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arxiv: 2605.12601 · v1 · submitted 2026-05-12 · ❄️ cond-mat.str-el · hep-th

Recognition: 2 theorem links

· Lean Theorem

Lattice Gauging Interfaces and Noninvertible Defects in Higher Dimensions

David Hofmeier , Giovanna Pimenta , Weiguang Cao
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Pith reviewed 2026-05-14 20:34 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-th
keywords gaugingdefectsinterfaceslatticemathbbsymmetrycondensationdimensions
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The pith

Explicit lattice constructions of gauging interfaces and condensation defects are given for higher-dimensional systems with higher-form symmetries, using movement operators to manage constrained Hilbert spaces.

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

In quantum systems, symmetries can be gauged, which means promoting them to local constraints that remove some degrees of freedom. When this gauging is done only on one side of an interface, the boundary between gauged and ungauged regions becomes a special defect. In higher dimensions these defects are hard to handle because the allowed quantum states change depending on where the interface sits. The authors introduce movement operators that move both the interface Hamiltonian and its constraints together across a common, unconstrained space. They apply this to build condensation defects from finite-region gauging and to reconstruct the full gauging map. They also treat subgroup gauging, such as gauging Z2 inside Z4, which produces a dual symmetry with a mixed anomaly that can be diagnosed by how the symmetry acts on the defects.

Core claim

We construct explicit interface Hamiltonians for gauging a Z2^(0) symmetry in (2+1)d and a Z2^(1) symmetry in (3+1)d. In higher dimensions... we resolve this by introducing movement operators acting on a common unconstrained Hilbert space, which transport both the interface Hamiltonians and the associated constraints.

Load-bearing premise

That the movement operators can be defined to act consistently on the common unconstrained Hilbert space while correctly transporting both the interface Hamiltonians and the location-dependent constraints without introducing new inconsistencies or anomalies.

Figures

Figures reproduced from arXiv: 2605.12601 by David Hofmeier, Giovanna Pimenta, Weiguang Cao.

Figure 1
Figure 1. Figure 1: We gauge the semi-infinite region B to localize a duality interface D on W that interpolates between a theory with global Z (0) 2 symmetry (blue, region A) and a theory with quantum Z (1) 2 symmetry (red, region B). The boundary sets WA and WB are indicated by darker shades. The arrow on D denotes its orientation and distinguishes it from its conjugate D† . Geometry of the interface. We divide the lattice … view at source ↗
Figure 2
Figure 2. Figure 2: Local action of the gauging map in region [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The movement operator λ 0,1 e0,v0 removes the edge e0 from region B and enlarges region A by the vertex v0. Construction of movement operator. A general movement operator Λ0,1 W,W′ can be constructed as a product of minimal movement operators λ 0,1 e0,v0 . These minimal movement operators move the gauging interface W to include an adjacent vertex in the gauged region B, as illustrated in [PITH_FULL_IMAGE:… view at source ↗
Figure 4
Figure 4. Figure 4: Up to rotations, there are four possible local movements in 2d. These [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Fusing two C1-defects located on Σ1 and Σ′ 1 . 3.4 Relating the Different Condensation Defects. With the movement operators, we can progressively enlarge the single vertex Σ0 to the non￾contractible cycle Σ1, thus relating the decomposable defect C1(Σ0) to the indecomposable defect C1(Σ1). The whole process is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: After successive application of the movement operator, we can relate [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Starting from a one-dimensional condensation defect [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Local action of the gauging map on the three-dimensional lattice. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Graphical depiction of 3d half gauging. We gauge in sublattice B such that the model hosts dual degrees of freedom on plaquette centers (depicted in red). The degrees of freedom participating in interactions across the boundary are shaded in a darker color. Together, they make up the interface, denoted by W = WA ⊔ WB. In this simple case of a flat interface, these interactions are all given by ZeXp where e… view at source ↗
Figure 10
Figure 10. Figure 10: An exemplary movement in 3d which removes the plaquette [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The bulk Z (1) 2 defect η1(Σ∨ 1 ) can be moved onto the condensation defect C1(Σ2), where it becomes a Wilson line We(Σ∨ 1 ) of the Z2 gauge theory defined on Σ2. One can revert back to the original condensation defect Hamiltonian by applying We(Σ∨ 1 ) = Q p∈Σ∨ 1 Zp. Note that We(Σ∨ 1 ) is the electric Wilson line of the 2d Z2 gauge theory local￾ized on Σ2. If Σ∨ 1 has endpoints, it creates charged partic… view at source ↗
Figure 12
Figure 12. Figure 12: Fusing two condensation defects produces a single condensation defect [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: In (i), we fuse two Ue operators in the presence of a Ve condensation defect. In (ii), two Ve operators are fused in the presence of a Ue condensation defect. In both cases, the symmetry operator fractionalizes on the condensation defect. 5.3 New Defect Diagnosis of Symmetry Fractionalization In Sec. 5.1, we have reviewed that both Z2 symmetries fractionalize in the presence of a symmetry twist of the oth… view at source ↗
Figure 14
Figure 14. Figure 14: Fusing two symmetry defects results in an operator that is charged [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (a) For the configuration in Figure 15a. The movement operator acts non-trivially on the following terms of the Hamiltonian Zv←Xe0 7→ Zv←Zv0 , (161a) Zv↓Xe↓ 7→ Zv↓Xe↓Zv0 , (161b) Av0 7→ Xv0 , (161c) Xe↑ 7→ Zv0Xe↑ , (161d) Xe→ 7→ Zv0Xe→ . (161e) It also changes the constraints 1 = Zv0 7→ Xe0 = 1 , (162a) 1 = Xe0Xe↓ 7→ Xe↓ = 1 , (162b) where the second constraint comes from considering Vγ0 with γ0 being the… view at source ↗
Figure 15
Figure 15. Figure 15: Up to rotations, there are four possible local movements in 2d. These [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
read the original abstract

We study gauging interfaces and their defect descendants in lattice models with generalized symmetries in higher dimensions. We construct explicit interface Hamiltonians for gauging a $\mathbb Z_2^{(0)}$ symmetry in $(2+1)d$ and a $\mathbb Z_2^{(1)}$ symmetry in $(3+1)d$. In higher dimensions, and especially in the presence of higher-form symmetries, the topological nature of gauging interfaces is obscured by the fact that the constrained Hilbert space depends on the location of the interface. We resolve this by introducing movement operators acting on a common unconstrained Hilbert space, which transport both the interface Hamiltonians and the associated constraints. As applications, we analyze condensation defects obtained from finite-region gauging and reconstruct the gauging map from movement operators. Finally, we apply the same framework to subgroup gauging, focusing on the example of gauging $\mathbb Z_2\subset \mathbb Z_4$. This produces a dual symmetry carrying a mixed anomaly, which we diagnose on the lattice through symmetry fractionalization on condensation defects. Our results provide an explicit lattice framework for studying topological interfaces, condensation defects, and the associated anomalies arising from gauging in higher-dimensional systems.

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

1 major / 1 minor

Summary. The manuscript constructs explicit interface Hamiltonians for gauging a Z_2^{(0)} symmetry in (2+1)d and a Z_2^{(1)} symmetry in (3+1)d. It introduces movement operators on a common unconstrained Hilbert space to transport both the interface Hamiltonians and the location-dependent constraints, thereby addressing the topological obstruction in higher dimensions. Applications include condensation defects from finite-region gauging, reconstruction of the gauging map from movement operators, and subgroup gauging of Z_2 inside Z_4, where a mixed anomaly is diagnosed via symmetry fractionalization on condensation defects.

Significance. If the algebraic consistency of the movement operators is established, the work supplies a concrete lattice framework for gauging interfaces and noninvertible defects with generalized symmetries, extending prior abstract treatments to explicit Hamiltonians and constraint transport. The explicit constructions for both 0-form and 1-form cases, together with the anomaly diagnosis on condensation defects, would be a useful addition to the literature on higher-form symmetries in condensed-matter models.

major comments (1)
  1. [Movement operators for higher-form symmetries] In the section defining and applying the movement operators to the (3+1)d Z_2^{(1)} case, the claim that these operators transport both the interface Hamiltonian and the position-dependent constraints (including higher-form Gauss laws) without introducing new anomalies or inconsistencies lacks an explicit algebraic verification. A direct check that the transported constraints remain mutually commuting and that the interface commutes with the dual symmetry operators in the expected way is required, as this step is load-bearing for resolving the topological obstruction.
minor comments (1)
  1. [Abstract] The abstract states that the framework applies 'in higher dimensions' but the explicit constructions are given only for (2+1)d and (3+1)d; a brief statement clarifying the scope for general d would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment below and will incorporate the requested algebraic verification in the revised version.

read point-by-point responses

  1. Referee: In the section defining and applying the movement operators to the (3+1)d Z_2^{(1)} case, the claim that these operators transport both the interface Hamiltonian and the position-dependent constraints (including higher-form Gauss laws) without introducing new anomalies or inconsistencies lacks an explicit algebraic verification. A direct check that the transported constraints remain mutually commuting and that the interface commutes with the dual symmetry operators in the expected way is required, as this step is load-bearing for resolving the topological obstruction.

    Authors: We agree that an explicit algebraic verification is needed to fully substantiate the claim. In the revised manuscript we will add a new subsection (or appendix) containing the direct calculation: we will explicitly compute the action of the movement operators on the position-dependent higher-form Gauss-law constraints, verify that the transported constraints remain mutually commuting, and confirm that the interface Hamiltonian commutes with the dual symmetry operators in the manner required by the gauging construction. These relations will be presented both in operator form and via their action on representative states in the unconstrained Hilbert space. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit lattice constructions stand independently

full rationale

The paper's core contribution consists of direct constructions of interface Hamiltonians for Z2^(0) gauging in (2+1)d and Z2^(1) gauging in (3+1)d, together with movement operators defined on a common unconstrained Hilbert space. These objects are introduced by explicit definition rather than by fitting parameters to data, renaming prior results, or reducing via self-citation chains. The movement operators are posited to transport both Hamiltonians and location-dependent constraints; this is a constructive step whose consistency is asserted by the definitions themselves, not derived tautologically from the target claim. No equations reduce to their own inputs by construction, and no load-bearing uniqueness theorems or ansatze are imported from the authors' prior work in a self-referential manner. The framework is therefore self-contained as a lattice model-building exercise.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not list explicit free parameters or invented entities; the framework appears to rest on standard lattice gauge theory assumptions plus the new movement operators.

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