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Anomaly-free symmetries with obstructions to gauging and onsiteability

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Pith's one-line read This paper constructs finite-group symmetries of 2D lattice systems that cannot be made on-site or coupled to gauge fields, yet are anomaly-free in the standard sense.

desk verdict A solid, explicit counterexample to the lore that non-onsiteable 2D symmetries must be anomalous, with a useful index and a couple of fixable soft spots. read the letter →

arxiv 2507.21267 v1 pith:LVOSSVDJ submitted 2025-07-28 cond-mat.str-el hep-thquant-ph

Wilbur Shirley , Carolyn Zhang , Wenjie Ji , Michael Levin This is my paper
classification cond-mat.str-elhep-thquant-ph
keywords anomaly-freesymmetryonsiteabilitygaugingobstructionquantumcellularautomatonGNVWindexgroupcohomologylatticeHamiltoniansymmetry-protectedtopologicalorder
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

The paper claims that the usual chain of implications—anomalous⇒ not gaugeable ⇒ not onsiteable—is not reversible in two dimensions. The authors construct finite, unitary, internal symmetries of 2D lattice Hamiltonians that admit symmetric, gapped Hamiltonians with unique, invertible ground states, yet cannot be rewritten as an on-site symmetry or coupled to a background or dynamical gauge field. These are counterexamples to the lore that any symmetry which resists gauging or on-site rewriting must be anomalous. The paper further identifies the mechanism: such symmetries carry a nontrivial index in the cohomology group $H^2(G,\mathbb{Q}_+)$, built from the GNVW indices of one-dimensional boundary operators obtained by restricting the symmetry to a disk. If correct, the standard no-go picture of anomalies in lattice systems is incomplete: anomaly-freedom by itself does not guarantee gaugeability or onsiteability.

What carries the argument

The central object is the $H^2(G,\mathbb{Q}_+)$ onsiteability index. For a symmetry $\{U_g\}$ written as a finite-depth quantum circuit and a restriction $U_{g,A}$ to a large disk $A$, the paper forms $\Omega_{g,h}=U_{g,A}U_{h,A}U^{-1}_{gh,A}$, a unitary supported near the boundary; its GNVW index—the positive rational number measuring how many degrees of freedom a one-dimensional quantum cellular automaton shifts across a cut—defines a cocycle $\omega(g,h)\in\mathbb{Q}_+$. The cohomology class $[\omega]$ is invariant under the choice of restriction. The GNVW index is load-bearing because it completely classifies one-dimensional quantum cellular automata up to finite-depth circuits and ancillas, so the boundary shift's index cannot be undone by local redefinitions.

What would settle it

Search for a one-dimensional quantum cellular automaton with GNVW index 1 that is not equivalent to a finite-depth circuit even after adding ancillas; the classification used here says none exists, and finding one would remove the ground on which the onsiteability obstruction stands. Alternatively, produce a 2D symmetry whose cocycle is a nontrivial element of $H^2(G,\mathbb{Q}_+)$ but that can be explicitly conjugated to an on-site symmetry, which would refute the paper's claim that the index obstructs onsiteability.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is a family of 2D $G$-symmetries, one for every finite group $G$ and every cohomology class $[\omega]\in H^2(G,\mathbb{Q}_+)$, that are anomaly-free but not onsiteable and not gaugeable. The basic example is a $\mathbb{Z}_2$ symmetry on the honeycomb lattice: $U_g = \hat{S}[\{\sigma^z_p\}] \prod_p \sigma^x_p$, where $\hat{S}$ translates vertex qubits along the domain walls of the plaquette configuration. Restricting $U_g$ to a disk and squaring yields an odd translation along the disk boundary, an operation whose GNVW index is $2$; this boundary shift cannot be removed by any choice of restriction or by conjugation, so the symmetry is not a finite-depth-conjugated on-site symmetry. The same symmetry has a commuting-projector parent Hamiltonian with a unique product ground state, so it is anomaly-free in the strongest sense. The paper proves that a nontrivial index obstructs background gauging, that this index is realized for every element of $H^2(G,\mathbb{Q}_+)$, and that dynamical gaugeability is equivalent to onsiteability, giving additional examples including a Clifford-circuit $\mathbb{Z}_4$ symmetry of the toric code that permutes the $e$ and $m$ anyons.

Load-bearing premise

The argument's load-bearing premise is the completeness of the GNVW index for one-dimensional quantum cellular automata: every 1D QCA is equivalent, up to finite-depth circuits and ancillas, to a translation with a rational index, and two QCAs are equivalent exactly when their indices agree, so a boundary shift with index 2 cannot be undone by local redefinitions.

Editorial extensions

If this is right

  • Every 2D symmetry that is a finite-depth quantum circuit carries a well-defined invariant $[\omega]\in H^2(G,\mathbb{Q}_+)$; a nontrivial class obstructs onsiteability and also background and dynamical gauging.
  • Anomaly-free is strictly weaker than onsiteable in two dimensions: the constructed symmetries admit symmetric gapped Hamiltonians with unique invertible ground states, yet cannot be disentangled to on-site operators.
  • The obstruction is not preserved under renormalization-group flow, so it can disappear in a low-energy subspace even though it was present in the original lattice model.
  • The paper realizes every class $[\omega]\in H^2(G,\mathbb{Q}_+)$ by an explicit 2D symmetry for any finite group $G$, and gives additional examples including a $\mathbb{Z}_4$ symmetry of the toric code that permutes $e$ and $m$ anyons and fermionic Gaussian symmetries.

Reading between the lines

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

  • The construction suggests that the $H^2$ index is a kinematic, lattice-level obstruction invisible to anomaly inflow and to low-energy topological field theory, since it is not RG-stable; this invites a systematic search for similar cohomology-valued obstructions in higher dimensions by decorating lower-dimensional junctions with QCAs, as the paper itself anticipates.
  • Because the index is defined through the GNVW classification of 1D QCAs, any future refinement or exception to that classification would directly change which symmetries are obstructed; the paper's conclusion is therefore tied to the completeness of that classification.
  • The fermionic Gaussian examples indicate that analogous obstructions occur for locally generated symmetries with algebraic tails and connect the index to Chern numbers, even though those symmetries are not strictly finite-depth circuits; formalizing the index for locally generated unitaries is a natural next step.
  • At a practical level, the boundary index of $U_{g,A}^n$ is a finite-size probe: a numerical or experimental measurement of the shift of operators across a cut could certify non-onsiteability of a symmetry without needing to solve for its anomalies.
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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

0 major / 7 minor

Summary. This paper addresses the question whether every non-onsiteable two-dimensional internal symmetry is necessarily anomalous. The main construction is a Z2 symmetry on a honeycomb lattice, defined by U_g = Ŝ[{σ^z}]∏_p σ^x_p, where Ŝ translates τ-qubits along the domain walls of the σ^z configuration. The authors prove that U_g is an FDQC, that the frustration-free Hamiltonian H = -∑_i τ^x_i -∑_p σ^x_p -∑_p U^{-1}_g σ^x_p U_g is gapped with |Ψ0⟩ = ⊗ |σ^x=+1, τ^x=+1⟩ as its unique ground state, and that the boundary operator U^2_{g,A} has GNVW index 2, giving a nontrivial element [ω]∈H^2(Z2,Q+). They show that a nontrivial [ω] obstructs both onsiteability and background gauging, and in the Supplemental Material they generalize the construction to arbitrary finite G and cocycle [ω]∈H^2(G,Q+) via a bulk-boundary map from 3D G-symmetric QCAs. Additional examples include a Z4 toric-code symmetry and fermionic Gaussian symmetries.

Significance. The Z2 example is a clean and explicit counterexample to the lore that an obstruction to gauging or onsiteability implies an anomaly. The argument is self-contained and uses the established GNVW classification of 1D QCAs as an external input rather than adjusting definitions to fit the conclusion; the gap and uniqueness proof for the parent Hamiltonian is complete. The general H^2(G,Q+) index is a natural and computable invariant, and the SM construction realizing every cohomology class is a substantial classification step. The main caveat is that the index is defined for FDQC symmetries; whether it extends to all QCA symmetries in 2D is left as a plausible but unproven assumption, so the characterization in the abstract should be read with that caveat.

minor comments (7)
  1. [SM V] The Z4 toric-code symmetry is introduced as 'anomaly-free,' but no symmetric gapped Hamiltonian with a unique invertible ground state is provided; the toric-code Hamiltonian itself has a topologically degenerate ground-state space and therefore does not satisfy the paper's own anomaly-free definition. Please either supply such a parent Hamiltonian or explicitly label this example as one whose anomaly-free status is not established.
  2. [SM VI.A] The fermionic Gaussian symmetries are not strictly locality-preserving QCAs, and the paper states that the index definition does not directly apply; the non-onsiteability conclusion therefore rests on an assumed extension of the H^2(G,Q+) formalism to locally generated symmetries. This conditional status should be stated prominently when these examples are summarized.
  3. [Obstruction to background gauging] After Eqs. (11) and (12), the claim that Ω_{g,h} is an FDQC needs one more step: from U_{g,A}V1U_{h,A}U^{-1}_{gh,A} ∝ V2 one should insert U_{g,A}^{-1}U_{g,A} and use that conjugation of an FDQC by a QCA is again an FDQC. Writing this out would remove the only non-obvious step in an otherwise clear argument.
  4. [H^2(G,Q+) onsiteability index] The sentence 'Likewise, the same is true for any onsiteable symmetry' is terse, since onsiteability allows conjugation by an arbitrary QCA rather than an FDQC. A short justification, or an explicit restriction of the claim to FDQC disentanglers, would make the obstruction statement precise.
  5. [SM III.C, Eq. (32)] In the definition of the modified Gauss law operators, the right-hand side contains indices s'_k and s_k that are not defined; the intent appears to be a map from site configurations to link configurations, but as written the equation is difficult to parse and should be corrected.
  6. [SM I] For readers not familiar with the GNVW machinery, it would help to state that for commuting subalgebras the overlap η(X,Y) is the square root of the dimension of the intersection algebra; this makes the index value 2 for a single-site translation immediately transparent.
  7. [Introduction, footnote 9] The abstract and introduction characterize all of the constructed symmetries by the H^2(G,Q+) index, but the index is defined only for FDQC symmetries and the equality with all QCA symmetries in 2D is presented as plausible rather than proven. This caveat should appear wherever the classification statement is made.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central Z2 counterexample and H^2(G,Q+) index are derived from the independent GNVW classification; only minor non-load-bearing self-citations appear.

full rationale

The central derivation chain is self-contained. The Z2 symmetry Ug (Eq. 1) is shown anomaly-free by an explicit symmetric, frustration-free Hamiltonian (Eq. 2), with gap and uniqueness proven by reducing to the sigma^x sector; this part does not use the H^2 index. Non-onsiteability follows from the boundary square U^2_{g,A} being an odd shift, and the external input is the GNVW classification of 1D QCAs (Ref. [1]), an established theorem used as stated: alternative restrictions differ by boundary 1D QCAs whose indices contribute only even shifts. The H^2(G,Q+) index (Eqs. 5-6) is defined from the GNVW index of Omega_{g,h}; nontriviality for the example is computed, not assumed. The background-gauging obstruction is proved by contrapositive from the defining properties of background gaugeability, and dynamical gaugeability is reduced to background gaugeability in SM III B by an explicit Gauss-law construction; neither step assumes the conclusion. The only self-reliance is minor and non-load-bearing: the supplementary proposition that dynamical gaugeability is equivalent to onsiteability (SM III C) invokes a construction from Ref. [6] (arXiv:2503.09717), co-authored by one of the present authors, and the general all-cocycle construction cites the bulk-boundary map [18,19] with a co-authored component. These do not support the core counterexample: the central obstruction to dynamical gauging is already implied by the background-gauging proof, and the general construction is verified explicitly in the SM. The toric-code and fermionic examples are supplementary and flagged as incomplete, so they carry no circularity burden. Overall, no prediction reduces to a fit, no definition is circular, and no load-bearing argument reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters and no new physical entities. The only inputs are the group G and a cocycle, and the index values are computed, not tuned. The assumptions are standard QCA classification theorems, domain restrictions to finite-dimensional local Hilbert spaces, and the stated restriction to FDQC symmetries.

assumptions (4)
  • standard math The GNVW index completely classifies 1D quantum cellular automata (Gross, Nesme, Vogts, Werner).
    Used to define the H^2(G,Q+) index and to argue boundary shifts are robust; see main text after Eq. (6) and SM Sec. I.
  • standard math For any FDQC symmetry, restrictions to a disk exist, and any two restrictions differ by a 1D QCA near the boundary.
    Invoked in the definition of the index and in the Z2 non-onsiteability argument; follows from Else-Nayak, Ref. [7].
  • domain assumption The Hilbert spaces are finite-dimensional local tensor products and symmetries are strict QCAs.
    Assumed in the definition of anomaly-free and in all index arguments; the paper notes fermionic examples only satisfy a weaker locality condition.
  • ad hoc to paper For finite G in 2D, every QCA symmetry is plausibly an FDQC, so the index applies to all QCA symmetries.
    The authors state this is plausible via Freedman-Hastings (Ref. [17]) but not proven; the classification is strictly for FDQC symmetries.

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Cite this review

Pith. "Pith review of Anomaly-free symmetries with obstructions to gauging and onsiteability." pith.science (2026). https://pith.science/paper/LVOSSVDJ

@misc{pith2026250721267,
  author       = {Pith},
  title        = {Pith review of: Anomaly-free symmetries with obstructions to gauging and onsiteability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVOSSVDJ}},
  note         = {Machine review of arXiv:2507.21267}
}
abstract

We present counterexamples to the lore that symmetries that cannot be gauged or made on-site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index $[\omega]\in H^2(G,\mathbb{Q}_+)$, where $\mathbb{Q}_+$ is the multiplicative group of rational numbers labeling one-dimensional quantum cellular automata.

Figures

Figures reproduced from arXiv: 2507.21267 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A fundamental property of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Action of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Graphical representation of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (9 more)
Figure 1
Figure 1. Figure 1: b. One can check that the above depth-6 circuit exactly realizes the operator Sˆ[{σ z p}].[3] To realize Ug, one simply needs to add one more layer to implement the spin flip operation Q p σ x p . B. Action of Ug on local operators We now discuss how the Pauli operator…
Figure 2
Figure 2. Figure 2: FIG. 2. The labeling of the vertices [PITH_FULL_IMAGE:figures/full_fig_p010_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The degrees of freedom of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. A triangulated surface [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Triangular lattice with a global vertex ordering. (b) 1D qubit chains living on edges of the dual lattice. Each dot represents a qubit. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Truncated unitary [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]

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Reference graph

Works this paper leans on

65 extracted references · 48 canonical work pages · cited by 2 Pith papers

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    First we choose a re- striction Ug,A of the symmetry operator Ug to a disk A: Ug,A = ˆS[{˜σz p}] ∏ p∈A σx p ˜σz p = { σz p, if p ∈ A 1, otherwise

    discussed earlier. First we choose a re- striction Ug,A of the symmetry operator Ug to a disk A: Ug,A = ˆS[{˜σz p}] ∏ p∈A σx p ˜σz p = { σz p, if p ∈ A 1, otherwise . (8) This is a valid restriction because Ug,A is a QCA that acts like Ug in the interior of region A and acts as the identity operator outside of A. Next, we compute Ωg,g . Using ( 8) we have...

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    If g′ i = gi except for i ∈ R for some subset R ⊂ Λ, then U{g′ i} ∝ W U{gi} for some unitary W supported within a finite distance of R. Here, we use ‘ ∝’ to denote equality up to a phase. This defi- nition deserves a few comments:

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    We conclude that the equivalence class [ω] ∈ H 2(G, Q+) is a well-defined quantity insensitive to the particular choice of restriction

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    Let {gij} be a background gauge field – that is, an as- signment of group elements to links ⟨ij⟩ with the identifica- tion gij = g−1 ji

    To motivate this definition, we note that the U{gi} opera- tors provide a canonical way to couple an arbitrary local sym- metric Hamiltonian H to a flat 10 spatial background gauge field. Let {gij} be a background gauge field – that is, an as- signment of group elements to links ⟨ij⟩ with the identifica- tion gij = g−1 ji . Suppose that gij is “flat”, that is, ...

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