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arxiv: 2606.11527 · v1 · pith:2MMUOUIOnew · submitted 2026-06-10 · ❄️ cond-mat.str-el · hep-th· quant-ph

Invariants of Sequential Circuits and Generalized Non-Abelian Statistics

Shintaro Sato , Yoshimasa Hidaka , Ryohei Kobayashi This is my paper

Pith reviewed 2026-06-27 08:36 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thquant-ph
keywords non-invertible symmetryBerry phasesequential circuitstopological ordernon-Abelian statisticsloop excitations't Hooft anomalyD4 topological order
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The pith

Berry phases from sequential circuits moving non-invertible defects define invariants that detect 't Hooft anomalies and classify non-Abelian loop statistics in 3D.

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

The paper shows that sequences of unitary circuits moving symmetry defects produce a Berry phase on defect states that remains unchanged under local deformations whenever the circuits preserve locality. This invariance supplies a diagnostic for 't Hooft anomalies of non-invertible symmetries by excluding short-range-entangled states that would otherwise respect the symmetry. When applied to loop excitations, the same construction identifies a previously unrecognized non-Abelian fermionic loop in the (3+1)D D4 topological order and determines its generalized statistics. The framework additionally locates a new mixed topological order whose long-range entanglement is protected by the circuit invariant.

Core claim

Sequential unitary circuits that move non-invertible defects generate a Berry phase evaluated on quantum states with defects. This phase defines an invariant under local deformations provided the sequential circuits preserve the locality of those deformations. The invariant rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly purely in terms of unitary operators acting on a state. Applied to loop excitations in three spatial dimensions, the invariant characterizes a new non-Abelian fermionic loop in the (3+1)D D4 topological order and its statistics; it also identifies a new (3+1)D mixed topological order with a single suc

What carries the argument

The Berry phase invariant generated by locality-preserving sequential unitary circuits that move non-invertible symmetry defects.

If this is right

  • The invariant rules out short-range-entangled states that preserve non-invertible symmetries.
  • It characterizes the generalized statistics of non-Abelian fermionic loops in (3+1)D topological orders.
  • It identifies and protects a new mixed topological order containing a single non-Abelian fermionic loop.
  • The construction applies directly to loop excitations in three spatial dimensions.

Where Pith is reading between the lines

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

  • The locality-preserving condition may allow similar invariants to be constructed for other classes of excitations or symmetries in higher dimensions.
  • Numerical evaluation of the Berry phase on lattice models with sequential circuits could provide concrete checks of the predicted statistics.
  • The new non-Abelian fermionic loop may impose additional constraints on possible fusion or braiding rules in related topological phases.

Load-bearing premise

Sequential circuits must preserve the locality of the deformations for the Berry phase to remain invariant under those deformations.

What would settle it

An explicit short-range-entangled state that preserves a given non-invertible symmetry yet yields a trivial or non-invariant Berry phase under the corresponding sequential circuits would falsify the anomaly-detection claim.

Figures

Figures reproduced from arXiv: 2606.11527 by Ryohei Kobayashi, Shintaro Sato, Yoshimasa Hidaka.

Figure 1
Figure 1. Figure 1: FIG. 1. A sequential circuit [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. We break a sequential circuit [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Labels and orientations of a vertex and faces on the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Definition of the operator [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The surface operator [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Open surfaces [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. 24-unitary sequence to define the invariant. Red lines denote the configuration of the loop excitations, and blue arrows [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The red face is a support of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Left: The red bold line shows the loop excitation [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Microscopic definitions of the local edge operators [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Black lines show edges of dual surfaces representing boundaries of [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

Non-invertible symmetries in quantum many-body systems generally give rise to sequential unitary circuits that move symmetry defects. In this paper, we investigate invariants defined by sequences of such circuits, which move non-invertible defects and generate a Berry phase evaluated on quantum states with defects. We show that this Berry phase generally defines an invariant under local deformations, provided that the sequential circuits preserve the locality of those deformations. This invariant also rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling the 't Hooft anomaly of a non-invertible symmetry purely in terms of unitary operators acting on a state. We then apply this framework to loop excitations in three spatial dimensions and identify a new loop excitation in the (3+1)D $\mathbb{D}_4$ topological order, which we dub a non-Abelian fermionic loop. Using the invariant of sequential circuits, we characterize the statistics of non-Abelian fermionic loops. In addition, we find a new (3+1)D mixed topological order with a single non-Abelian fermionic loop, whose long-range entanglement is protected by an invariant of sequential circuits.

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 / 1 minor

Summary. The paper claims that sequences of unitary circuits moving non-invertible symmetry defects generate a Berry phase on defect states that defines an invariant under local deformations, provided the circuits preserve locality of the deformations. This invariant detects 't Hooft anomalies of non-invertible symmetries by ruling out short-range-entangled states that preserve the symmetry. The framework is applied to loop excitations in (3+1)D, identifying a new non-Abelian fermionic loop in D4 topological order whose statistics are characterized via the invariant, and a new mixed topological order whose long-range entanglement is protected by the same invariant.

Significance. If the invariance holds with the stated condition verified, the work supplies an operator-centric method for diagnosing anomalies and generalized non-Abelian statistics without reference to field-theory or lattice Hamiltonians, which could aid classification of 3+1D topological orders with non-invertible symmetries. The concrete identification of the non-Abelian fermionic loop and the new mixed order provides falsifiable examples.

major comments (2)
  1. [Abstract / Berry phase invariant section] Abstract and the paragraph defining the Berry phase invariant: the invariance under local deformations is asserted only when sequential circuits preserve locality of deformations, yet no explicit support-size tracking, locality bound, or step-by-step verification is supplied for the circuits constructed in the (3+1)D D4 or mixed topological order examples; this condition is load-bearing for both the invariance claim and the subsequent anomaly-detection argument.
  2. [D4 topological order application] Section applying the invariant to the D4 topological order: the characterization of the non-Abelian fermionic loop statistics is stated to follow from the invariant, but the manuscript supplies neither the explicit sequential circuit construction nor the Berry phase evaluation on the defect state that would confirm the invariance holds for this case.
minor comments (1)
  1. [Introduction of new excitation] Notation for the non-Abelian fermionic loop is introduced without a clear comparison table to existing loop excitations in the same order.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / Berry phase invariant section] Abstract and the paragraph defining the Berry phase invariant: the invariance under local deformations is asserted only when sequential circuits preserve locality of deformations, yet no explicit support-size tracking, locality bound, or step-by-step verification is supplied for the circuits constructed in the (3+1)D D4 or mixed topological order examples; this condition is load-bearing for both the invariance claim and the subsequent anomaly-detection argument.

    Authors: We agree that the invariance relies on the locality-preservation condition and that the general argument is presented in the Berry phase invariant section. However, the manuscript does not include explicit support-size tracking, locality bounds, or step-by-step verification for the circuits in the (3+1)D examples. We will add this verification in the revised manuscript. revision: yes

  2. Referee: [D4 topological order application] Section applying the invariant to the D4 topological order: the characterization of the non-Abelian fermionic loop statistics is stated to follow from the invariant, but the manuscript supplies neither the explicit sequential circuit construction nor the Berry phase evaluation on the defect state that would confirm the invariance holds for this case.

    Authors: The manuscript applies the general framework to characterize the statistics, but does not supply the explicit sequential circuit constructions or Berry phase evaluations for the D4 case. We will include these explicit constructions and evaluations in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; central invariant derived from Berry phase with explicit condition

full rationale

The abstract and described framework define the Berry phase invariant directly from sequences of unitary circuits acting on defect states, with invariance under local deformations holding only under the stated locality-preservation condition on the circuits. No equations or steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the anomaly-detection argument follows from applying this definition to rule out SRE states. The derivation chain remains self-contained against the provided text, with the locality condition serving as an explicit assumption rather than a hidden reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard quantum-mechanical properties of Berry phases and unitary evolution together with the domain assumption that sequential circuits can be chosen to preserve locality. The non-Abelian fermionic loop is introduced as a new entity whose statistics are defined via the invariant.

axioms (1)
  • domain assumption Berry phase generated by sequential circuits is invariant under local deformations when the circuits preserve locality of the deformations
    Explicitly stated as the condition under which the invariant is well-defined.
invented entities (1)
  • non-Abelian fermionic loop no independent evidence
    purpose: New loop excitation in (3+1)D D4 topological order whose statistics are characterized by the circuit invariant
    Introduced and named in the abstract as a discovery within the D4 order; no independent falsifiable prediction outside the invariant is given.

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Reference graph

Works this paper leans on

104 extracted references · 18 canonical work pages · 1 internal anchor

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    In- variance under these shifts corresponds to the lin- ear constraints X a,D′ ϵ(Σ, a;D,D ′)− X a,D′ ϵ(Σ, a∗;D ′,D) = 0.(7)

    By redefining each state by a phase|D⟩ → eiϕ(D) |D⟩, each phase factor is shifted as θ(Σ, a;D,D ′)→θ(Σ, a;D,D ′) +ϕ(D)−ϕ(D ′). In- variance under these shifts corresponds to the lin- ear constraints X a,D′ ϵ(Σ, a;D,D ′)− X a,D′ ϵ(Σ, a∗;D ′,D) = 0.(7)

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    We can redefine each operator by a phase V(Σ, a;D,D ′)→e iϕ(Σ,a;D,D′)V(Σ, a;D,D ′). The redefinitions have to be consistent with the con- straints (3), (4); for instance, when Σ D1,D2 = ΣD3,D4 = Σ and (D 1,D 2)|∂Σ = (D 3,D 4)|∂Σ, the phase shift must be identical:ϕ(Σ, a;D 1,D 2) = ϕ(Σ, a;D 3,D 4). The invariance under such phase redefinitions corresponds ...

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    thickening

    We also require the Berry phase to be invariant un- der admissible local deformations of a movement operatorV(Σ, a;D 1,D 2). Such deformations may occur near the boundary∂Σ and are associated with possible redefinitions of defect states along the defect network. Unlike the case of invertible symmetries, however, local deformations of sequen- tial circuits...

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    For each 1d line operator support Σ of the sequential circuit eV(Σ, a;D,D ′), we consider a partition of a line that separates Σ into small fractions{L r′}

    When defects are points We first consider the case when the defects are 0d point objects; the defect state eD E has a local 0d state along the defect network, eD E =|ψ⟩ D ⊗ |0⟩,(18) 5 where|ψ⟩ D is regarded as a union of 0d local states lo- calized at each defect. For each 1d line operator support Σ of the sequential circuit eV(Σ, a;D,D ′), we consider a ...

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    patchwork

    When defects are lines Let us now consider the case when the defects are 1d objects; the defect state eD E has a 1d localized state along the defect network, eD E =|ψ⟩ D ⊗ |0⟩,(24) where|ψ⟩ D is regarded as a 1d gapped state localized at the defect network, which admits a matrix product state (MPS) representation. For each 2d operator support Σ of the seq...

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    Lattice Hamiltonian The Hamiltonian on a 3D cubic lattice is defined as H=− X v Av − X f Bf (32) where the vertex termA v and the face termB f are given by Av := 1 |D4| X g∈D4 − →X g E(v) − →X g N(v) − →X g U(v) ← −X g W(v) ← −X g S(v) ← −X g D(v), (33) Bf :=δ g01g13g−1 23 g−1 02 ,e,(34) whereeis an identity element in a group, and an orien- tation of an ...

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    Definition of local unitary for sequential circuits We first introduce a local unitary to expand the defect by a unit open surface on the dual cube generated by a single edge (see Fig. 8). Let us take an edgeeat∂ ˆΣ, on which a magnetic flux excitation is supported (where∂ denotes the boundary of a surface). Then, take an edge e′ adjacent toe, and let ˆσ′...

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