Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries

TL;DR

This paper explores how gauging higher-form symmetries in quantum spin models reveals dualities, mixed anomalies, and topological phases, providing new insights into symmetry fractionalization and emergent non-invertible symmetries in 2D and 3D.

Contribution

It introduces a framework for gauging higher-form symmetries in quantum spin models, uncovering dualities and anomalies that deepen understanding of topological phases and symmetry fractionalization.

Findings

Gauging higher-form symmetries leads to dual models with higher-group symmetries.

Identification of mixed 't Hooft anomalies as symmetry fractionalization.

Discovery of self-dual Hamiltonians with emergent non-invertible symmetries.

Abstract

We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions $d=2$ and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed 't Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension $d=2$ and 3. In $d=2$, these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In $d=3$, among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.