Average Categorical Symmetries in One-Dimensional Disordered Systems

TL;DR

The paper develops a topological-holographic framework to study 1D disordered systems with average non-invertible (categorical) symmetries described by a $G$-graded fusion category $\mathcal{B}$ with exact identity component $\mathcal{A}$. By mapping the 1D system to a 2D bulk $\mathcal{Z}[\mathcal{A}]$ enriched by $G$, it classifies anomalies and average SPT phases via the existence of a $G$-invariant magnetic Lagrangian algebra in $\mathcal{Z}[\mathcal{A}]$; absence signals an average anomaly, while its presence implies anomaly-free behavior and allows construction of solvable disordered ensembles. A key result is that, when the full symmetry is anomalous, a single disorder realization yields a long-range entangled ground state with probability one and Griffiths-type low-energy singularities, whereas average symmetry without an exact component is always anomaly-free. The authors corroborate the theory with explicit solvable lattice models based on symmetry-enriched string-net constructions, illustrating both anomaly-free and anomalous regimes and clarifying the physical consequences for disordered 1D systems. The work thus provides a concrete, computable framework for understanding how average non-invertible symmetries constrain ground-state structures and low-energy physics in disordered quantum many-body 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.

Figures (7)

• Figure 1: Illustration of the defintion of $F$-symbol
• Figure 2: Illustration of the pentagon equation.
• Figure 3: The lattice model in $\mathcal{C}$-SSB phase on a honeycomb stripe. The $\mathcal{C}$ symmetry is implemented by fusing a string of type $a\in \mathcal{C}$ into the strip from its top edge. When $\mathcal{C}$ is the Fibonacci fusion category, each edge carries a two‐dimensional Hilbert space spanned by the orthonormal basis $\{|1\rangle,|\tau\rangle\}$. A local operator $O_i^\alpha$ can be defined as the insertion of an anyon of type $\alpha\in{\mathcal{L}}_e$ through the stripe, from the top, reference boundary to a trivial dashed line.
• Figure 4: The coupled pentagon equation
• Figure 5: The symmetry is implemented by fusing a left-oriented $s_{g}\in\mathcal{B}_{g}$ string in from the top, together with a unitary $U^g$ acts on the $G$ spins at plaquettes. In the clean case, the dual strings in red are fixed to be $\beta_i = \mathbf{1}\in\mathcal{B}^*_{\mathcal{M}}$, thus can be neglected. In the disordered setting, we add $\{ Q_b\}$ terms that project the right half of each module edge onto the subspace with trivial $G$-grading [Eq. \ref{['eq:disorderedH']}]. Consequently, in the ground space, $\beta_i$ carries grading $g_i g_{i-1}^{-1}$.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2