Average Categorical Symmetries in One-Dimensional Disordered Systems

Abstract: We study one-dimensional disordered systems with average non-invertible symmetries, where quenched disorder may locally break part of the symmetry while preserving it upon disorder averaging. A canonical example is the random transverse-field Ising model, which at criticality exhibits an average Kramers-Wannier duality. We consider the general setting in which the full symmetry is described by a $G$-graded fusion category $\mathcal{B}$, whose identity component $\mathcal{A}$ remains exact, while the components with nontrivial $G$-grading are realized either exactly or only on average. We develop a topological holographic framework that encodes the symmetry data of the 1D system in a 2D topological order $\mathcal{Z}[\mathcal{A}]$ (the Drinfeld center of $\mathcal{A}$), enriched by an exact or, respectively, average $G$ symmetry. Within this framework, we obtain a complete classification of anomalies and average symmetry-protected topological (SPT) phases: when the components with nontrivial $G$-grading are realized only on average, the symmetry is anomaly-free if and only if $\mathcal{Z}[\mathcal{A}]$ admits a magnetic Lagrangian algebra that is invariant under the permutation action of $G$ on anyons. When an anomaly is present, we show that the ground state of a single disorder realization is long-range entangled with probability one in the thermodynamic limit, and is expected to exhibit power-law Griffiths singularities in the low-energy spectrum. Finally, we present an explicit, exactly solvable lattice model based on a symmetry-enriched string-net construction. It yields trivial ground state ensemble in the anomaly-free case, and exhibits exotic low-energy behavior in the presence of an average anomaly.

• The paper establishes that an average categorical symmetry is anomalous exactly when the Drinfeld center of the exact symmetry lacks a G-invariant magnetic Lagrangian algebra.
• The framework shows that average anomalies force individual disorder realizations to be long-range entangled with probability approaching one, supported by flux-insertion arguments and solvable lattice models.
• The construction identifies anomaly-free ensembles with finite-depth realizations and connects anomalous cases to extensive degeneracy, logarithmic circuit depth, and possible Griffiths or infinite-randomness behavior.

Overview

This paper develops a systematic framework for non-invertible (categorical) symmetries in one-dimensional disordered systems, where quenched disorder locally breaks part of the symmetry while the full symmetry is restored only upon averaging over disorder realizations. The central object is a $G$-graded fusion category $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$, whose identity component $\mathcal{A}=\mathcal{B}_1$ remains an exact symmetry of every disorder realization, while components with nontrivial $G$-grading are realized only on average. The main results are: a complete criterion for when such an average categorical symmetry is anomalous; a proof that average anomalies force long-range entanglement in individual disorder realizations; and explicit exactly solvable lattice models realizing both anomaly-free and anomalous cases. The canonical motivating example is the random transverse-field Ising chain, whose critical point exhibits an average Kramers–Wannier (KW) duality.

The paper extends prior work on average symmetries for group-like symmetries to the categorical setting. This extension is physically important because many categorical symmetries intrinsically mix with lattice translation, which impurities generically break except on average.

Average symmetries and SRE ensembles

The setting is an ensemble of local Hamiltonians $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$ with spatially homogeneous, short-range-correlated disorder distribution $P[h]$. The exact symmetry $\mathcal{A}$ commutes with each realization; a transformation $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$ maps $H[h]$ to $H[{}^{\mathsf D}h]$ within the same ensemble and hence is a symmetry of the probability measure rather than of any single Hamiltonian. Consistency requires that applying an average symmetry does not destroy the exact one, which holds when $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$0 is a $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$1-graded extension of $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$2 (so that $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$3 is "normal" in $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$4).

Two definitional choices deserve emphasis. First, the 3-cocycle twist $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$5 of a $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$6-graded category by $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$7 is regarded as physically equivalent to $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$8 in the disordered setting, because sectors with nontrivial grading do not support local defects in any single realization. Second, short-range entangledness is defined for the ensemble: every ground state must be $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$9-symmetric and preparable by a finite-depth circuit whose depth is uniformly bounded across realizations (with a relaxed probabilistic version). An average anomaly is then defined as the absence of any such SRE ensemble.

Anomalous average KW duality

The paper first establishes a concrete instance of the phenomenon using the random transverse-field Ising chain, $\mathcal{A}=\mathcal{B}_1$0, where the Ising $\mathcal{A}=\mathcal{B}_1$1 is exact and KW duality $\mathcal{A}=\mathcal{B}_1$2 (a locality-preserving automorphism of the symmetric operator algebra satisfying $\mathcal{A}=\mathcal{B}_1$3) holds only on average. At the self-dual critical point the model flows to Fisher's infinite-randomness fixed point, with typical correlations decaying as $\mathcal{A}=\mathcal{B}_1$4 but averaged correlations following a power law $\mathcal{A}=\mathcal{B}_1$5 with $\mathcal{A}=\mathcal{B}_1$6, and gaps closing as $\mathcal{A}=\mathcal{B}_1$7.

The key argument is a partial flux insertion: given a hypothetical SRE realization with ground state $\mathcal{A}=\mathcal{B}_1$8, apply $\mathcal{A}=\mathcal{B}_1$9 to all local terms inside a large interval $G$0. Because $G$1 is an average symmetry, the resulting Hamiltonian occurs with equal probability. Deep inside $G$2 the ground state matches $G$3—which lies in the $G$4-broken phase since $G$5 exchanges ordered and disordered phases—while deep outside it matches $G$6. This forces long-range $G$7-odd correlations inside $G$8, so the preparation circuit depth grows beyond any bound. Since such regions can be placed anywhere and in arbitrary number without changing probabilities, the probability of an SRE ground state vanishes in the thermodynamic limit. This yields Proposition 1: in the thermodynamic limit, the $G$9-symmetric ground state of a single realization is long-range entangled with probability going to 1. The result also admits an Imry–Ma interpretation: an SRE realization would amount to spontaneous breaking of the average KW symmetry, which cannot occur in 1D typical samples. The authors note that how average anomalies constrain disorder-averaged observables more quantitatively remains open, though they expect either spontaneous breaking of the exact symmetry or power-law Edwards–Anderson correlations.

Topological holography and the classification of average anomalies

The general framework models the 1D system as a slab of a 2D topological order $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$0—the Drinfeld center of $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$1—enriched by a $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$2 symmetry (a braided $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$3-crossed extension, i.e., a $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$4-SET). By the ENO theorem, there is a unique such $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$5-SET whose $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$6 domain walls form $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$7; specifying $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$8 fixes the anyon permutations, fractionalization data, and possible stacking with 2D $H[h]=H_0+\sum_i h_i^\alpha O_i^\alpha$9-SPTs. Gapped 1D phases correspond to gapped bottom boundaries, classified by Lagrangian algebras of $P[h]$0. A magnetic Lagrangian algebra $P[h]$1 (sharing only the vacuum with the electric algebra $P[h]$2) corresponds to an $P[h]$3-unbroken, unique-ground-state phase—an $P[h]$4-SPT.

In the clean case with fully exact $P[h]$5, three obstructions can forbid a $P[h]$6-symmetric gapped boundary: (1) absence of a $P[h]$7-invariant $P[h]$8 as an object; (2) projective $P[h]$9 quantum numbers of condensed anyons preventing consistent charge assignment; (3) an $\mathcal{A}$0 obstruction from defect fusion. These are illustrated with the $\mathcal{A}$1-enriched toric code (where the charged-$\mathcal{A}$2 boundary carries a nontrivial $\mathcal{A}$3 obstruction) and the $\mathcal{A}$4 Tambara–Yamagami category, for which the analysis yields exactly 3 SPT phases.

In the disordered case, the classification collapses dramatically:

• Average anomaly criterion: $\mathcal{A}$5 is anomalous if and only if $\mathcal{A}$6 admits no magnetic Lagrangian algebra invariant under the permutation action of $\mathcal{A}$7 on anyons.
• Obstructions 2 and 3 drop out entirely, because they concern whether a single state can be both $\mathcal{A}$8-symmetric and SRE, whereas in a disordered realization the ground state is not a $\mathcal{A}$9-eigenstate. Correspondingly, only the anyon-permutation data remain topological invariants of an average $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$0-SET; fractionalization and SPT-stacking data cease to be invariants.
• When the criterion is satisfied, distinct $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$1 average-SPT phases are in one-to-one correspondence with $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$2-invariant magnetic Lagrangian algebras.

The physical consequence of an average anomaly is established via a generalized flux-insertion argument (Proposition 2): if no $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$3-invariant $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$4 exists, then either the partially transformed region lies in an $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$5-broken phase (case a) or in a different $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$6-SPT phase (case b), in which case junctions between inequivalent fiber functors carry degenerate zero modes that must be entangled across the interval. In both cases the single-realization ground state is LRE with probability 1. The paper further argues that average anomalies provide a topological origin for Griffiths singularities: rare regions uniformly in one phase occur with probability $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$7 while their gaps close exponentially in $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$8, and the compensation yields power-law low-energy singularities $\mathsf{D}\in\mathcal{B}\setminus\mathcal{A}$9; depending on the Harris criterion ($H[h]$0 versus $H[h]$1), the system flows to infinite randomness or retains finite-$H[h]$2 Griffiths behavior. This claim about Griffiths physics is stated as an expectation grounded in scaling arguments, not proven from the anomaly alone.

A complementary result shows that an average categorical symmetry with no exact component is always anomaly-free: starting from the fully $H[h]$3-broken phase of a string-net strip, charged disorder lifts the $H[h]$4-fold degeneracy, and the Imry–Ma theorem in $H[h]$5 guarantees destruction of order in typical realizations. For the Fibonacci example, the probability of a unique gapped ground state tends to 1 as $H[h]$6.

Solvable lattice models

The anomaly-free direction is proven constructively using a symmetry-enriched string-net model on a square-lattice strip, with bulk edges labeled by objects of $H[h]$7, module edges labeled by a $H[h]$8-module category $H[h]$9 induced from a fiber functor $H[{}^{\mathsf D}h]$0 of $H[{}^{\mathsf D}h]$1, and plaquette $H[{}^{\mathsf D}h]$2-spins. In the disordered version, the $H[{}^{\mathsf D}h]$3-spins are quenched, the plaquette term sums only over trivially graded strings, and ancillary strings valued in the Morita dual $H[{}^{\mathsf D}h]$4 absorb the grading mismatch at domain walls. The Hamiltonian is a sum of commuting projectors—the commutativity of modified plaquette terms relies on the coupled pentagon identity of the module category—and a conjugation by an explicitly constructed finite-depth isometry $H[{}^{\mathsf D}h]$5 removes the disorder, reducing each realization to an $H[{}^{\mathsf D}h]$6 string-net strip with fixed boundary condition. If $H[{}^{\mathsf D}h]$7 is $H[{}^{\mathsf D}h]$8-invariant as an $H[{}^{\mathsf D}h]$9-module category, this reduced model has a unique gapped ground state, proving that a $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$00-invariant fiber functor (equivalently, a $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$01-invariant magnetic Lagrangian algebra) implies anomaly-freeness. Together with Proposition 2, this completes the if-and-only-if classification.

Applying the same construction to the anomalous $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$02 symmetry produces a fixed-point ensemble with sharply characterized exotic properties: a typical realization has extensive ground-state degeneracy, with the number of ferromagnetic sectors asymptotically normal and centered at $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$03; and the probability that a realization is preparable by depth-$\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$04 circuit decays as $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$05, with the required depth concentrated near $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$06. These quantitative statements make precise the sense in which average anomalies enforce long-range entanglement in individual samples.

Limitations and open questions

Several caveats are acknowledged within the paper. The flux-insertion arguments rely on the assumption that local reduced density matrices are insensitive to symmetric Hamiltonian modifications far away—a disordered-system generalization of "local perturbations perturb locally" that is asserted but not rigorously proven, particularly for systems with only mobility gaps. The connection between average anomalies and Griffiths singularities, including the flow to infinite-randomness fixed points, is argued on physical grounds rather than derived from the anomaly structure alone, and the paper notes that constraints on disorder-averaged correlation functions from average anomalies remain poorly understood. The lattice construction assumes fusion multiplicity-free categories ($\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$07) for simplicity. The scope is restricted to 1D bosonic systems with $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$08-graded extensions having a nontrivial exact component; fermionic systems, higher dimensions, and mixed-state settings are identified as open directions, as is the question of whether strong/weak non-invertible symmetry anomalies forbid gapped Markovian states in mixed-state phases.

Conclusion

The paper establishes that average categorical symmetries in 1D have a strictly simpler anomaly structure than their clean counterparts: the 't Hooft anomaly is determined solely by whether the Drinfeld center of the exact symmetry possesses a $\mathcal{B}=\bigoplus_{g\in G}\mathcal{B}_g$09-invariant magnetic Lagrangian algebra, with all fractionalization and cocycle data becoming physically irrelevant under disorder averaging. When this criterion fails, every individual disorder realization is long-range entangled with probability one, and solvable models exhibit extensive degeneracy and logarithmic-depth entanglement; when it holds, explicit commuting-projector ensembles realize trivial average-SPT phases. The framework thus provides a computable classification and a concrete microscopic handle on the interplay between non-invertible symmetry, quenched disorder, and rare-region physics.