Average Categorical Symmetries in One-Dimensional Disordered Systems

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.

💡 Research Summary

This paper investigates one‑dimensional quantum spin chains in which a non‑invertible (categorical) symmetry is preserved only on average in the presence of quenched disorder. The authors begin by extending the familiar notions of exact and average symmetries from group‑like cases to fusion‑category symmetries. They consider a G‑graded fusion category 𝔅 = ⊕_{g∈G} 𝔅_g, where the identity component 𝔄 = 𝔅_1 is an exact symmetry of every disorder realization, while the non‑trivial graded components 𝔅_g (g≠1) act only on the ensemble level because the disorder distribution is invariant under the finite group G.

To capture the long‑distance physics, the authors develop a “topological holographic” framework: the 1D system is viewed as the boundary of a 2D topological order 𝒵