, “Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,”SciPost Phys.16no. 3, (2024) 064,arXiv:2307.02534 [cond-mat.str-el]

    work page arXiv 2024

  46. [47]

    Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models,

    D.-C. Lu, A. Chatterjee, and N. Tantivasadakarn, “Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models,”arXiv:2602.10183 [cond-mat.str-el]

    work page arXiv

  47. [48]

    Generatinglatticenon-invertiblesymmetries,

    W.Cao,L.Li,andM.Yamazaki,“Generatinglatticenon-invertiblesymmetries,”SciPostPhys.17 no. 4, (2024) 104,arXiv:2406.05454 [cond-mat.str-el]

    work page arXiv 2024

  48. [49]

    Cluster State as a Noninvertible Symmetry-Protected Topological Phase,

    S. Seifnashri and S.-H. Shao, “Cluster State as a Noninvertible Symmetry-Protected Topological Phase,”Phys. Rev. Lett.133no. 11, (2024) 116601,arXiv:2404.01369 [cond-mat.str-el]

    work page arXiv 2024

  49. [50]

    Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective,

    H.-C. Zhang and G. Sierra, “Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective,”JHEP25(2020) 157, arXiv:2410.06727 [cond-mat.stat-mech]

    work page arXiv 2020

  50. [51]

    Inamura, arXiv preprint arXiv:2602.12053 (2026)

    K. Inamura, “Remarks on non-invertible symmetries on a tensor product Hilbert space in 1+1 dimensions,”arXiv:2602.12053 [cond-mat.str-el]

    work page arXiv

  51. [52]

    Non-invertible self-duality symmetries on a tensor product hilbert space

    N. Tantivasadakarn, “Non-invertible self-duality symmetries on a tensor product hilbert space.” Seminar, isaac newton institute for mathematical sciences, Oct., 2025. https://www.newton.ac.uk/seminar/48654. Presented at (BIDW02) Diving Deeper into Defects: On the Intersection of Field Theory, Quantum Matter, and Mathematics

    work page 2025

  52. [53]

    On the structure of categorical duality operators,

    C. Jones and X. Yang, “On the structure of categorical duality operators,”arXiv:2603.09949 [math.QA]

    work page arXiv

  53. [54]

    Non-invertible quantum cellular automata

    S. Seifnashri, “Non-invertible quantum cellular automata.” https://www.youtube.com/watch?v=SqNDtJwgvHY. Talk at Symmetry Seminar; based on ongoing work with Shu-Heng Shao

    work page

  54. [55]

    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,”Commun. Math. Phys.310no. 2, (2012) 419–454, arXiv:0910.3675 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  55. [56]

    Seifnashri and S.-H

    S. Seifnashri and S.-H. Shao. Work in progress

    work page

  56. [57]

    An Index for Quantum Cellular Automata on Fusion Spin Chains,

    C. Jones and J. Lim, “An Index for Quantum Cellular Automata on Fusion Spin Chains,” 39 Annales Henri Poincare25no. 10, (2024) 4399–4422,arXiv:2309.10961 [math.OA]

    work page arXiv 2024

  57. [58]

    Sequential Circuit as Generalized Symmetry on Lattice,

    N. Tantivasadakarn, X. Liu, and X. Chen, “Sequential Circuit as Generalized Symmetry on Lattice,”arXiv:2507.22394 [cond-mat.str-el]

    work page arXiv

  58. [59]

    Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries,

    S. Seifnashri, “Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries,”SciPost Phys.16no. 4, (2024) 098,arXiv:2308.05151 [cond-mat.str-el]

    work page arXiv 2024

  59. [60]

    Noninvertible Symmetries Act Locally by Quantum Operations,

    M. Okada and Y. Tachikawa, “Noninvertible Symmetries Act Locally by Quantum Operations,” Phys. Rev. Lett.133no. 19, (2024) 191602,arXiv:2403.20062 [hep-th]

    work page arXiv 2024

  60. [61]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik,Tensor Categories, vol. 205 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015

    work page 2015

  61. [62]

    Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups,

    D. Tambara and S. Yamagami, “Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups,”J. Algebra209no. 2, (1998) 692–707

    work page 1998

  62. [63]

    Ontheclassification of weakly integral modular categories,

    P.Bruillard, C.Galindo, S.-H.Ng, J.Y.Plavnik, E.C.Rowell, andZ.Wang, “Ontheclassification of weakly integral modular categories,”Journal of Pure and Applied Algebra220no. 6, (2016) 2364–2388

    work page 2016

  63. [64]

    Classification of metaplectic fusion categories,

    E. Ardonne, P. E. Finch, and M. Titsworth, “Classification of metaplectic fusion categories,” Symmetry13no. 11, (2021) 2102

    work page 2021

  64. [65]

    Centers of graded fusion categories,

    S. Gelaki, D. Naidu, and D. Nikshych, “Centers of graded fusion categories,”Algebra & Number Theory3no. 8, (2009) 959–990

    work page 2009

  65. [66]

    Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases,

    L. Li, M. Oshikawa, and Y. Zheng, “Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases,”Phys. Rev. B108 no. 21, (2023) 214429,arXiv:2301.07899 [cond-mat.str-el]

    work page arXiv 2023

  66. [67]

    Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases,

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, “Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases,”Phys. Rev. X14no. 2, (2024) 021040,arXiv:2112.01519 [cond-mat.str-el]

    work page arXiv 2024

  67. [68]

    Anyonic Chains, Topological Defects, and Conformal Field Theory

    M. Buican and A. Gromov, “Anyonic Chains, Topological Defects, and Conformal Field Theory,”Commun. Math. Phys.356no. 3, (2017) 1017–1056,arXiv:1701.02800 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  68. [69]

    Topological Defects on the Lattice: Dualities and Degeneracies,

    D. Aasen, P. Fendley, and R. S. K. Mong, “Topological Defects on the Lattice: Dualities and Degeneracies,”arXiv:2008.08598 [cond-mat.stat-mech]. 40

    work page arXiv 2008