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arxiv: 2606.06343 · v1 · pith:KQWES3S3new · submitted 2026-06-04 · ❄️ cond-mat.str-el · hep-th· math-ph· math.MP

E_infty^(1,2)-type Lieb-Schultz-Mattis anomalies, deconfined quantum critical points, and non-invertible symmetry breaking

Hao-Ran Zhang , Hanlin Lin , Shuo Yang , Qing-Rui Wang This is my paper

Pith reviewed 2026-06-27 23:24 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thmath-phmath.MP
keywords Lieb-Schultz-Mattis anomalydeconfined quantum critical pointnon-invertible symmetryspectral sequencespin chainD8 symmetrygauging
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The pith

Gauging an E∞^{1,2}-type LSM anomaly necessarily produces a non-invertible dual symmetry.

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

The paper characterizes Lieb-Schultz-Mattis anomalies in one-dimensional spin chains via the Lyndon-Hochschild-Serre spectral sequence, locating a class of anomalies in the E∞^{1,2} term that decorates translation defects with projective representations of an internal symmetry group. It demonstrates that gauging this internal symmetry under such an anomaly condition yields a non-invertible dual symmetry. This mechanism accounts for type-II deconfined quantum critical points, which are dual to spontaneous breaking of non-invertible symmetries, in contrast to type-I points that break ordinary group symmetries. The argument is worked out explicitly for a spin-1/2 chain with anomalous D8 symmetry, where gauging produces the non-invertible Rep(H8) symmetry and a candidate dimer-to-ferromagnet transition appears with central charge near 1.

Core claim

Gauging the internal symmetry in the presence of an E∞^{1,2}-type anomaly necessarily produces a non-invertible dual symmetry. This supplies the general mechanism for type-II DQCP: in contrast to type-I examples with E∞^{2,1}-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry.

What carries the argument

The E∞^{1,2} term of the spectral sequence, equal to H^1(Z_trans, H^2(G_int, U(1))) inside H^3(G_int ⋊_ρ Z_trans, U(1)), which decorates a translation defect with a projective representation of the internal symmetry; the gauging map that converts this anomaly into a non-invertible dual symmetry.

If this is right

  • Type-II DQCPs correspond to spontaneous breaking of non-invertible symmetries rather than ordinary symmetries.
  • In the D8 example the dual symmetry after gauging is the representation category Rep(H8).
  • The critical theory at the dimer-to-ferromagnet point has central charge approximately 1.
  • Numerical evidence on the lattice model supports the existence of this critical point.

Where Pith is reading between the lines

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

  • The same spectral-sequence placement may classify non-invertible dualities in models with mixed translation and internal symmetries beyond one dimension.
  • Explicit lattice constructions of the non-invertible symmetry could be used to test duality relations in other candidate DQCP models.
  • Category-theoretic descriptions of the dual symmetry may connect to classification schemes for topological phases protected by non-invertible symmetries.

Load-bearing premise

That an anomaly placed in the E∞^{1,2} position of the spectral sequence must produce a non-invertible symmetry when the internal symmetry is gauged.

What would settle it

An explicit lattice model or category-theoretic calculation in which an E∞^{1,2}-type LSM anomaly is gauged yet only an invertible (group-like) dual symmetry appears.

Figures

Figures reproduced from arXiv: 2606.06343 by Hanlin Lin, Hao-Ran Zhang, Qing-Rui Wang, Shuo Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: Effect of LHS anomaly components under gauging an Abelian normal subgroup [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Decorated-domain-wall representatives for a 2+1D bulk SPT with 1 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The peak of the MPS correlation length [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The MPS correlation length [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The MPS entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Growing peak of the MPS correlation length [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗
read the original abstract

We study deconfined quantum critical points (DQCP) associated with Lieb-Schultz-Mattis (LSM) anomalies in one-dimensional spin chains. Our starting point is a structural characterization of the LSM anomaly in the Lyndon-Hochschild-Serre spectral sequence: $\omega_{\mathrm{LSM}}\in E_\infty^{1,2}= H^1(\mathbb Z_{\mathrm{trans}},H^2(G_{\mathrm{int}},\mathrm{U}(1)))\subseteq H^3(G_{\mathrm{int}}\rtimes_{\rho}\mathbb Z_{\mathrm{trans}},\mathrm{U}(1))$. Physically, this class decorates a translation defect with a projective representation of the internal symmetry $G_\mathrm{int}$. We show that gauging the internal symmetry in the presence of an $E_\infty^{1,2}$-type anomaly necessarily produces a non-invertible dual symmetry. This gives a general mechanism for type-II DQCP: in contrast to type-I examples with $E_\infty^{2,1}$-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry. We illustrate the mechanism using a spin-$1/2$ chain with an anomalous $D_8$ LSM symmetry. We construct a dimer-to-ferromagnet DQCP candidate, provide numerical evidence for a critical theory with central charge $c\approx 1$, and show, using both category theory and explicit lattice constructions, that gauging the internal symmetry yields the non-invertible $\mathrm{Rep}(H_8)$ dual symmetry.

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 manuscript claims that LSM anomalies classified in the E_∞^{1,2} term of the Lyndon-Hochschild-Serre spectral sequence (ω_LSM ∈ H^1(Z_trans, H^2(G_int, U(1))) ⊆ H^3(G_int ⋊_ρ Z_trans, U(1))) decorate translation defects with projective G_int representations; gauging the internal symmetry in the presence of such an anomaly necessarily produces a non-invertible dual symmetry. This supplies a general mechanism for type-II DQCPs, distinct from type-I cases with E_∞^{2,1} anomalies that are dual to ordinary group-like symmetry breaking. The claim is illustrated by an explicit spin-1/2 chain with anomalous D_8 LSM symmetry, including lattice constructions of a dimer-to-ferromagnet transition, category-theoretic identification of the dual Rep(H_8) symmetry, and numerical evidence for a critical theory with c ≈ 1.

Significance. If the central claims hold, the work supplies a cohomological organizing principle that distinguishes type-I and type-II DQCPs via the anomaly class and directly ties the latter to spontaneous breaking of non-invertible symmetries. Credit is due for the explicit D_8 lattice model, the category-theoretic and lattice constructions of the dual symmetry, and the accompanying numerical data; these elements make the proposed mechanism falsifiable and provide a concrete benchmark for future studies of non-invertible symmetries in one dimension.

major comments (2)
  1. [§3] §3 (general gauging argument): the assertion that an E_∞^{1,2}-type anomaly 'necessarily' produces a non-invertible dual symmetry upon gauging rests on the defect-decoration interpretation of the class; the manuscript does not supply an explicit general computation of the dual fusion category (or a proof that it cannot be group-like) that is independent of the specific extension class ρ or the choice of G_int, which is load-bearing for the type-II DQCP mechanism.
  2. [§5.2] §5.2 (D_8 numerical evidence): the reported central charge c ≈ 1 is consistent with a critical theory, but the finite-size scaling analysis of the dimerization order parameter and the correlation-length exponent across the putative transition is not presented in sufficient detail to rule out a conventional (non-deconfined) critical point, weakening support for the DQCP interpretation.
minor comments (2)
  1. The notation E_∞^{1,2} and the embedding into H^3(G_int ⋊_ρ Z_trans, U(1)) would benefit from a short self-contained reminder of the filtration and convergence properties of the spectral sequence in the introduction.
  2. Figure captions for the numerical data should explicitly state the system sizes used and the fitting procedure for the central charge extraction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments, which help clarify the presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (general gauging argument): the assertion that an E_∞^{1,2}-type anomaly 'necessarily' produces a non-invertible dual symmetry upon gauging rests on the defect-decoration interpretation of the class; the manuscript does not supply an explicit general computation of the dual fusion category (or a proof that it cannot be group-like) that is independent of the specific extension class ρ or the choice of G_int, which is load-bearing for the type-II DQCP mechanism.

    Authors: The defect-decoration interpretation follows directly from the definition of the E_∞^{1,2} class in the Lyndon-Hochschild-Serre spectral sequence and is the physical content of the anomaly. This decoration of translation defects by projective G_int representations implies that, after gauging G_int, the dual symmetry must fuse in a manner incompatible with a group-like structure for any choice of ρ and G_int; this is the cohomological distinction from E_∞^{2,1} anomalies. The D_8 example provides an explicit realization, but the argument itself is independent of the specific group because it relies only on the existence of the projective decoration. We will add a clarifying paragraph in §3 making this cohomological obstruction explicit without reference to a particular model. revision: partial

  2. Referee: [§5.2] §5.2 (D_8 numerical evidence): the reported central charge c ≈ 1 is consistent with a critical theory, but the finite-size scaling analysis of the dimerization order parameter and the correlation-length exponent across the putative transition is not presented in sufficient detail to rule out a conventional (non-deconfined) critical point, weakening support for the DQCP interpretation.

    Authors: We agree that additional finite-size scaling details would strengthen the case for deconfined criticality. In the revised manuscript we will include data-collapse plots for the dimerization order parameter and estimates of the correlation-length exponent obtained from multiple system sizes, allowing a clearer comparison with conventional critical points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on standard spectral sequence and external category/lattice constructions.

full rationale

The paper takes the E_∞^{1,2} characterization of the LSM anomaly as its starting point via the Lyndon-Hochschild-Serre spectral sequence (a standard tool in group cohomology) and then derives the non-invertible dual symmetry upon gauging. This is illustrated concretely for the D8 case with explicit category-theoretic arguments, lattice constructions, and numerics (c≈1), without any step reducing by definition or self-citation to the target result itself. No fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing uniqueness theorems imported from the authors' own prior work appear in the derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities are identifiable from the abstract alone.

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Reference graph

Works this paper leans on

139 extracted references · 4 canonical work pages

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    This gauging is qualitatively different from gaugingZ XY 2

    GaugingZ a2 2 : self-dual invertible symmetry We next gauge the central subgroupZ a2 2 ⊂G int, which is generated byU XY UY X = Q i Zi. This gauging is qualitatively different from gaugingZ XY 2 . The subgroup is central and diagonal in the internal symmetry, so the operation is closer to an ordinary Kramers-Wannier relabeling of order and disorder variab...

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    It can be implemented sequentially by first gauging the diagonal subgroupZ a2 2 and then gauging the remainingZ XY 2 subgroup

    Gauging the full internal symmetryZ XY 2 ×Z Y X 2 Gauging the full internal symmetry means gauging the normal Abelian subgroupG int =⟨U XY , UY X ⟩=⟨ax, a 2⟩. It can be implemented sequentially by first gauging the diagonal subgroupZ a2 2 and then gauging the remainingZ XY 2 subgroup. The resulting dual symmetry is Rep(H 8). This sequential implementation...

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    We now consider the dual Hamilto- nian of Eq

    Dual model of the dimer-to-ferromagnet DQCP In the preceding discussion, we derived the lattice gauging procedure in detail. We now consider the dual Hamilto- nian of Eq. 71 after gauging anomaly-free subgroups ofD 8. The dual Hamiltonian is obtained by tracking the gauge image of each local symmetric operator. The advantage of the bond-algebra constructi...

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    Fusion-category symmetries and gapped phases LetCbe a fusion category overC. Physically, in a 1+1D systemTwith fusion-category symmetryC, objects of Care topological defect lines, the tensor productX⊗Yis the fusion of two defect lines, and the associator is the F-symbol that tells us how three lines reassociate [57, 60, 61, 63, 64]. A simple object is an ...

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    Gauging algebras and dual symmetries Generalized gauging is a different use of condensable algebras. To gauge, one chooses a condensable algebraA∈ C, or equivalently the corresponding interface module categoryC A. This is operational data, not merely a name for a gapped phase: the network to be summed over is built from the multiplication and unit ofA. Ph...

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    Algebras inVec ω G The basic pointed example is Vec ω G. Its simple objects are labeled by elementsg∈G, and they fuse by group multiplication,g⊗h=gh. The associator is multiplication by a normalized 3-cocycleω(g, h, k). Thus Vec G describes an ordinary non-anomalousGsymmetry, while Vec ω G describes aGsymmetry with ’t Hooft anomaly [ω]∈H 3(G,U(1)), as for...

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    FromE 1,2 ∞ decomposition to type-II DQCP We now choose the rank-two normal subgroupA=Z a 2 ×Z b 2 and quotientQ=G/A=Z c 2, giving the short exact sequence 1→Z a 2 ×Z b 2 →Z a 2 ×Z b 2 ×Z c 2 →Z c 2 →1.(B5) This choice is the one most directly adapted to the lattice construction below. In the corresponding LHS spectral sequence Ep,q 2 =H p(Zc 2, Hq(Za 2 ×...

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    A candidate type-II DQCP We start from the boundary-action formula withω 3 =ω III 3 . In the group-variable basis|{g i}⟩, the anomalous actions of the generatorsa,b, andcofZ a 2 ×Z b 2 ×Z c 2 are Ua|{gi}⟩= Y ⟨ij⟩ ωsij 3 (g−1 i gj, g−1 j , a)|{gi +a}⟩=|{g i +a}⟩,(B8) Ub|{gi}⟩= Y ⟨ij⟩ ωsij 3 (g−1 i gj, g−1 j , b)|{gi +b}⟩=|{g i +b}⟩,(B9) Uc|{gi}⟩= Y ⟨ij⟩ ωs...

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