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REVIEW 2 major objections 5 minor 61 references

Fermionic Anomalies of Finite Symmetries on Lattices

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper develops a lattice-level characterization of fermionic 't Hooft anomalies and shows that a lattice symmetry with trivial continuum anomaly can still forbid a symmetric short-range-entangled state.

desk verdict A solid lattice-anomaly paper with a genuinely interesting mismatch example, but the 2+1D result leans on an unpublished Haah theorem and the explicit Z_4^F operator may not be a finite-depth circuit. read the letter →

arxiv 2608.12455 v1 pith:7CWUHF5J submitted 2026-08-12 cond-mat.str-el hep-thquant-ph

Ameya Chavda , Ryohei Kobayashi This is my paper · ORCID
classification cond-mat.str-elhep-thquant-ph
keywords fermionic'tHooftanomalieslatticesymmetriesshort-range-entangledstatesquantumcellularautomataMajoranatranslationanomalyindicescentralextensionbyfermionparitytensor-productHilbertspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a lattice-level characterization of fermionic 't Hooft anomalies for finite internal symmetries, phrased as obstructions to symmetric short-range-entangled states. In one spatial dimension an exact lattice symmetry carries a pair of cohomological indices $(n_2,\nu_3)$; in two dimensions it carries a triple $(n_2,n_3,\nu_4)$. For a symmetry whose bosonic group and fermion parity factorize, the paper proves that vanishing indices imply the symmetry can be rewritten as a strictly local product of single-site operations, hence admits a symmetric short-range-entangled state, while any nonvanishing index obstructs such a state. In the non-factorized case, a lattice index can obstruct a symmetric short-range-entangled state even when the continuum 't Hooft anomaly is trivial, as shown by an explicit $\mathbb Z_4^F$ symmetry with $U_g^2=(-1)^F$ and Majorana-layer index $n_2=\omega_2$.

What carries the argument

The machinery is the truncation of the symmetry operator to a finite region and the study of the boundary operators that make the truncated group law hold. In one dimension the fermion parity of the left endpoint operator defines $n_2$, and the phase violation defines $\nu_3$. In two dimensions the boundary operator's fermionic quantum-cellular-automaton index, taking values in $\mathbb Q_+$ or $\sqrt 2\,\mathbb Q_+$ according to whether it contains a Majorana translation, defines the Majorana layer $n_2$; after that layer is trivialized, an interval truncation of the boundary circuit defines the complex-fermion layer $n_3$ and then the bosonic layer $\nu_4$. In the explicit example the symmetry generator is a domain-wall-dependent translation of Majorana fermions by one lattice site whose square is fermion parity, producing $n_2=\omega_2$.

What would settle it

A concrete falsifier would be an explicit symmetric short-range-entangled ground state for the $\mathbb Z_4^F$ lattice symmetry of Sec. 7, which the paper's Majorana-layer index $n_2=\omega_2$ says cannot exist.

Watch

Extended reading notes

Core claim

The central discovery is that anomaly data extracted from an exact lattice symmetry do not have to match the anomaly classification of continuum quantum field theory. The paper obtains its indices by truncating a finite-depth-circuit symmetry operator to a disk or interval and reading the fermion parity or fermionic quantum-cellular-automaton index of the boundary operators that restore the group law. In (1+1)D, for $G_f=G_b\times\mathbb Z_2^F$, trivial $(n_2,\nu_3)$ implies the symmetry is onsiteable, so the pair faithfully detects the lattice anomaly; but the continuum layer $H^1(BG_b,\mathbb Z_2)$ cannot be realized by an exact lattice symmetry, and a $\mathbb Z_2$ symmetry realizes only the even $\mathbb Z_4$ subgroup of the continuum $\mathbb Z_8$ classification. In (2+1)D each nonzero layer of $(n_2,n_3,\nu_4)$ obstructs symmetric short-range-entangled states, and for a factorized symmetry it obstructs all symmetric invertible states. The central example is a $\mathbb Z_4^F$ lattice symmetry with $U_g^2=(-1)^F$ and $n_2=\omega_2$: it forbids a symmetric short-range-entangled state, yet its continuum 't Hooft anomaly is trivial because $n_2=\omega_2$ is equivalent to zero in the continuum classification.

Load-bearing premise

The load-bearing premise is that every finite internal fermionic symmetry on a two-dimensional tensor-product lattice can be written as a finite-depth quantum circuit up to a product of local circuits, a fact the paper cites to an unpublished theorem; if that fact fails, the boundary operators carrying the anomaly indices are not guaranteed to exist.

Editorial extensions

If this is right

  • In (1+1)D, for $G_f=G_b\times\mathbb Z_2^F$, any exact symmetry with $(n_2,\nu_3)=(0,0)$ is onsiteable and therefore admits a symmetric short-range-entangled state.
  • Continuum anomalies with a nonzero $H^1(BG_b,\mathbb Z_2)$ layer, such as the odd elements of the $\mathbb Z_8$ classification of $\mathbb Z_2$ symmetry in (1+1)D, cannot be realized as exact internal symmetries on a tensor-product lattice; four copies of any exact $\mathbb Z_2$ symmetry carry trivial lattice anomaly.
  • In (2+1)D, a nonzero value of any of $n_2,n_3,\nu_4$ obstructs a symmetric short-range-entangled state, and when $G_f=G_b\times\mathbb Z_2^F$ it obstructs every symmetric invertible state.
  • The $\mathbb Z_4^F$ symmetry with $U_g^2=(-1)^F$ and $n_2=\omega_2$ forbids a symmetric short-range-entangled state while having a trivial continuum anomaly, so exact lattice symmetries can be incompatible with symmetric trivial phases for reasons invisible to continuum QFT.
  • The paper conjectures that in (2+1)D triviality of all three indices is also sufficient for onsiteability; if that conjecture holds, the triple is a faithful lattice anomaly classification, and anti-unitary anomalies with only a $p+ip$ layer would be unrealizable by exact symmetries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same truncation machinery should generalize to (3+1)D, where boundary operators on a two-dimensional surface carry fermionic quantum-cellular-automaton indices; a four-layer hierarchy analogous to $(n_2,n_3,\nu_4)$ would likely exhibit further lattice-continuum mismatches.
  • A direct test of the paper's picture is to build the predicted symmetric chiral invertible state with $c_-=1$ for the $\mathbb Z_4^F$ symmetry; success would confirm that the lattice obstruction blocks short-range-entangled states but not all gapped symmetric states.
  • One can read the unattainable continuum layers, namely the $H^1$ layer in (1+1)D and the $p+ip$ layer under time reversal in (2+1)D, as a general rule: anomaly classes that require spatial translation or long-range entanglement in their microscopic realization cannot be exact internal symmetries of tensor-product lattices and must instead appear as emanant symmetries.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a lattice characterization of fermionic 't Hooft anomalies for finite internal symmetries in (1+1)D and (2+1)D, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. In (1+1)D the authors define an Else--Nayak-type pair of anomaly indices (n2, ν3), derive their gauge transformations and stacking rules, and prove for G_f = G_b × Z_2^F that trivial indices imply the symmetry is onsiteable after adding ancillas. They conclude that exact lattice symmetries realize only a subgroup of the continuum anomaly classification, for example only the even Z_4 subgroup of the Z_8 classification for G_b = Z_2. In (2+1)D they define three anomaly layers (n2, n3, ν4), prove that a nontrivial value of any layer obstructs a symmetric SRE state, and prove a stronger obstruction to symmetric invertible states when the extension class ω2 is trivial. The central example is a Z_4^F lattice symmetry in (2+1)D carrying n2 = ω2, which they argue forbids an SRE state even though its continuum 't Hooft anomaly is trivial, thereby demonstrating a mismatch between lattice and continuum fermionic anomalies.

Significance. If the main claims hold, this is a valuable and timely contribution. It gives a concrete lattice framework for extracting fermionic anomaly data directly from symmetry operators, and it provides a nontrivial example in which an exact lattice symmetry obstructs an SRE state despite having a trivial continuum anomaly. The 1+1D completeness result (trivial lattice indices imply onsiteability) is also useful and carefully argued. The paper is generally careful: cocycle identities, gauge transformations, stacking rules, and obstruction proofs are written out in detail, and the authors are explicit about which statements are conjectural. The main weakness is that the most striking 2+1D claim rests on an unpublished and currently unverifiable theorem about 2D fermionic quantum cellular automata, and that theorem is load-bearing for the construction of the Majorana-layer index and for the Z_4^F example.

major comments (2)
  1. [Sec. 5.1, Eq. (41) and Sec. 7.1, Eq. (80)] The definition of the Majorana-layer index n2 and the computation for the Z_4^F example both require the symmetry operator U(g) to be a finite-depth quantum circuit, so that the disk truncation U_A(g) in Eq. (41) and the boundary operator Ω_{∂A} are well defined. The paper justifies this with the statement 'the fermionic QCA in 2D space is trivial up to circuit multiplications [56]', where [56] is an unpublished manuscript ('J. Haah, To appear'). This is not merely a citation problem: the explicit Z_4^F generator in Eq. (80) contains bT[{σ^z_p}], a conditional translation of Majoranas along domain-wall loops of arbitrary perimeter, and no circuit decomposition of this operator is provided. For a single closed loop of length L this is a cyclic Majorana shift, which is not manifestly a constant-depth nearest-neighbor circuit, so the theorem from [56] is doing essential work for both the definition of n2 in Sec. 5.1 and the obstruction proof in Sec. 6.1.1. The authors should state the precise theorem, supply a proof or a published reference, and verify that it applies to the conditional loop translation in Eq. (80). Without this, the central claim that a continuum-trivial Z_4^F symmetry forbids an SRE state is not established.
  2. [Sec. 6.1.2, after Eq. (69)] The proof that a nontrivial complex-fermion layer n3 obstructs SRE uses the statement 'Because n2 = 0, the 1D state |Φ>_{g;∂A} ... becomes a trivial invertible phase, hence SRE.' From the previous subsection, however, n2 = 0 only gives δχ = 0 for the cochain χ(g) defined by the Kitaev-chain class of |Φ>_{g;∂A}; χ can be a nontrivial element of H^1(G_b, Z_2), e.g. for G_b = Z_2 one may have χ(g) = 1. To reach the conclusion one must use the freedom to multiply U_A(g) by a boundary FQCA to shift χ by an arbitrary 1-cochain, a gauge freedom that is not stated in Sec. 5.1 or in Sec. 6.1. The gap is local and easily repaired, but it should be made explicit because it is needed for the reduction from the n3 layer to the bosonic layer.
minor comments (5)
  1. [Sec. 5.1] The phrase 'trivial up to circuit multiplications' is ambiguous; if every 2D fermionic QCA is a circuit, the 'up to' qualifier should be removed or explained.
  2. [Sec. 2, Eq. (14)] The notation [(-1)^{F_I}]^{ω2(g,h)} should use parentheses around the exponent to avoid ambiguity, for instance [(-1)^{F_I}]^{ω2(g,h)}.
  3. [Sec. 5.1] There is a typo in 'quantum cellular autamata' which should read 'quantum cellular automata'.
  4. [References] Reference [56] is listed only as 'J. Haah, To appear.'; if a preprint or published version exists, it should be cited with full details, since the present version is not verifiable.
  5. [Sec. 7.2] The statement that the continuum Z_4^F anomaly is trivial is supported by [20,58]; it would help the reader if the precise equivalence relation that identifies n2 = ω2 with the trivial class were quoted or derived.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the anomaly indices are computed from operator data, and the obstruction proofs derive coboundary conditions rather than assuming the conclusions.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The 1D anomaly pair (n2, ν3) is defined from the truncated symmetry algebra in Eqs. (14)-(18), and the onsiteability theorem in Sec. 3 is a constructive proof using the explicit disentangler W of Eq. (38), not a restatement of the definition of the index. In (2+1)D, the three layers n2, n3, ν4 are defined from independent QCA and endpoint-operator data (Eqs. 41-43, 47-48, 59), and the obstruction proofs in Sec. 6 show that existence of a symmetric SRE state forces each cochain to be a coboundary (n2 = δχ in Eq. (68), n3 = δχ, ν4 = δφ), so the claims that nontrivial indices forbid SRE are derived consequences rather than built-in equivalences. The central Z_4^F example computes n2 = ω2 from the explicit operator in Eq. (80) through the truncated algebra in Eq. (85); it is not fitted or assumed. Its continuum triviality is benchmarked to external continuum classifications [20,58] and to the equivalence relation of the continuum anomaly triple, not to the paper's own lattice index. The main load-bearing external assumption is the unpublished theorem cited as [56] (J. Haah, 'To appear') that 2D fermionic QCAs are trivial up to circuit multiplication, on which the 2D truncation and Majorana-layer index rely; similarly, the 1D statement that finite internal symmetries are generated by FDQC is assumed. These are genuine correctness/verifiability risks, but they are not self-citations, not definitional reductions, and not fitted-input predictions, so they do not constitute circularity under the stated criteria.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard group cohomology and prior anomaly/SPT classification results, plus two domain assumptions that carry real risk: the unpublished 2D FQCA triviality theorem [56] and the continuum anomaly classifications used as benchmarks. There are no fitted free parameters and no invented physical entities.

assumptions (6)
  • domain assumption Finite internal symmetries in 1D are generated by finite-depth quantum circuits, and in 2D fermionic QCA are trivial up to circuit multiplication.
    Used in Secs. 2 and 5.1 to justify truncating U(g) to intervals or disks and reading off boundary QCA indices. The 2D part depends on [56], an unpublished 'To appear' reference.
  • standard math 1D QCA index theory: bosonic QCA index in Q+, fermionic Majorana translation gives sqrt(2) Q+.
    Basis for defining n2 in Secs. 2 and 5.1; cited to [52-54].
  • standard math A 1D invertible state with Z_2^F symmetry is classified by Z_2, with the nontrivial case being Kitaev's Majorana chain.
    Used in Sec. 6.1.1 to turn a boundary state into the 1-cochain chi and prove [n2]=0 for SRE states; cited to [57].
  • domain assumption Continuum fermionic anomaly classifications, including the (1+1)D Z_8 for G_b=Z_2 and the (2+1)D equivalence (n2,n3,nu4)~(n2+omega2,n3,nu4) for Z_4^F, are correct.
    Used in Secs. 4 and 7.2 to conclude that the n1 layer is missing on the lattice and that the Z_4^F example has trivial continuum anomaly; these comparisons come from [20,58].
  • standard math A Kasteleyn orientation exists on the graph Gamma formed by honeycomb-edge Majorana fermions.
    Needed in Sec. 7.1 to define the complex fermion c_e and the Majorana translation bT with signs s_ij.
  • domain assumption Bosonic QCA indices [nu2] in H^2(BG_b,Q+) do not lead to dynamical anomalies, and an anomaly-free bosonic symmetry with inverse index exists.
    Used in Sec. 5.2 to tensor with ancillary symmetry U'_b(g) and reduce to the complex-fermion layer; cited to [7].

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Pith. "Pith review of Fermionic Anomalies of Finite Symmetries on Lattices." pith.science (2026). https://pith.science/paper/7CWUHF5J

@misc{pith2026260812455,
  author       = {Pith},
  title        = {Pith review of: Fermionic Anomalies of Finite Symmetries on Lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CWUHF5J}},
  note         = {Machine review of arXiv:2608.12455}
}
abstract

We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in (1+1)D and (2+1)D, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. We consider lattice systems formed by tensor product of onsite fermionic and bosonic Hilbert spaces, and finite internal symmetry given by a central extension $\mathbb Z_2^F\to G_f\to G_b$. We extract a hierarchy of fermionic anomaly indices for a given symmetry operator. In (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data $(n_2,\nu_3)$. For $G_f=G_b\times\mathbb Z_2^F$, we show that a symmetry with trivial anomaly indices $(n_2,\nu_3)$ is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum QFT, we find that exact lattice symmetries do not realize the additional $H^1(BG_b,\mathbb Z_2)$ anomaly layer in continuum QFT. In particular, for $G_b=\mathbb Z_2$, exact lattice symmetries realize only the even $\mathbb Z_4$ subgroup of the continuum $\mathbb Z_8$ classification. In (2+1)D, we identify three successive anomaly layers of cohomological data $(n_2,n_3,\nu_4)$. We show that a nontrivial value of any layer obstructs symmetric SRE. For $G_f=G_b\times\mathbb Z_2^F$, it also forbids a symmetric invertible state. When the bosonic group $G_b$ is non-trivially extended by fermion parity, the lattice obstruction to SRE states does not generally coincide with the continuum 't Hooft anomaly. We explicitly construct a $\mathbb Z_4^F$ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids symmetric SRE states, even though its continuum anomaly is trivial. Our results highlight a mismatch between lattice and continuum anomalies and motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.

Figures

Figures reproduced from arXiv: 2608.12455 by the authors.

Figure 1
Figure 1. A disk region A and an interval I (thick red line), separated into IL = [l, m] and IR = [m, r]. 6 Dynamical consequence of anomalies in (2+1)D 6.1 Obstruction to SRE 6.1.1 Majorana layer Suppose that an SRE state |Ψ⟩ = V |0⟩ is Gf symmetric: U(g)|Ψ⟩ ∝ |Ψ⟩, where V is FDQC and |0⟩ is a product state with Fock vacuum of fermions. We show that Gf symmetry then must have trivial [n2] ∈ H2 (Gb, Z2) index, therefore the n… view at source ↗
Figure 2
Figure 2. The lattice Γ. Its edges will be endowed with Kasteleyn orientation. Each vertex of Γ has a [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

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Works this paper leans on

61 extracted references · 37 canonical work pages

  1. [56]

    To appear

    J. Haah, “To appear.”

    work page

  2. [1]

    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

    work page 2014

  3. [2]

    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

    work page 2023

  4. [3]

    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 Physics16no. 4, (Apr., 2024) . 17

    work page 2024

  5. [4]

    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]

    arXiv

  6. [5]

    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]

    arXiv

  7. [6]

    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]

    arXiv

  8. [7]

    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]

    arXiv

Show all 61 references
  1. [8]

    Higher-form anomalies on lattices,

    Y. Feng, R. Kobayashi, Y.-A. Chen, and S. Ryu, “Higher-form anomalies on lattices,” Physical Review Letters136no. 4, (Jan., 2026)

    2026

  2. [9]

    When can higher-form symmetries be made on site,

    Y. Feng, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “When can higher-form symmetries be made on site,” Physical Review Letters137no. 2, (2026)

    2026

  3. [10]

    Anomalies of global symmetries on the lattice,

    Y.-T. Tu, D. M. Long, and D. V. Else, “Anomalies of global symmetries on the lattice,” Physical Review X16no. 1, (Feb., 2026)

    2026

  4. [11]

    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

    2025

  5. [12]

    Anomalies on the lattice, homotopy of quantum cellular automata, and a spectrum of invertible states,

    A. M. Czajka, R. Geiko, and R. Thorngren, “Anomalies on the lattice, homotopy of quantum cellular automata, and a spectrum of invertible states,”arXiv:2512.02105 [cond-mat.str-el]

  6. [13]

    Topological phases of fermions in one dimension,

    L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B83 (Feb, 2011) 075103

    2011

  7. [14]

    Fermionic symmetry protected topological phases and cobordisms,

    A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, “Fermionic symmetry protected topological phases and cobordisms,” Journal of High Energy Physics2015no. 12, (Dec., 2015) 1–21

    2015

  8. [15]

    The ”Parity

    E. Witten, “The ”Parity” Anomaly on an Unorientable Manifold,” Phys. Rev.B94no. 19, (2016) 195150,arXiv:1605.02391 [hep-th]

    2016 arXiv

  9. [16]

    Anomalies in (2+1)d fermionic topological phases and (3+1)d path integral state sums for fermionic spts,

    S. Tata, R. Kobayashi, D. Bulmash, and M. Barkeshli, “Anomalies in (2+1)d fermionic topological phases and (3+1)d path integral state sums for fermionic spts,” Communications in Mathematical Physics397no. 1, (Dec., 2022) 199–336

    2022

  10. [17]

    Notes on 8 majorana fermions,

    D. Tong and C. Turner, “Notes on 8 majorana fermions,” SciPost Physics Lecture Notes (Mar., 2020)

    2020

  11. [18]

    Anomaly matching in the symmetry broken phase: Domain walls, cpt, and the smith isomorphism,

    I. Hason, Z. Komargodski, and R. Thorngren, “Anomaly matching in the symmetry broken phase: Domain walls, cpt, and the smith isomorphism,” SciPost Physics8no. 4, (Apr., 2020)

    2020

  12. [19]

    Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory,

    Z.-C. Gu and X.-G. Wen, “Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory,” Phys. Rev. B90(Sep,

  13. [20]

    Construction and classification of symmetry-protected topological phases in interacting fermion systems,

    Q.-R. Wang and Z.-C. Gu, “Construction and classification of symmetry-protected topological phases in interacting fermion systems,” Physical Review X10no. 3, (2020)

    2020

  14. [21]

    Fermionic SPT phases in higher dimensions and bosonization,

    A. Kapustin and R. Thorngren, “Fermionic SPT phases in higher dimensions and bosonization,” JHEP10(2017) 080,arXiv:1701.08264 [cond-mat.str-el]

    2017 arXiv

  15. [22]

    Classification of (2 + 1)d invertible fermionic topological phases with symmetry,

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and N. Manjunath, “Classification of (2 + 1)d invertible fermionic topological phases with symmetry,” Phys. Rev. B105(Jun, 2022) 235143

    2022

  16. [23]

    State sum constructions of spin-tfts and string net constructions of fermionic phases of matter,

    L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-tfts and string net constructions of fermionic phases of matter,” Journal of High Energy Physics2017no. 96, (2017) . 18

    2017

  17. [24]

    Anomalies and bosonization,

    R. Thorngren, “Anomalies and bosonization,”arXiv:1810.04414 [cond-mat.str-el]

    arXiv

  18. [25]

    Global anomalies on the hilbert space,

    D. Delmastro, D. Gaiotto, and J. Gomis, “Global anomalies on the hilbert space,” Journal of High Energy Physics2021no. 11, (Nov., 2021)

    2021

  19. [26]

    The pontrjagin dual of 3-dimensional spin bordism,

    G. Brumfiel and J. Morgan, “The pontrjagin dual of 3-dimensional spin bordism,”arXiv:1612.02860 [math.AT]

    arXiv

  20. [27]

    The pontrjagin dual of 4-dimensional spin bordism,

    G. Brumfiel and J. Morgan, “The pontrjagin dual of 4-dimensional spin bordism,”arXiv:1803.08147 [math.GT]

    arXiv

  21. [28]

    Anomaly cascade in (2+1)-dimensional fermionic topological phases,

    D. Bulmash and M. Barkeshli, “Anomaly cascade in (2+1)-dimensional fermionic topological phases,” Physical Review B105no. 15, (Apr., 2022)

    2022

  22. [29]

    Symmetry protected topological phases and generalized cohomology,

    D. Gaiotto and T. Johnson-Freyd, “Symmetry protected topological phases and generalized cohomology,” Journal of High Energy Physics2019no. 5, (May, 2019)

    2019

  23. [30]

    Symmetry protected topological orders and the group cohomology of their symmetry group,

    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

    2013

  24. [31]

    Quantized axial charge of staggered fermions and the chiral anomaly,

    A. Chatterjee, S. D. Pace, and S.-H. Shao, “Quantized axial charge of staggered fermions and the chiral anomaly,” Physical Review Letters134no. 2, (Jan., 2025)

    2025

  25. [32]

    Lattice t-duality from non-invertible symmetries in quantum spin chains,

    S. D. Pace, A. Chatterjee, and S.-H. Shao, “Lattice t-duality from non-invertible symmetries in quantum spin chains,” SciPost Physics18no. 4, (Apr., 2025)

    2025

  26. [33]

    Lsm and cpt,

    N. Seiberg, S.-H. Shao, and W. Zhang, “Lsm and cpt,” Journal of High Energy Physics2025no. 11, (Nov., 2025)

    2025

  27. [34]

    Tori, klein bottles, and modulo 8 parity/time-reversal anomalies of 2+1d staggered fermions,

    N. Seiberg and W. Zhang, “Tori, klein bottles, and modulo 8 parity/time-reversal anomalies of 2+1d staggered fermions,”arXiv:2601.01191 [hep-th]

    arXiv

  28. [35]

    Non-invertible bosonic chiral symmetry on the lattice,

    L. Fidkowski, C. Xu, and C. Zhang, “Non-invertible bosonic chiral symmetry on the lattice,” arXiv:2510.17969 [cond-mat.str-el]

  29. [36]

    Chiral lattice gauge theories from symmetry disentanglers,

    R. Thorngren, J. Preskill, and L. Fidkowski, “Chiral lattice gauge theories from symmetry disentanglers,”arXiv:2601.04304 [hep-th]

  30. [37]

    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 Physics16no. 3, (Mar., 2024)

    2024

  31. [38]

    Exact chiral symmetries of 3+1d hamiltonian lattice fermions,

    L. Gioia and R. Thorngren, “Exact chiral symmetries of 3+1d hamiltonian lattice fermions,” Physical Review Letters136no. 6, (Feb., 2026)

    2026

  32. [39]

    Exact lattice chiral symmetry in 2d gauge theory,

    E. Berkowitz, A. Cherman, and T. Jacobson, “Exact lattice chiral symmetry in 2d gauge theory,” arXiv:2310.17539 [hep-lat]

    arXiv

  33. [40]

    Symmetry-enforced fermi surfaces,

    M. L. Kim, S. D. Pace, and S.-H. Shao, “Symmetry-enforced fermi surfaces,” Physical Review Letters 136no. 17, (Apr., 2026)

    2026

  34. [41]

    Parity anomaly from a lieb-schultz-mattis theorem: Exact valley symmetries on the lattice,

    S. D. Pace, M. L. Kim, A. Chatterjee, and S.-H. Shao, “Parity anomaly from a lieb-schultz-mattis theorem: Exact valley symmetries on the lattice,” Physical Review Letters135no. 23, (Dec., 2025)

    2025

  35. [42]

    Anomalies in quantum spin systems and nielsen-ninomiya type theorems,

    R. Liu, “Anomalies in quantum spin systems and nielsen-ninomiya type theorems,” arXiv:2602.13948 [math-ph]

  36. [43]

    Fermionic villain model with exact lattice chiral symmetries,

    Z. Lu and S.-H. Shao, “Fermionic villain model with exact lattice chiral symmetries,” arXiv:2608.02728 [hep-th]

    arXiv

  37. [44]

    Infinite-order lattice chiral anomalies and cpt,

    E. Lew-Smith, S. D. Pace, and S.-H. Shao, “Infinite-order lattice chiral anomalies and cpt,” arXiv:2606.12510 [hep-th]. 19

    arXiv

  38. [45]

    Fermionic matrix product operators and topological phases of matter,

    D. J. Williamson, N. Bultinck, J. Haegeman, and F. Verstraete, “Fermionic matrix product operators and topological phases of matter,”arXiv:1609.02897 [quant-ph]

    arXiv

  39. [46]

    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]

    arXiv

  40. [47]

    Remarks on non-invertible symmetries on a tensor product hilbert space in 1+1 dimensions,

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

  41. [48]

    Non-invertible symmetries on tensor-product hilbert spaces and quantum cellular automata,

    R. Wen, K. Inamura, and S. Schafer-Nameki, “Non-invertible symmetries on tensor-product hilbert spaces and quantum cellular automata,”arXiv:2605.15194 [cond-mat.str-el]

    arXiv

  42. [49]

    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 Physics18no. 6, (2025)

    2025

  43. [50]

    Lieb-schultz-mattis type theorems for majorana models with discrete symmetries,

    O. M. Aksoy, A. Tiwari, and C. Mudry, “Lieb-schultz-mattis type theorems for majorana models with discrete symmetries,” Physical Review B104no. 7, (Aug., 2021)

    2021

  44. [51]

    Higher-group symmetry in finite gauge theory and stabilizer codes,

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes,” SciPost Physics16no. 4, (Apr., 2024) ,arXiv:2211.11764 [cond-mat.str-el]

    2024 arXiv

  45. [52]

    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

    2012

  46. [53]

    Chiral floquet phases of many-body localized bosons,

    H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, and A. Vishwanath, “Chiral floquet phases of many-body localized bosons,” Phys. Rev. X6(Dec, 2016) 041070

    2016

  47. [54]

    Interacting invariants for floquet phases of fermions in two dimensions,

    L. Fidkowski, H. C. Po, A. C. Potter, and A. Vishwanath, “Interacting invariants for floquet phases of fermions in two dimensions,” Physical Review B99no. 8, (Feb., 2019)

    2019

  48. [55]

    Fermionic cellular automata in one dimension,

    L. S. Trezzini, M. Lugli, P. Meda, A. Bisio, P. Perinotti, and A. Tosini, “Fermionic cellular automata in one dimension,”arXiv:2501.05349 [quant-ph]

    arXiv

  49. [57]

    Unpaired majorana fermions in quantum wires,

    A. Y. Kitaev, “Unpaired majorana fermions in quantum wires,” Physics-Uspekhi44no. 10S, (Oct.,

  50. [58]

    Dai-freed anomalies in particle physics,

    I. Garcia-Etxebarria and M. Montero, “Dai-freed anomalies in particle physics,” Journal of High Energy Physics2019no. 8, (Aug., 2019)

    2019

  51. [59]

    Symmetry fractionalization, defects, and gauging of topological phases,

    M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, “Symmetry fractionalization, defects, and gauging of topological phases,” Phys. Rev. B100(Sep, 2019) 115147

    2019

  52. [60]

    On 2-Group Global Symmetries and their Anomalies,

    F. Benini, C. C´ ordova, and P.-S. Hsin, “On 2-Group Global Symmetries and their Anomalies,” JHEP 03(2019) 118,arXiv:1803.09336 [hep-th]

    2019 arXiv

  53. [61]

    Symmetry transmutation and anomaly matching,

    N. Seiberg and S. Seifnashri, “Symmetry transmutation and anomaly matching,” Journal of High Energy Physics2025no. 9, (2025) . 20

    2025

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Reviewed August 16, 2026 · model on record in the stance chip above.