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arXiv:2602.09083

Average Categorical Symmetries in One-Dimensional Disordered Systems

Yabo Li, Meng Cheng, Ruochen Ma

Published Feb 9, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2602.09083

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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 GGG-graded fusion category B\mathcal{B}B, whose identity component A\mathcal{A}A remains exact, while the components with nontrivial GGG-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 Z[A]\mathcal{Z}[\mathcal{A}]Z[A] (the Drinfeld center of A\mathcal{A}A), enriched by an exact or, respectively, average GGG symmetry. Within this framework, we obtain a complete classification of anomalies and average symmetry-protected topological (SPT) phases: when the components with nontrivial GGG-grading are realized only on average, the symmetry is anomaly-free if and only if Z[A]\mathcal{Z}[\mathcal{A}]Z[A] admits a magnetic Lagrangian algebra that is invariant under the permutation action of GGG 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.
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We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic Z4F\mathbb Z_4^FZ4F​ symmetry in (2+1) dimensions. For an onsite Z4F\mathbb Z_4^FZ4F​ symmetry, symmetric invertible phases have integer chiral central charge c−∈Zc_-\in\mathbb Zc−​∈Z, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local Z4F\mathbb Z_4^FZ4F​ symmetry of a p+ipp+ipp+ip superconductor with c−=1/2c_-=1/2c−​=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c−∈Zc_-\in \mathbb{Z}c−​∈Z invertible states. The allowed invertible phases are consequently shifted to c−∈Z+1/2c_-\in\mathbb Z+1/2c−​∈Z+1/2. By gauging the fermion parity of the p+ipp+ipp+ip superconductor, one obtains an Ising topological order enriched by a Z4\mathbb Z_4Z4​ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ipp+ipp+ip state admits an enlarged non-onsite U(1)fU(1)^fU(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)fU(1)^fU(1)f symmetry again forbids Chern insulators carrying c−∈Zc_-\in\mathbb{Z}c−​∈Z, while being compatible with invertible states with c−∈Z+1/2c_-\in\mathbb{Z}+1/2c−​∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
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Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra.
We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category A\mathcal{A}A describes a (2+1)(2+1)(2+1)-dimensional topological order whose gapped boundaries enumerate all (1+1)(1+1)(1+1)-dimensional gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the electric Lagrangian algebra, that describes a phase with A\mathcal{A}A fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in A\mathcal{A}A. We observe that if there exists a different gapped boundary, given by a magnetic Lagrangian algebra, then there exists a gapped phase where A\mathcal{A}A is not spontaneously broken at all, which means that A\mathcal{A}A is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to A\mathcal{A}A being anomaly free. As an application, we consider the Drinfeld centers of ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.